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List of Dancing with the Stars (U.S.) competitors

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Dancing with the Stars is an American reality television show in which celebrity contestants and professional dance partners compete to be the best dancers, as determined by the show's judges and public voting. The series first broadcast in the 2005, and twenty-six complete seasons have aired on ABC through Spring 2018. During each season, competitors are progressively eliminated on the basis of public voting and scores received from the judges until only a few contestants remain. These finalists participate in a finale, from which a winner is determined.[1] Celebrities appearing on Dancing with the Stars include "actors, singers, comedians, musicians, entrepreneurs, athletes, reality stars, journalists, internet personalities, newsmakers, and where-are-they-now personalities".[2]

As of season 27, 309 celebrities have competed. Seven of those withdrew from the competition: Sara Evans of season three left the show to "give her family full attention" after filing for divorce; Misty May-Treanor of season seven was forced to pull out after an ankle injury; Tom DeLay of season nine reportedly withdrew "due to stress fractures in both of his feet"; Dorothy Hamill of season sixteen withdrew due to a previous injury that was unrelated to the competition; Billy Dee Williams of season eighteen quit due to a back injury; Kim Zolciak-Biermann of season twenty-one was forced to withdraw after suffering a mini-stroke, which although cleared her to dance, prevented her from flying to California from Atlanta; and Tamar Braxton of season twenty-one withdrew due to pulmonary embolisms in her lungs.[3][4][5] At age 14, actress Willow Shields of season twenty was the youngest contestant to compete on the show. At age 82, actress Cloris Leachman of season seven was the oldest contestant to compete on the show.[6] At age 51, singer Donny Osmond of season nine was the oldest contestant to win the competition.[7] At age 16 years, Laurie Hernandez of season twenty-three was the youngest contestant to win. The first (and, as of now, only) contestant to pass away after appearing on the show is Florence Henderson of season eleven, who died in November 2016, six years after her participation. Forty-five professional dancers have partnered with the celebrities. The twenty-six winners of the show, in chronological order, are Kelly Monaco, Drew Lachey, Emmitt Smith, Apolo Anton Ohno, Hélio Castroneves, Kristi Yamaguchi, Brooke Burke, Shawn Johnson, Donny Osmond, Nicole Scherzinger, Jennifer Grey, Hines Ward, J.R. Martinez, Donald Driver, Melissa Rycroft, Kellie Pickler, Amber Riley, Meryl Davis, Alfonso Ribeiro, Rumer Willis, Bindi Irwin, Nyle DiMarco, Laurie Hernandez, Rashad Jennings, Jordan Fisher and Adam Rippon. The fifteen professional partners who have won are Alec Mazo, Cheryl Burke (twice), Julianne Hough (twice), Mark Ballas (twice), Derek Hough (six times), Kym Johnson (twice), Karina Smirnoff, Peta Murgatroyd (twice), Tony Dovolani, Maksim Chmerkovskiy, Witney Carson, Valentin Chmerkovskiy (twice), Emma Slater, Lindsay Arnold and Jenna Johnson.

Competitors[edit]

In the following list, the winner of each season is shown in gold, second place in silver and third place in bronze.

Contents
Season: 1 · 2 · 3 · 4 · 5 · 6 · 7 · 8 · 9 · 10 · 11 · 12 · 13 · 14 · 15 · 16 · 17 · 18 · 19 · 20 · 21 · 22 · 23 · 24 · 25 · 26 · 27 · Jr.

Drew Lachey, winner of season 2
Emmitt Smith, winner of season 3
Apolo Anton Ohno, winner of season 4
Hélio Castroneves, winner of season 5
Kristi Yamaguchi, winner of season 6
Brooke Burke Charvet, winner of season 7
Shawn Johnson, winner of season 8 and runner-up of season 15
Donny Osmond, winner of season 9
Nicole Scherzinger, winner of season 10
Jennifer Grey, winner of season 11
Hines Ward, winner of season 12
J.R. Martinez, winner of season 13
Donald Driver, winner of season 14
Kellie Pickler, winner of season 16
Amber Riley, winner of season 17
Meryl Davis (winner) and Charlie White (5th place) from season 18
Alfonso Ribeiro, winner of season 19
Rumer Willis, winner of season 20
Bindi Irwin, winner of season 21
Nyle DiMarco, winner of season 22
Laurie Hernandez, winner of season 23
Rashad Jennings, winner of season 24
Jordan Fisher, winner of season 25 and host of Dancing with the Stars: Juniors
Adam Rippon, winner of season 26
Season Celebrity Professional partner Finish
1 Trista Sutter Louis van Amstel 6th
Evander Holyfield Edyta Śliwińska 5th
Rachel Hunter Jonathan Roberts 4th
Joey McIntyre Ashly DelGrosso 3rd
John O'Hurley Charlotte Jørgensen 2nd
Kelly Monaco Alec Mazo 1st
2 Kenny Mayne Andrea Hale 10th
Tatum O'Neal Nick Kosovich 9th
Giselle Fernandez Jonathan Roberts 8th
Master P Ashly DelGrosso 7th
Tia Carrere Maksim Chmerkovskiy 6th
George Hamilton Edyta Śliwińska 5th
Lisa Rinna Louis van Amstel 4th
Stacy Keibler Tony Dovolani 3rd
Jerry Rice Anna Trebunskaya 2nd
Drew Lachey Cheryl Burke 1st
3 Tucker Carlson Elena Grinenko 11th
Shanna Moakler Jesse DeSoto 10th
Harry Hamlin Ashly DelGrosso 9th
Vivica A. Fox Nick Kosovich 8th
Willa Ford Maksim Chmerkovskiy 7th
Sara Evans Tony Dovolani 6th*
Jerry Springer Kym Johnson 5th
Monique Coleman Louis van Amstel 4th
Joey Lawrence Edyta Śliwińska 3rd
Mario Lopez Karina Smirnoff 2nd
Emmitt Smith Cheryl Burke 1st
4 Paulina Porizkova Alec Mazo 11th
Shandi Finnessey Brian Fortuna 10th
Leeza Gibbons Tony Dovolani 9th
Clyde Drexler Elena Grinenko 8th
Heather Mills Jonathan Roberts 7th
John Ratzenberger Edyta Śliwińska 6th
Billy Ray Cyrus Karina Smirnoff 5th
Ian Ziering Cheryl Burke 4th
Laila Ali Maksim Chmerkovskiy 3rd
Joey Fatone Kym Johnson 2nd
Apolo Anton Ohno Julianne Hough 1st
5 Josie Maran Alec Mazo 12th
Albert Reed Anna Trebunskaya 11th
Wayne Newton Cheryl Burke 10th
Floyd Mayweather Karina Smirnoff 9th
Mark Cuban Kym Johnson 8th
Sabrina Bryan Mark Ballas 7th
Jane Seymour Tony Dovolani 6th
Cameron Mathison Edyta Śliwińska 5th
Jennie Garth Derek Hough 4th
Marie Osmond Jonathan Roberts 3rd
Mel B Maksim Chmerkovskiy 2nd
Hélio Castroneves Julianne Hough 1st
6 Penn Jillette Kym Johnson 12th
Monica Seles Jonathan Roberts 11th
Steve Guttenberg Anna Trebunskaya 10th
Adam Carolla Julianne Hough 9th
Priscilla Presley Louis van Amstel 8th
Marlee Matlin Fabian Sanchez 7th
Shannon Elizabeth Derek Hough 6th
Mario Karina Smirnoff 5th
Marissa Jaret Winokur Tony Dovolani 4th
Cristián de la Fuente Cheryl Burke 3rd
Jason Taylor Edyta Śliwińska 2nd
Kristi Yamaguchi Mark Ballas 1st
7 Jeffrey Ross Edyta Śliwińska 13th
Ted McGinley Inna Brayer 12th
Kim Kardashian Mark Ballas 11th
Misty May-Treanor Maksim Chmerkovskiy 10th*
Rocco DiSpirito Karina Smirnoff 9th
Toni Braxton Alec Mazo 8th
Cloris Leachman Corky Ballas 7th
Susan Lucci Tony Dovolani 6th
Maurice Greene Cheryl Burke 5th
Cody Linley Julianne Hough 4th
Lance Bass Lacey Schwimmer 3rd
Warren Sapp Kym Johnson 2nd
Brooke Burke Derek Hough 1st
8 Belinda Carlisle Jonathan Roberts 13th
Denise Richards Maksim Chmerkovskiy 12th
Holly Madison Dmitry Chaplin 11th
Steve Wozniak Karina Smirnoff 10th
David Alan Grier Kym Johnson 9th
Steve-O Lacey Schwimmer 8th
Lawrence Taylor Edyta Śliwińska 7th
Chuck Wicks Julianne Hough 6th
Lil' Kim Derek Hough 5th
Ty Murray Chelsie Hightower 4th
Melissa Rycroft Tony Dovolani 3rd
Gilles Marini Cheryl Burke 2nd
Shawn Johnson Mark Ballas 1st
9 Ashley Hamilton Edyta Śliwińska 16th
Macy Gray Jonathan Roberts 15th
Kathy Ireland Tony Dovolani 14th
Tom DeLay Cheryl Burke 13th*
Debi Mazar Maksim Chmerkovskiy 12th
Chuck Liddell Anna Trebunskaya 11th
Natalie Coughlin Alec Mazo 10th
Melissa Joan Hart Mark Ballas 9th
Louie Vito Chelsie Hightower 8th
Michael Irvin Anna Demidova 7th
Mark Dacascos Lacey Schwimmer 6th
Aaron Carter Karina Smirnoff 5th
Joanna Krupa Derek Hough 4th
Kelly Osbourne Louis van Amstel 3rd
Mýa Dmitry Chaplin 2nd
Donny Osmond Kym Johnson 1st
10 Shannen Doherty Mark Ballas 11th
Buzz Aldrin Ashly DelGrosso-Costa 10th
Aiden Turner Edyta Śliwińska 9th
Kate Gosselin Tony Dovolani 8th
Jake Pavelka Chelsie Hightower 7th
Pamela Anderson Damian Whitewood 6th
Niecy Nash Louis van Amstel 5th
Chad Ochocinco Cheryl Burke 4th
Erin Andrews Maksim Chmerkovskiy 3rd
Evan Lysacek Anna Trebunskaya 2nd
Nicole Scherzinger Derek Hough 1st
11 David Hasselhoff Kym Johnson 12th
Michael Bolton Chelsie Hightower 11th
Margaret Cho Louis van Amstel 10th
Michael "The Situation" Sorrentino Karina Smirnoff 9th
Florence Hendersondagger Corky Ballas 8th
Audrina Patridge Tony Dovolani 7th
Rick Fox Cheryl Burke 6th
Kurt Warner Anna Trebunskaya 5th
Brandy Maksim Chmerkovskiy 4th
Bristol Palin Mark Ballas 3rd
Kyle Massey Lacey Schwimmer 2nd
Jennifer Grey Derek Hough 1st
12 Mike Catherwood Lacey Schwimmer 11th
Wendy Williams Tony Dovolani 10th
Sugar Ray Leonard Anna Trebunskaya 9th
Petra Nemcova Dmitry Chaplin 8th
Chris Jericho Cheryl Burke 7th
Kendra Wilkinson Louis van Amstel 6th
Romeo Chelsie Hightower 5th
Ralph Macchio Karina Smirnoff 4th
Chelsea Kane Mark Ballas 3rd
Kirstie Alley Maksim Chmerkovskiy 2nd
Hines Ward Kym Johnson 1st
13 Metta World Peace Peta Murgatroyd 12th
Elisabetta Canalis Valentin Chmerkovskiy 11th
Kristin Cavallari Mark Ballas 10th
Chynna Phillips Tony Dovolani 9th
Carson Kressley Anna Trebunskaya 8th
Chaz Bono Lacey Schwimmer 7th
David Arquette Kym Johnson 6th
Nancy Grace Tristan MacManus 5th
Hope Solo Maksim Chmerkovskiy 4th
Ricki Lake Derek Hough 3rd
Rob Kardashian Cheryl Burke 2nd
J.R. Martinez Karina Smirnoff 1st
14 Martina Navratilova Tony Dovolani 12th
Jack Wagner Anna Trebunskaya 11th
Sherri Shepherd Valentin Chmerkovskiy 10th
Gavin DeGraw Karina Smirnoff 9th
Gladys Knight Tristan MacManus 8th
Jaleel White Kym Johnson 7th
Roshon Fegan Chelsie Hightower 6th
Melissa Gilbert Maksim Chmerkovskiy 5th
Maria Menounos Derek Hough 4th
William Levy Cheryl Burke 3rd
Katherine Jenkins Mark Ballas 2nd
Donald Driver Peta Murgatroyd 1st
15
(All-Stars)
Pamela Anderson Tristan MacManus 13th
Joey Fatone Kym Johnson 12th
Drew Lachey Anna Trebunskaya 11th
Hélio Castroneves Chelsie Hightower 10th
Bristol Palin Mark Ballas 9th
Sabrina Bryan Louis van Amstel 8th
Kirstie Alley Maksim Chmerkovskiy 7th
Gilles Marini Peta Murgatroyd 6th
Apolo Anton Ohno Karina Smirnoff 5th
Emmitt Smith Cheryl Burke 4th
Kelly Monaco Valentin Chmerkovskiy 3rd
Shawn Johnson Derek Hough 2nd
Melissa Rycroft Tony Dovolani 1st
16 Dorothy Hamill Tristan MacManus 12th*
Wynonna Judd Tony Dovolani 11th
Lisa Vanderpump Gleb Savchenko 10th
D.L. Hughley Cheryl Burke 9th
Victor Ortiz Lindsay Arnold 8th
Andy Dick Sharna Burgess 7th
Sean Lowe Peta Murgatroyd 6th
Ingo Rademacher Kym Johnson 5th
Alexandra Raisman Mark Ballas 4th
Jacoby Jones Karina Smirnoff 3rd
Zendaya Valentin Chmerkovskiy 2nd
Kellie Pickler Derek Hough 1st
17 Keyshawn Johnson Sharna Burgess 12th
Bill Nye Tyne Stecklein 11th
Valerie Harper Tristan MacManus 10th
Christina Milian Mark Ballas 9th
Nicole "Snooki" Polizzi Sasha Farber 8th
Brant Daugherty Peta Murgatroyd 7th
Elizabeth Berkley-Lauren Valentin Chmerkovskiy 6th
Leah Remini Tony Dovolani 5th
Bill Engvall Emma Slater 4th
Jack Osbourne Cheryl Burke 3rd
Corbin Bleu Karina Smirnoff 2nd
Amber Riley Derek Hough 1st
18 Diana Nyad Henry Byalikov 12th
Sean Avery Karina Smirnoff 11th
Billy Dee Williams Emma Slater 10th*
Cody Simpson Witney Carson 9th
Drew Carey Cheryl Burke 8th
NeNe Leakes Tony Dovolani 7th
Danica McKellar Valentin Chmerkovskiy 6th
Charlie White Sharna Burgess 5th
James Maslow Peta Murgatroyd 4th
Candace Cameron Bure Mark Ballas 3rd
Amy Purdy Derek Hough 2nd
Meryl Davis Maksim Chmerkovskiy 1st
19 Lolo Jones Keo Motsepe 13th
Tavis Smiley Sharna Burgess 12th
Randy Couture Karina Smirnoff 11th
Betsey Johnson Tony Dovolani 10th
Jonathan Bennett Allison Holker 9th
Antonio Sabàto, Jr. Cheryl Burke 8th
Michael Waltrip Emma Slater 7th
Lea Thompson Artem Chigvintsev 6th
Tommy Chong Peta Murgatroyd 5th
Bethany Mota Derek Hough 4th
Janel Parrish Valentin Chmerkovskiy 3rd
Sadie Robertson Mark Ballas 2nd
Alfonso Ribeiro Witney Carson 1st
20 Redfoo Emma Slater 12th
Charlotte McKinney Keo Motsepe 11th
Michael Sam Peta Murgatroyd 10th
Suzanne Somers Tony Dovolani 9th
Patti LaBelle Artem Chigvintsev 8th
Willow Shields Mark Ballas 7th
Robert Herjavec Kym Johnson 6th
Chris Soules Witney Carson 5th
Nastia Liukin Derek Hough 4th
Noah Galloway Sharna Burgess 3rd
Riker Lynch Allison Holker 2nd
Rumer Willis Valentin Chmerkovskiy 1st
21 Chaka Khan Keo Motsepe 13th
Victor Espinoza Karina Smirnoff 12th
Kim Zolciak-Biermann Tony Dovolani 11th*
Gary Busey Anna Trebunskaya 10th
Paula Deen Louis van Amstel 9th
Hayes Grier Emma Slater 8th
Andy Grammer Allison Holker 7th
Alexa PenaVega Mark Ballas 6th
Tamar Braxton Valentin Chmerkovskiy 5th*
Carlos PenaVega Witney Carson 4th
Alek Skarlatos Lindsay Arnold 3rd
Nick Carter Sharna Burgess 2nd
Bindi Irwin Derek Hough 1st
22 Geraldo Rivera Edyta Śliwińska 12th
Mischa Barton Artem Chigvintsev 11th
Marla Maples Tony Dovolani 10th
Doug Flutie Karina Smirnoff 9th
Kim Fields Sasha Farber 7th
Von Miller Witney Carson
Jodie Sweetin Keo Motsepe 6th
Antonio Brown Sharna Burgess 4th
Wanyá Morris Lindsay Arnold
Ginger Zee Valentin Chmerkovskiy 3rd
Paige VanZant Mark Ballas 2nd
Nyle DiMarco Peta Murgatroyd 1st
23 Jake T. Austin Jenna Johnson 13th
Rick Perry Emma Slater 12th
Kenny "Babyface" Edmonds Allison Holker 11th
Vanilla Ice Witney Carson 10th
Amber Rose Maksim Chmerkovskiy 9th
Maureen McCormick Artem Chigvintsev 8th
Ryan Lochte Cheryl Burke 7th
Marilu Henner Derek Hough 6th
Terra Jolé Sasha Farber 5th
Jana Kramer Gleb Savchenko 4th
Calvin Johnson, Jr. Lindsay Arnold 3rd
James Hinchcliffe Sharna Burgess 2nd
Laurie Hernandez Valentin Chmerkovskiy 1st
24 Chris Kattan Witney Carson 12th
Charo Keo Motsepe 11th
Mr. T Kym Johnson-Herjavec 10th
Erika Jayne Gleb Savchenko 9th
Heather Morris Maksim Chmerkovskiy 8th
Nancy Kerrigan Artem Chigvintsev 6th
Nick Viall Peta Murgatroyd
Bonner Bolton Sharna Burgess 5th
Simone Biles Sasha Farber 4th
Normani Kordei Valentin Chmerkovskiy 3rd
David Ross Lindsay Arnold 2nd
Rashad Jennings Emma Slater 1st
25 Barbara Corcoran Keo Motsepe 13th
Debbie Gibson Alan Bersten 12th
Derek Fisher Sharna Burgess 11th
Sasha Pieterse Gleb Savchenko 10th
Nick Lachey Peta Murgatroyd 9th
Nikki Bella Artem Chigvintsev 7th
Vanessa Lachey Maksim Chmerkovskiy
Terrell Owens Cheryl Burke 6th
Victoria Arlen Valentin Chmerkovskiy 5th
Drew Scott Emma Slater 4th
Frankie Muniz Witney Carson 3rd
Lindsey Stirling Mark Ballas 2nd
Jordan Fisher Lindsay Arnold 1st
26
(Athletes)
Jamie Anderson Artem Chigvintsev 9th
Johnny Damon Emma Slater
Kareem Abdul-Jabbar Lindsay Arnold 7th
Arike Ogunbowale Gleb Savchenko
Jennie Finch Daigle Keo Motsepe 4th
Chris Mazdzer Witney Carson
Mirai Nagasu Alan Bersten
Tonya Harding Sasha Farber 3rd
Josh Norman Sharna Burgess 2nd
Adam Rippon Jenna Johnson 1st
27 Nikki Glaser Gleb Savchenko 13th
Danelle Umstead Artem Chigvintsev 12th
Nancy McKeon Valentin Chmerkovskiy 11th
Tinashe Brandon Armstrong 10th
Mary Lou Retton Sasha Farber 9th
Joe Amabile Jenna Johnson TBD
Bobby Bones Sharna Burgess TBD
Juan Pablo Di Pace Cheryl Burke TBD
Evanna Lynch Keo Motsepe TBD
Milo Manheim Witney Carson TBD
Alexis Ren Alan Bersten TBD
John Schneider Emma Slater TBD
DeMarcus Ware Lindsay Arnold TBD
Juniors Tripp Johnston Palin Hailey Bills 11th
Addison Osta Smith Lev Khmelev
Hudson West Kameron Couch 10th
Sophia Pippen Jake Monreal 9th
Alana "Honey Boo Boo" Thompson Tristan Ianiero 8th
Miles Brown Rylee Arnold TBD
Sky Brown JT Church TBD
Ariana Greenblatt Artyon Celestine TBD
Jason Maybaum Elliana Walmsley TBD
Mandla Morris Brightyn Brems TBD
Akash Vukoti Kamri Peterson TBD
Mackenzie Ziegler Sage Rosen TBD
*  Indicates contestant withdrew from competition for personal/health reasons.
dagger Indicates contestant died after competing.

Notes[edit]

  1. Jump up ^ "Voting -- How It Works". American Broadcasting Company. Archived from the original on October 8, 2009. Retrieved October 24, 2009.
  2. Jump up ^ Weber, Joseph (August 17, 2009). "Tom DeLay joins 'Dancing With the Stars'". The Washington Times. News World Communications. Retrieved October 24, 2009.
  3. Jump up ^ "Divorce Forces Sara Evans to Quit 'Dancing with the Stars'". Fox News Channel. October 13, 2006. Retrieved October 23, 2009.
  4. Jump up ^ Reardanz, Karen (October 8, 2008). "Misty May-Treanor Has Surgery for Dance Injury". San Francisco Chronicle. Retrieved October 23, 2009.
  5. Jump up ^ Rizzo, Monica (October 6, 2009). "Tom DeLay to Quit Dancing with the Stars". People. Time Inc. Archived from the original on October 9, 2009. Retrieved October 23, 2009.
  6. Jump up ^ Gorman, Steve (August 25, 2008). "Cloris Leachman, 82, joins "Dancing with Stars"". Reuters. Retrieved October 24, 2009.
  7. Jump up ^ "'Dancing With The Stars' crowns a winner". USA Today. Gannett Company. November 24, 2009. Retrieved November 25, 2009.

References[edit]

  • "Dancing With The Stars". American Broadcasting Company. Retrieved October 24, 2009.
  • ""Dancing With The Stars" Timeline". CBS. May 19, 2009. Retrieved October 21, 2009.
  • Lee, Allyssa (March 1, 2010). "'Dancing with the Stars': The Season 10 cast is revealed!". Los Angeles Times. Tribune Company. Retrieved March 7, 2010.
hide
  • v
  • t
  • e
Dancing with the Stars (U.S. TV series)
Seasons
  • 1
  • 2
  • 3
  • 4
  • 5
  • 6
  • 7
  • 8
  • 9
  • 10
  • 11
  • 12
  • 13
  • 14
  • 15: All-Stars
  • 16
  • 17
  • 18
  • 19
  • 20
  • 21
  • 22
  • 23
  • 24
  • 25
  • 26: Athletes
  • 27
  • Juniors
Participants
List of competitors
Winners
Celebrity
  • Kelly Monaco
  • Drew Lachey
  • Emmitt Smith
  • Apolo Anton Ohno
  • Hélio Castroneves
  • Kristi Yamaguchi
  • Brooke Burke
  • Shawn Johnson
  • Donny Osmond
  • Nicole Scherzinger
  • Jennifer Grey
  • Hines Ward
  • J.R. Martinez
  • Donald Driver
  • Melissa Rycroft
  • Kellie Pickler
  • Amber Riley
  • Meryl Davis
  • Alfonso Ribeiro
  • Rumer Willis
  • Bindi Irwin
  • Nyle DiMarco
  • Laurie Hernandez
  • Rashad Jennings
  • Jordan Fisher
  • Adam Rippon
Professional
  • Alec Mazo
  • Cheryl Burke
  • Julianne Hough
  • Mark Ballas
  • Derek Hough
  • Kym Johnson
  • Karina Smirnoff
  • Peta Murgatroyd
  • Tony Dovolani
  • Maksim Chmerkovskiy
  • Witney Carson
  • Valentin Chmerkovskiy
  • Emma Slater
  • Lindsay Arnold
  • Jenna Johnson
Games
  • Dancing with the Stars
  • We Dance!

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Equations of motion

Equations of motion

From Wikipedia, the free encyclopedia
Jump to: navigation, search
Classical mechanics
F → = m a → {\displaystyle {\vec {F}}=m{\vec {a}}} {\vec {F}}=m{\vec {a}}
Second law of motion
  • History
  • Timeline
Branches[show]
  • Applied
  • Celestial
  • Continuum
  • Dynamics
  • Kinematics
  • Kinetics
  • Statics
  • Statistical
Fundamentals[show]
  • Acceleration
  • Angular momentum
  • Couple
  • D'Alembert's principle
  • Energy
    • kinetic
    • potential
  • Force
  • Frame of reference
  • Impulse
  • Inertia / Moment of inertia
  • Mass

  • Mechanical power
  • Mechanical work

  • Moment
  • Momentum
  • Space
  • Speed
  • Time
  • Torque
  • Velocity
  • Virtual work
Formulations[show]
  • Newton's laws of motion
  • Analytical mechanics
    • Lagrangian mechanics
    • Hamiltonian mechanics
    • Routhian mechanics
    • Hamilton–Jacobi equation
    • Appell's equation of motion
    • Udwadia–Kalaba equation
    • Koopman–von Neumann mechanics
Core topics[show]
  • Damping (ratio)
  • Displacement
  • Equations of motion
  • Euler's laws of motion
  • Fictitious force
  • Friction
  • Harmonic oscillator
  • Inertial / Non-inertial reference frame
  • Mechanics of planar particle motion
  • Motion (linear)
  • Newton's law of universal gravitation
  • Newton's laws of motion
  • Relative velocity
  • Rigid body
    • dynamics
    • Euler's equations
  • Simple harmonic motion
  • Vibration
Rotation[show]
  • Circular motion
  • Rotating reference frame
  • Centripetal force
  • Centrifugal force
    • reactive
  • Coriolis force
  • Pendulum
  • Tangential speed
  • Rotational speed
  • Angular acceleration / displacement / frequency / velocity
Scientists[show]
  • Galileo
  • Newton
  • Kepler
  • Horrocks
  • Halley
  • Euler
  • d'Alembert
  • Clairaut
  • Lagrange
  • Laplace
  • Hamilton
  • Poisson
  • Daniel Bernoulli
  • Johann Bernoulli
  • Cauchy
  • v
  • t
  • e

In mathematical physics, equations of motion are equations that describe the behaviour of a physical system in terms of its motion as a function of time.[1] More specifically, the equations of motion describe the behaviour of a physical system as a set of mathematical functions in terms of dynamic variables: normally spatial coordinates and time are used, but others are also possible, such as momentum components and time. The most general choice are generalized coordinates which can be any convenient variables characteristic of the physical system.[2] The functions are defined in a Euclidean space in classical mechanics, but are replaced by curved spaces in relativity. If the dynamics of a system is known, the equations are the solutions to the differential equations describing the motion of the dynamics.

There are two main descriptions of motion: dynamics and kinematics. Dynamics is general, since momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.

However, kinematics is simpler as it concerns only variables derived from the positions of objects, and time. In circumstances of constant acceleration, these simpler equations of motion are usually referred to as the SUVAT equations, arising from the definitions of kinematic quantities: displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t).

Equations of motion can therefore be grouped under these main classifiers of motion. In all cases, the main types of motion are translations, rotations, oscillations, or any combinations of these.

A differential equation of motion, usually identified as some physical law and applying definitions of physical quantities, is used to set up an equation for the problem. Solving the differential equation will lead to a general solution with arbitrary constants, the arbitrariness corresponding to a family of solutions. A particular solution can be obtained by setting the initial values, which fixes the values of the constants.

To state this formally, in general an equation of motion M is a function of the position r of the object, its velocity (the first time derivative of r, v = dr/dt), and its acceleration (the second derivative of r, a = d2r/dt2), and time t. Euclidean vectors in 3D are denoted throughout in bold. This is equivalent to saying an equation of motion in r is a second order ordinary differential equation (ODE) in r,

M [ r ( t ) , r ˙ ( t ) , r ¨ ( t ) , t ] = 0 , {\displaystyle M\left[\mathbf {r} (t),\mathbf {\dot {r}} (t),\mathbf {\ddot {r}} (t),t\right]=0\,,} M\left[\mathbf {r} (t),\mathbf {\dot {r}} (t),\mathbf {\ddot {r}} (t),t\right]=0\,,

where t is time, and each overdot denotes one time derivative. The initial conditions are given by the constant values at t = 0,

r ( 0 ) , r ˙ ( 0 ) . {\displaystyle \mathbf {r} (0)\,,\quad \mathbf {\dot {r}} (0)\,.} \mathbf {r} (0)\,,\quad \mathbf {\dot {r}} (0)\,.

The solution r(t) to the equation of motion, with specified initial values, describes the system for all times t after t = 0. Other dynamical variables like the momentum p of the object, or quantities derived from r and p like angular momentum, can be used in place of r as the quantity to solve for from some equation of motion, although the position of the object at time t is by far the most sought-after quantity.

Sometimes, the equation will be linear and is more likely to be exactly solvable. In general, the equation will be non-linear, and cannot be solved exactly so a variety of approximations must be used. The solutions to nonlinear equations may show chaotic behavior depending on how sensitive the system is to the initial conditions.

Contents

 [hide]
  • 1 History
  • 2 Kinematic equations for one particle
    • 2.1 Kinematic quantities
    • 2.2 Uniform acceleration
      • 2.2.1 Constant translational acceleration in a straight line
      • 2.2.2 Constant linear acceleration in any direction
      • 2.2.3 Applications
      • 2.2.4 Constant circular acceleration
    • 2.3 General planar motion
    • 2.4 General 3D motion
  • 3 Dynamic equations of motion
    • 3.1 Newtonian mechanics
    • 3.2 Applications
  • 4 Analytical mechanics
  • 5 Electrodynamics
  • 6 General relativity
    • 6.1 Geodesic equation of motion
    • 6.2 Spinning objects
  • 7 Analogues for waves and fields
    • 7.1 Field equations
    • 7.2 Wave equations
    • 7.3 Quantum theory
  • 8 See also
  • 9 References

History[edit]

Historically, equations of motion first appeared in classical mechanics to describe the motion of massive objects, a notable application was to celestial mechanics to predict the motion of the planets as if they orbit like clockwork (this was how Neptune was predicted before its discovery), and also investigate the stability of the solar system.

It is important to observe that the huge body of work involving kinematics, dynamics and the mathematical models of the universe developed in baby steps – faltering, getting up and correcting itself – over three millennia and included contributions of both known names and others who have since faded from the annals of history.

In antiquity, notwithstanding the success of priests, astrologers and astronomers in predicting solar and lunar eclipses, the solstices and the equinoxes of the Sun and the period of the Moon, there was nothing other than a set of algorithms to help them. Despite the great strides made in the development of geometry made by Ancient Greeks and surveys in Rome, we were to wait for another thousand years before the first equations of motion arrive.

The exposure of Europe to the collected works by the Muslims of the Greeks, the Indians and the Islamic scholars, such as Euclid’s Elements, the works of Archimedes, and Al-Khwārizmī's treatises [3] began in Spain, and scholars from all over Europe went to Spain, read, copied and translated the learning into Latin. The exposure of Europe to Arabic numerals and their ease in computations encouraged first the scholars to learn them and then the merchants and invigorated the spread of knowledge throughout Europe.

By the 13th century the universities of Oxford and Paris had come up, and the scholars were now studying mathematics and philosophy with lesser worries about mundane chores of life—the fields were not as clearly demarcated as they are in the modern times. Of these, compendia and redactions, such as those of Johannes Campanus, of Euclid and Aristotle, confronted scholars with ideas about infinity and the ratio theory of elements as a means of expressing relations between various quantities involved with moving bodies. These studies led to a new body of knowledge that is now known as physics.

Of these institutes Merton College sheltered a group of scholars devoted to natural science, mainly physics, astronomy and mathematics, of similar in stature to the intellectuals at the University of Paris. Thomas Bradwardine, one of those scholars, extended Aristotelian quantities such as distance and velocity, and assigned intensity and extension to them. Bradwardine suggested an exponential law involving force, resistance, distance, velocity and time. Nicholas Oresme further extended Bradwardine's arguments. The Merton school proved that the quantity of motion of a body undergoing a uniformly accelerated motion is equal to the quantity of a uniform motion at the speed achieved halfway through the accelerated motion.

For writers on kinematics before Galileo, since small time intervals could not be measured, the affinity between time and motion was obscure. They used time as a function of distance, and in free fall, greater velocity as a result of greater elevation. Only Domingo de Soto, a Spanish theologian, in his commentary on Aristotle's Physics published in 1545, after defining "uniform difform" motion (which is uniformly accelerated motion) – the word velocity wasn't used – as proportional to time, declared correctly that this kind of motion was identifiable with freely falling bodies and projectiles, without his proving these propositions or suggesting a formula relating time, velocity and distance. De Soto's comments are shockingly correct regarding the definitions of acceleration (acceleration was a rate of change of motion (velocity) in time) and the observation that during the violent motion of ascent acceleration would be negative.

Discourses such as these spread throughout Europe and definitely influenced Galileo and others, and helped in laying the foundation of kinematics.[4] Galileo deduced the equation s = 1/2gt2 in his work geometrically,[5] using the Merton rule, now known as a special case of one of the equations of kinematics. He couldn't use the now-familiar mathematical reasoning. The relationships between speed, distance, time and acceleration was not known at the time.

Galileo was the first to show that the path of a projectile is a parabola. Galileo had an understanding of centrifugal force and gave a correct definition of momentum. This emphasis of momentum as a fundamental quantity in dynamics is of prime importance. He measured momentum by the product of velocity and weight; mass is a later concept, developed by Huygens and Newton. In the swinging of a simple pendulum, Galileo says in Discourses[6] that "every momentum acquired in the descent along an arc is equal to that which causes the same moving body to ascend through the same arc." His analysis on projectiles indicates that Galileo had grasped the first law and the second law of motion. He did not generalize and make them applicable to bodies not subject to the earth's gravitation. That step was Newton's contribution.

The term "inertia" was used by Kepler who applied it to bodies at rest. (The first law of motion is now often called the law of inertia.)

Galileo did not fully grasp the third law of motion, the law of the equality of action and reaction, though he corrected some errors of Aristotle. With Stevin and others Galileo also wrote on statics. He formulated the principle of the parallelogram of forces, but he did not fully recognize its scope.

Galileo also was interested by the laws of the pendulum, his first observations was when he was a young man. In 1583, while he was praying in the cathedral at Pisa, his attention was arrested by the motion of the great lamp lighted and left swinging, referencing his own pulse for time keeping. To him the period appeared the same, even after the motion had greatly diminished, discovering the isochronism of the pendulum.

More careful experiments carried out by him later, and described in his Discourses, revealed the period of oscillation varies with the square root of length but is independent of the mass the pendulum.

Thus we arrive at René Descartes, Isaac Newton, Gottfried Leibniz, et al.; and the evolved forms of the equations of motion that begin to be recognized as the modern ones.

Later the equations of motion also appeared in electrodynamics, when describing the motion of charged particles in electric and magnetic fields, the Lorentz force is the general equation which serves as the definition of what is meant by an electric field and magnetic field. With the advent of special relativity and general relativity, the theoretical modifications to spacetime meant the classical equations of motion were also modified to account for the finite speed of light, and curvature of spacetime. In all these cases the differential equations were in terms of a function describing the particle's trajectory in terms of space and time coordinates, as influenced by forces or energy transformations.[7]

However, the equations of quantum mechanics can also be considered "equations of motion", since they are differential equations of the wavefunction, which describes how a quantum state behaves analogously using the space and time coordinates of the particles. There are analogs of equations of motion in other areas of physics, for collections of physical phenomena that can be considered waves, fluids, or fields.

Kinematic equations for one particle[edit]

Kinematic quantities[edit]

Kinematic quantities of a classical particle of mass m: position r, velocity v, acceleration a.

From the instantaneous position r = r(t), instantaneous meaning at an instant value of time t, the instantaneous velocity v = v(t) and acceleration a = a(t) have the general, coordinate-independent definitions;[8]

v = d r d t , a = d v d t = d 2 r d t 2 {\displaystyle \mathbf {v} ={\frac {d\mathbf {r} }{dt}}\,,\quad \mathbf {a} ={\frac {d\mathbf {v} }{dt}}={\frac {d^{2}\mathbf {r} }{dt^{2}}}\,\!} \mathbf {v} ={\frac {d\mathbf {r} }{dt}}\,,\quad \mathbf {a} ={\frac {d\mathbf {v} }{dt}}={\frac {d^{2}\mathbf {r} }{dt^{2}}}\,\!

Notice that velocity always points in the direction of motion, in other words for a curved path it is the tangent vector. Loosely speaking, first order derivatives are related to tangents of curves. Still for curved paths, the acceleration is directed towards the center of curvature of the path. Again, loosely speaking, second order derivatives are related to curvature.

The rotational analogues are the "angular vector" (angle the particle rotates about some axis) θ = θ(t), angular velocity ω = ω(t), and angular acceleration α = α(t):

θ = θ n ^ , ω = d θ d t , α = d ω d t , {\displaystyle {\boldsymbol {\theta }}=\theta {\hat {\mathbf {n} }}\,,\quad {\boldsymbol {\omega }}={\frac {d{\boldsymbol {\theta }}}{dt}}\,,\quad {\boldsymbol {\alpha }}={\frac {d{\boldsymbol {\omega }}}{dt}}\,,} {\boldsymbol {\theta }}=\theta {\hat {\mathbf {n} }}\,,\quad {\boldsymbol {\omega }}={\frac {d{\boldsymbol {\theta }}}{dt}}\,,\quad {\boldsymbol {\alpha }}={\frac {d{\boldsymbol {\omega }}}{dt}}\,,

where n̂ is a unit vector in the direction of the axis of rotation, and θ is the angle the object turns through about the axis.

The following relation holds for a point-like particle, orbiting about some axis with angular velocity ω:[9]

v = ω × r {\displaystyle \mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {r} \,\!} \mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {r} \,\!

where r is the position vector of the particle (radial from the rotation axis) and v the tangential velocity of the particle. For a rotating continuum rigid body, these relations hold for each point in the rigid body.

Uniform acceleration[edit]

The differential equation of motion for a particle of constant or uniform acceleration in a straight line is simple: the acceleration is constant, so the second derivative of the position of the object is constant. The results of this case are summarized below.

Constant translational acceleration in a straight line[edit]

These equations apply to a particle moving linearly, in three dimensions in a straight line with constant acceleration.[10] Since the position, velocity, and acceleration are collinear (parallel, and lie on the same line) – only the magnitudes of these vectors are necessary, and because the motion is along a straight line, the problem effectively reduces from three dimensions to one.

v = a t + v 0 [ 1 ] {\displaystyle {\begin{aligned}v&=at+v_{0}\quad [1]\\\end{aligned}}} {\begin{aligned}v&=at+v_{0}\quad [1]\\\end{aligned}}
r = r 0 + v 0 t + 1 2 a t 2 [ 2 ] {\displaystyle {\begin{aligned}r&=r_{0}+v_{0}t+{\tfrac {1}{2}}{a}t^{2}\quad [2]\\\end{aligned}}} {\displaystyle {\begin{aligned}r&=r_{0}+v_{0}t+{\tfrac {1}{2}}{a}t^{2}\quad [2]\\\end{aligned}}}
r = r 0 + 1 2 ( v + v 0 ) t [ 3 ] v 2 = v 0 2 + 2 a ( r − r 0 ) [ 4 ] r = r 0 + v t − 1 2 a t 2 [ 5 ] {\displaystyle {\begin{aligned}r&=r_{0}+{\tfrac {1}{2}}\left(v+v_{0}\right)t\quad [3]\\v^{2}&=v_{0}^{2}+2a\left(r-r_{0}\right)\quad [4]\\r&=r_{0}+vt-{\tfrac {1}{2}}{a}t^{2}\quad [5]\\\end{aligned}}} {\displaystyle {\begin{aligned}r&=r_{0}+{\tfrac {1}{2}}\left(v+v_{0}\right)t\quad [3]\\v^{2}&=v_{0}^{2}+2a\left(r-r_{0}\right)\quad [4]\\r&=r_{0}+vt-{\tfrac {1}{2}}{a}t^{2}\quad [5]\\\end{aligned}}}

where:

  • r0 is the particle's initial position
  • r is the particle's final position
  • v0 is the particle's initial velocity
  • v is the particle's final velocity
  • a is the particle's acceleration
  • t is the time interval
[show]
  Derivation

Equations [1] and [2] are from integrating the definitions of velocity and acceleration,[10] subject to the initial conditions r(t0) = r0 and v(t0) = v0;

v = ∫ a d t = a t + v 0 , [ 1 ] r = ∫ ( a t + v 0 ) d t = a t 2 2 + v 0 t + r 0 , [ 2 ] {\displaystyle {\begin{aligned}\mathbf {v} &=\int \mathbf {a} dt=\mathbf {a} t+\mathbf {v} _{0}\,,\quad [1]\\\mathbf {r} &=\int (\mathbf {a} t+\mathbf {v} _{0})dt={\frac {\mathbf {a} t^{2}}{2}}+\mathbf {v} _{0}t+\mathbf {r} _{0}\,,\quad [2]\\\end{aligned}}} {\begin{aligned}\mathbf {v} &=\int \mathbf {a} dt=\mathbf {a} t+\mathbf {v} _{0}\,,\quad [1]\\\mathbf {r} &=\int (\mathbf {a} t+\mathbf {v} _{0})dt={\frac {\mathbf {a} t^{2}}{2}}+\mathbf {v} _{0}t+\mathbf {r} _{0}\,,\quad [2]\\\end{aligned}}

in magnitudes,

v = a t + v 0 , [ 1 ] r = a t 2 2 + v 0 t + r 0 . [ 2 ] {\displaystyle {\begin{aligned}v&=at+v_{0}\,,\quad [1]\\r&={\frac {{a}t^{2}}{2}}+v_{0}t+r_{0}\,.\quad [2]\\\end{aligned}}} {\begin{aligned}v&=at+v_{0}\,,\quad [1]\\r&={\frac {{a}t^{2}}{2}}+v_{0}t+r_{0}\,.\quad [2]\\\end{aligned}}

Equation [3] involves the average velocity v + v0/2. Intuitively, the velocity increases linearly, so the average velocity multiplied by time is the distance traveled while increasing the velocity from v0 to v, as can be illustrated graphically by plotting velocity against time as a straight line graph. Algebraically, it follows from solving [1] for

a = ( v − v 0 ) t {\displaystyle \mathbf {a} ={\frac {(\mathbf {v} -\mathbf {v} _{0})}{t}}} \mathbf {a} ={\frac {(\mathbf {v} -\mathbf {v} _{0})}{t}}

and substituting into [2]

r = r 0 + v 0 t + t 2 ( v − v 0 ) , {\displaystyle \mathbf {r} =\mathbf {r} _{0}+\mathbf {v} _{0}t+{\frac {t}{2}}(\mathbf {v} -\mathbf {v} _{0})\,,} \mathbf {r} =\mathbf {r} _{0}+\mathbf {v} _{0}t+{\frac {t}{2}}(\mathbf {v} -\mathbf {v} _{0})\,,

then simplifying to get

r = r 0 + t 2 ( v + v 0 ) {\displaystyle \mathbf {r} =\mathbf {r} _{0}+{\frac {t}{2}}(\mathbf {v} +\mathbf {v} _{0})} \mathbf {r} =\mathbf {r} _{0}+{\frac {t}{2}}(\mathbf {v} +\mathbf {v} _{0})

or in magnitudes

r = r 0 + ( v + v 0 2 ) t [ 3 ] {\displaystyle r=r_{0}+\left({\frac {v+v_{0}}{2}}\right)t\quad [3]} r=r_{0}+\left({\frac {v+v_{0}}{2}}\right)t\quad [3]

From [3],

t = ( r − r 0 ) ( 2 v + v 0 ) {\displaystyle t=\left(r-r_{0}\right)\left({\frac {2}{v+v_{0}}}\right)} t=\left(r-r_{0}\right)\left({\frac {2}{v+v_{0}}}\right)

substituting for t in [1]:

v = a ( r − r 0 ) ( 2 v + v 0 ) + v 0 v ( v + v 0 ) = 2 a ( r − r 0 ) + v 0 ( v + v 0 ) v 2 + v v 0 = 2 a ( r − r 0 ) + v 0 v + v 0 2 v 2 = v 0 2 + 2 a ( r − r 0 ) [ 4 ] {\displaystyle {\begin{aligned}v&=a\left(r-r_{0}\right)\left({\frac {2}{v+v_{0}}}\right)+v_{0}\\v\left(v+v_{0}\right)&=2a\left(r-r_{0}\right)+v_{0}\left(v+v_{0}\right)\\v^{2}+vv_{0}&=2a\left(r-r_{0}\right)+v_{0}v+v_{0}^{2}\\v^{2}&=v_{0}^{2}+2a\left(r-r_{0}\right)\quad [4]\\\end{aligned}}} {\begin{aligned}v&=a\left(r-r_{0}\right)\left({\frac {2}{v+v_{0}}}\right)+v_{0}\\v\left(v+v_{0}\right)&=2a\left(r-r_{0}\right)+v_{0}\left(v+v_{0}\right)\\v^{2}+vv_{0}&=2a\left(r-r_{0}\right)+v_{0}v+v_{0}^{2}\\v^{2}&=v_{0}^{2}+2a\left(r-r_{0}\right)\quad [4]\\\end{aligned}}

From [3],

2 ( r − r 0 ) − v t = v 0 t {\displaystyle 2\left(r-r_{0}\right)-vt=v_{0}t} 2\left(r-r_{0}\right)-vt=v_{0}t

substituting into [2]:

r = a t 2 2 + 2 r − 2 r 0 − v t + r 0 0 = a t 2 2 + r − r 0 − v t r = r 0 + v t − a t 2 2 [ 5 ] {\displaystyle {\begin{aligned}r&={\frac {{a}t^{2}}{2}}+2r-2r_{0}-vt+r_{0}\\0&={\frac {{a}t^{2}}{2}}+r-r_{0}-vt\\r&=r_{0}+vt-{\frac {{a}t^{2}}{2}}\quad [5]\end{aligned}}} {\begin{aligned}r&={\frac {{a}t^{2}}{2}}+2r-2r_{0}-vt+r_{0}\\0&={\frac {{a}t^{2}}{2}}+r-r_{0}-vt\\r&=r_{0}+vt-{\frac {{a}t^{2}}{2}}\quad [5]\end{aligned}}

Usually only the first 4 are needed, the fifth is optional.

Here a is constant acceleration, or in the case of bodies moving under the influence of gravity, the standard gravity g is used. Note that each of the equations contains four of the five variables, so in this situation it is sufficient to know three out of the five variables to calculate the remaining two.

In elementary physics the same formulae are frequently written in different notation as:

v = u + a t [ 1 ] s = u t + 1 2 a t 2 [ 2 ] s = 1 2 ( u + v ) t [ 3 ] v 2 = u 2 + 2 a s [ 4 ] s = v t − 1 2 a t 2 [ 5 ] {\displaystyle {\begin{aligned}v&=u+at\quad [1]\\s&=ut+{\tfrac {1}{2}}at^{2}\quad [2]\\s&={\tfrac {1}{2}}(u+v)t\quad [3]\\v^{2}&=u^{2}+2as\quad [4]\\s&=vt-{\tfrac {1}{2}}at^{2}\quad [5]\\\end{aligned}}} {\displaystyle {\begin{aligned}v&=u+at\quad [1]\\s&=ut+{\tfrac {1}{2}}at^{2}\quad [2]\\s&={\tfrac {1}{2}}(u+v)t\quad [3]\\v^{2}&=u^{2}+2as\quad [4]\\s&=vt-{\tfrac {1}{2}}at^{2}\quad [5]\\\end{aligned}}}

where u has replaced v0, s replaces r, and s0 = 0. They are often referred to as the SUVAT equations, where "SUVAT" is an acronym from the variables: s = displacement (s0 = initial displacement), u = initial velocity, v = final velocity, a = acceleration, t = time.[11][12]

Constant linear acceleration in any direction[edit]

Trajectory of a particle with initial position vector r0 and velocity v0, subject to constant acceleration a, all three quantities in any direction, and the position r(t) and velocity v(t) after time t.

The initial position, initial velocity, and acceleration vectors need not be collinear, and take an almost identical form. The only difference is that the square magnitudes of the velocities require the dot product. The derivations are essentially the same as in the collinear case,

v = a t + v 0 [ 1 ] r = r 0 + v 0 t + 1 2 a t 2 [ 2 ] r = r 0 + 1 2 ( v + v 0 ) t [ 3 ] v 2 = v 0 2 + 2 a ⋅ ( r − r 0 ) [ 4 ] r = r 0 + v t − 1 2 a t 2 [ 5 ] {\displaystyle {\begin{aligned}\mathbf {v} &=\mathbf {a} t+\mathbf {v} _{0}\quad [1]\\\mathbf {r} &=\mathbf {r} _{0}+\mathbf {v} _{0}t+{\tfrac {1}{2}}\mathbf {a} t^{2}\quad [2]\\\mathbf {r} &=\mathbf {r} _{0}+{\tfrac {1}{2}}\left(\mathbf {v} +\mathbf {v} _{0}\right)t\quad [3]\\v^{2}&=v_{0}^{2}+2\mathbf {a} \cdot \left(\mathbf {r} -\mathbf {r} _{0}\right)\quad [4]\\\mathbf {r} &=\mathbf {r} _{0}+\mathbf {v} t-{\tfrac {1}{2}}\mathbf {a} t^{2}\quad [5]\\\end{aligned}}} {\displaystyle {\begin{aligned}\mathbf {v} &=\mathbf {a} t+\mathbf {v} _{0}\quad [1]\\\mathbf {r} &=\mathbf {r} _{0}+\mathbf {v} _{0}t+{\tfrac {1}{2}}\mathbf {a} t^{2}\quad [2]\\\mathbf {r} &=\mathbf {r} _{0}+{\tfrac {1}{2}}\left(\mathbf {v} +\mathbf {v} _{0}\right)t\quad [3]\\v^{2}&=v_{0}^{2}+2\mathbf {a} \cdot \left(\mathbf {r} -\mathbf {r} _{0}\right)\quad [4]\\\mathbf {r} &=\mathbf {r} _{0}+\mathbf {v} t-{\tfrac {1}{2}}\mathbf {a} t^{2}\quad [5]\\\end{aligned}}}

although the Torricelli equation [4] can be derived using the distributive property of the dot product as follows:

v 2 = v ⋅ v = ( v 0 + a t ) ⋅ ( v 0 + a t ) = v 0 2 + 2 t ( a ⋅ v 0 ) + a 2 t 2 {\displaystyle v^{2}=\mathbf {v} \cdot \mathbf {v} =(\mathbf {v} _{0}+\mathbf {a} t)\cdot (\mathbf {v} _{0}+\mathbf {a} t)=v_{0}^{2}+2t(\mathbf {a} \cdot \mathbf {v} _{0})+a^{2}t^{2}} v^{2}=\mathbf {v} \cdot \mathbf {v} =(\mathbf {v} _{0}+\mathbf {a} t)\cdot (\mathbf {v} _{0}+\mathbf {a} t)=v_{0}^{2}+2t(\mathbf {a} \cdot \mathbf {v} _{0})+a^{2}t^{2}
( 2 a ) ⋅ ( r − r 0 ) = ( 2 a ) ⋅ ( v 0 t + 1 2 a t 2 ) = 2 t ( a ⋅ v 0 ) + a 2 t 2 = v 2 − v 0 2 {\displaystyle (2\mathbf {a} )\cdot (\mathbf {r} -\mathbf {r} _{0})=(2\mathbf {a} )\cdot \left(\mathbf {v} _{0}t+{\tfrac {1}{2}}\mathbf {a} t^{2}\right)=2t(\mathbf {a} \cdot \mathbf {v} _{0})+a^{2}t^{2}=v^{2}-v_{0}^{2}} {\displaystyle (2\mathbf {a} )\cdot (\mathbf {r} -\mathbf {r} _{0})=(2\mathbf {a} )\cdot \left(\mathbf {v} _{0}t+{\tfrac {1}{2}}\mathbf {a} t^{2}\right)=2t(\mathbf {a} \cdot \mathbf {v} _{0})+a^{2}t^{2}=v^{2}-v_{0}^{2}}
∴ v 2 = v 0 2 + 2 ( a ⋅ ( r − r 0 ) ) {\displaystyle \therefore v^{2}=v_{0}^{2}+2(\mathbf {a} \cdot (\mathbf {r} -\mathbf {r} _{0}))} \therefore v^{2}=v_{0}^{2}+2(\mathbf {a} \cdot (\mathbf {r} -\mathbf {r} _{0}))

Applications[edit]

Elementary and frequent examples in kinematics involve projectiles, for example a ball thrown upwards into the air. Given initial speed u, one can calculate how high the ball will travel before it begins to fall. The acceleration is local acceleration of gravity g. At this point one must remember that while these quantities appear to be scalars, the direction of displacement, speed and acceleration is important. They could in fact be considered as unidirectional vectors. Choosing s to measure up from the ground, the acceleration a must be in fact −g, since the force of gravity acts downwards and therefore also the acceleration on the ball due to it.

At the highest point, the ball will be at rest: therefore v = 0. Using equation [4] in the set above, we have:

s = v 2 − u 2 − 2 g . {\displaystyle s={\frac {v^{2}-u^{2}}{-2g}}.} s={\frac {v^{2}-u^{2}}{-2g}}.

Substituting and cancelling minus signs gives:

s = u 2 2 g . {\displaystyle s={\frac {u^{2}}{2g}}.} s={\frac {u^{2}}{2g}}.

Constant circular acceleration[edit]

The analogues of the above equations can be written for rotation. Again these axial vectors must all be parallel to the axis of rotation, so only the magnitudes of the vectors are necessary,

ω = ω 0 + α t θ = θ 0 + ω 0 t + 1 2 α t 2 θ = θ 0 + 1 2 ( ω 0 + ω ) t ω 2 = ω 0 2 + 2 α ( θ − θ 0 ) θ = θ 0 + ω t − 1 2 α t 2 {\displaystyle {\begin{aligned}\omega &=\omega _{0}+\alpha t\\\theta &=\theta _{0}+\omega _{0}t+{\tfrac {1}{2}}\alpha t^{2}\\\theta &=\theta _{0}+{\tfrac {1}{2}}(\omega _{0}+\omega )t\\\omega ^{2}&=\omega _{0}^{2}+2\alpha (\theta -\theta _{0})\\\theta &=\theta _{0}+\omega t-{\tfrac {1}{2}}\alpha t^{2}\\\end{aligned}}} {\begin{aligned}\omega &=\omega _{0}+\alpha t\\\theta &=\theta _{0}+\omega _{0}t+{\tfrac {1}{2}}\alpha t^{2}\\\theta &=\theta _{0}+{\tfrac {1}{2}}(\omega _{0}+\omega )t\\\omega ^{2}&=\omega _{0}^{2}+2\alpha (\theta -\theta _{0})\\\theta &=\theta _{0}+\omega t-{\tfrac {1}{2}}\alpha t^{2}\\\end{aligned}}

where α is the constant angular acceleration, ω is the angular velocity, ω0 is the initial angular velocity, θ is the angle turned through (angular displacement), θ0 is the initial angle, and t is the time taken to rotate from the initial state to the final state.

General planar motion[edit]

Main article: General planar motion
Position vector r, always points radially from the origin.
Velocity vector v, always tangent to the path of motion.
Acceleration vector a, not parallel to the radial motion but offset by the angular and Coriolis accelerations, nor tangent to the path but offset by the centripetal and radial accelerations.
Kinematic vectors in plane polar coordinates. Notice the setup is not restricted to 2D space, but a plane in any higher dimension.

These are the kinematic equations for a particle traversing a path in a plane, described by position r = r(t).[13] They are simply the time derivatives of the position vector in plane polar coordinates using the definitions of physical quantities above for angular velocity ω and angular acceleration α. These are instantaneous quantities which change with time.

The position of the particle is

r = r ( r ( t ) , θ ( t ) ) = r e ^ r {\displaystyle \mathbf {r} =\mathbf {r} \left(r(t),\theta (t)\right)=r\mathbf {\hat {e}} _{r}} {\displaystyle \mathbf {r} =\mathbf {r} \left(r(t),\theta (t)\right)=r\mathbf {\hat {e}} _{r}}

where êr and êθ are the polar unit vectors. Differentiating with respect to time gives the velocity

v = e ^ r d r d t + r ω e ^ θ {\displaystyle \mathbf {v} =\mathbf {\hat {e}} _{r}{\frac {dr}{dt}}+r\omega \mathbf {\hat {e}} _{\theta }} {\displaystyle \mathbf {v} =\mathbf {\hat {e}} _{r}{\frac {dr}{dt}}+r\omega \mathbf {\hat {e}} _{\theta }}

with radial component dr/dt and an additional component rω due to the rotation. Differentiating with respect to time again obtains the acceleration

a = ( d 2 r d t 2 − r ω 2 ) e ^ r + ( r α + 2 ω d r d t ) e ^ θ {\displaystyle \mathbf {a} =\left({\frac {d^{2}r}{dt^{2}}}-r\omega ^{2}\right)\mathbf {\hat {e}} _{r}+\left(r\alpha +2\omega {\frac {dr}{dt}}\right)\mathbf {\hat {e}} _{\theta }} {\displaystyle \mathbf {a} =\left({\frac {d^{2}r}{dt^{2}}}-r\omega ^{2}\right)\mathbf {\hat {e}} _{r}+\left(r\alpha +2\omega {\frac {dr}{dt}}\right)\mathbf {\hat {e}} _{\theta }}

which breaks into the radial acceleration d2r/dt2, centripetal acceleration –rω2, Coriolis acceleration 2ωdr/dt, and angular acceleration rα.

Special cases of motion described be these equations are summarized qualitatively in the table below. Two have already been discussed above, in the cases that either the radial components or the angular components are zero, and the non-zero component of motion describes uniform acceleration.

State of motion Constant r r linear in t r quadratic in t r non-linear in t
Constant θ Stationary Uniform translation (constant translational velocity) Uniform translational acceleration Non-uniform translation
θ linear in t Uniform angular motion in a circle (constant angular velocity) Uniform angular motion in a spiral, constant radial velocity Angular motion in a spiral, constant radial acceleration Angular motion in a spiral, varying radial acceleration
θ quadratic in t Uniform angular acceleration in a circle Uniform angular acceleration in a spiral, constant radial velocity Uniform angular acceleration in a spiral, constant radial acceleration Uniform angular acceleration in a spiral, varying radial acceleration
θ non-linear in t Non-uniform angular acceleration in a circle Non-uniform angular acceleration in a spiral, constant radial velocity Non-uniform angular acceleration in a spiral, constant radial acceleration Non-uniform angular acceleration in a spiral, varying radial acceleration

General 3D motion[edit]

Main article: Spherical coordinate system

In 3D space, the equations in spherical coordinates (r, θ, φ) with corresponding unit vectors êr, êθ and êφ, the position, velocity, and acceleration generalize respectively to

r = r ( t ) = r e ^ r v = v e ^ r + r d θ d t e ^ θ + r d φ d t sin  θ e ^ φ a = ( a − r ( d θ d t ) 2 − r ( d φ d t ) 2 sin 2  θ ) e ^ r + ( r d 2 θ d t 2 + 2 v d θ d t − r ( d φ d t ) 2 sin  θ cos  θ ) e ^ θ + ( r d 2 φ d t 2 sin  θ + 2 v d φ d t sin  θ + 2 r d θ d t d φ d t cos  θ ) e ^ φ {\displaystyle {\begin{aligned}\mathbf {r} &=\mathbf {r} \left(t\right)=r\mathbf {\hat {e}} _{r}\\\mathbf {v} &=v\mathbf {\hat {e}} _{r}+r\,{\frac {d\theta }{dt}}\mathbf {\hat {e}} _{\theta }+r\,{\frac {d\varphi }{dt}}\,\sin \theta \mathbf {\hat {e}} _{\varphi }\\\mathbf {a} &=\left(a-r\left({\frac {d\theta }{dt}}\right)^{2}-r\left({\frac {d\varphi }{dt}}\right)^{2}\sin ^{2}\theta \right)\mathbf {\hat {e}} _{r}\\&+\left(r{\frac {d^{2}\theta }{dt^{2}}}+2v{\frac {d\theta }{dt}}-r\left({\frac {d\varphi }{dt}}\right)^{2}\sin \theta \cos \theta \right)\mathbf {\hat {e}} _{\theta }\\&+\left(r{\frac {d^{2}\varphi }{dt^{2}}}\,\sin \theta +2v\,{\frac {d\varphi }{dt}}\,\sin \theta +2r\,{\frac {d\theta }{dt}}\,{\frac {d\varphi }{dt}}\,\cos \theta \right)\mathbf {\hat {e}} _{\varphi }\end{aligned}}\,\!} {\displaystyle {\begin{aligned}\mathbf {r} &=\mathbf {r} \left(t\right)=r\mathbf {\hat {e}} _{r}\\\mathbf {v} &=v\mathbf {\hat {e}} _{r}+r\,{\frac {d\theta }{dt}}\mathbf {\hat {e}} _{\theta }+r\,{\frac {d\varphi }{dt}}\,\sin \theta \mathbf {\hat {e}} _{\varphi }\\\mathbf {a} &=\left(a-r\left({\frac {d\theta }{dt}}\right)^{2}-r\left({\frac {d\varphi }{dt}}\right)^{2}\sin ^{2}\theta \right)\mathbf {\hat {e}} _{r}\\&+\left(r{\frac {d^{2}\theta }{dt^{2}}}+2v{\frac {d\theta }{dt}}-r\left({\frac {d\varphi }{dt}}\right)^{2}\sin \theta \cos \theta \right)\mathbf {\hat {e}} _{\theta }\\&+\left(r{\frac {d^{2}\varphi }{dt^{2}}}\,\sin \theta +2v\,{\frac {d\varphi }{dt}}\,\sin \theta +2r\,{\frac {d\theta }{dt}}\,{\frac {d\varphi }{dt}}\,\cos \theta \right)\mathbf {\hat {e}} _{\varphi }\end{aligned}}\,\!}

In the case of a constant φ this reduces to the planar equations above.

Dynamic equations of motion[edit]

Newtonian mechanics[edit]

Main article: Newtonian mechanics

The first general equation of motion developed was Newton's second law of motion, in its most general form states the rate of change of momentum p = p(t) = mv(t) of an object equals the force F = F(x(t), v(t), t) acting on it,[14]

F = d p d t {\displaystyle \mathbf {F} ={\frac {d\mathbf {p} }{dt}}} \mathbf {F} ={\frac {d\mathbf {p} }{dt}}

The force in the equation is not the force the object exerts. Replacing momentum by mass times velocity, the law is also written more famously as

F = m a {\displaystyle \mathbf {F} =m\mathbf {a} } \mathbf {F} =m\mathbf {a}

since m is a constant in Newtonian mechanics.

Newton's second law applies to point-like particles, and to all points in a rigid body. They also apply to each point in a mass continua, like deformable solids or fluids, but the motion of the system must be accounted for, see material derivative. In the case the mass is not constant, it is not sufficient to use the product rule for the time derivative on the mass and velocity, and Newton's second law requires some modification consistent with conservation of momentum, see variable-mass system.

It may be simple to write down the equations of motion in vector form using Newton's laws of motion, but the components may vary in complicated ways with spatial coordinates and time, and solving them is not easy. Often there is an excess of variables to solve for the problem completely, so Newton's laws are not always the most efficient way to determine the motion of a system. In simple cases of rectangular geometry, Newton's laws work fine in Cartesian coordinates, but in other coordinate systems can become dramatically complex.

The momentum form is preferable since this is readily generalized to more complex systems, generalizes to special and general relativity (see four-momentum).[14] It can also be used with the momentum conservation. However, Newton's laws are not more fundamental than momentum conservation, because Newton's laws are merely consistent with the fact that zero resultant force acting on an object implies constant momentum, while a resultant force implies the momentum is not constant. Momentum conservation is always true for an isolated system not subject to resultant forces.

For a number of particles (see many body problem), the equation of motion for one particle i influenced by other particles is[8][15]

d p i d t = F E + ∑ i ≠ j F i j {\displaystyle {\frac {d\mathbf {p} _{i}}{dt}}=\mathbf {F} _{E}+\sum _{i\neq j}\mathbf {F} _{ij}\,\!} {\frac {d\mathbf {p} _{i}}{dt}}=\mathbf {F} _{E}+\sum _{i\neq j}\mathbf {F} _{ij}\,\!

where pi is the momentum of particle i, Fij is the force on particle i by particle j, and FE is the resultant external force due to any agent not part of system. Particle i does not exert a force on itself.

Euler's laws of motion are similar to Newton's laws, but they are applied specifically to the motion of rigid bodies. The Newton–Euler equations combine the forces and torques acting on a rigid body into a single equation.

Newton's second law for rotation takes a similar form to the translational case,[16]

τ = d L d t , {\displaystyle {\boldsymbol {\tau }}={\frac {d\mathbf {L} }{dt}}\,,} {\boldsymbol {\tau }}={\frac {d\mathbf {L} }{dt}}\,,

by equating the torque acting on the body to the rate of change of its angular momentum L. Analogous to mass times acceleration, the moment of inertia tensor I depends on the distribution of mass about the axis of rotation, and the angular acceleration is the rate of change of angular velocity,

τ = I ⋅ α . {\displaystyle {\boldsymbol {\tau }}=\mathbf {I} \cdot {\boldsymbol {\alpha }}.} {\boldsymbol {\tau }}=\mathbf {I} \cdot {\boldsymbol {\alpha }}.

Again, these equations apply to point like particles, or at each point of a rigid body.

Likewise, for a number of particles, the equation of motion for one particle i is[17]

d L i d t = τ E + ∑ i ≠ j τ i j , {\displaystyle {\frac {d\mathbf {L} _{i}}{dt}}={\boldsymbol {\tau }}_{E}+\sum _{i\neq j}{\boldsymbol {\tau }}_{ij}\,,} {\frac {d\mathbf {L} _{i}}{dt}}={\boldsymbol {\tau }}_{E}+\sum _{i\neq j}{\boldsymbol {\tau }}_{ij}\,,

where Li is the angular momentum of particle i, τij the torque on particle i by particle j, and τE is resultant external torque (due to any agent not part of system). Particle i does not exert a torque on itself.

Applications[edit]

Some examples[18] of Newton's law include describing the motion of a simple pendulum,

− m g sin  θ = m d 2 ( l θ ) d t 2 ⇒ d 2 θ d t 2 = − g l sin  θ , {\displaystyle -mg\sin \theta =m{\frac {d^{2}(l\theta )}{dt^{2}}}\quad \Rightarrow \quad {\frac {d^{2}\theta }{dt^{2}}}=-{\frac {g}{l}}\sin \theta \,,} {\displaystyle -mg\sin \theta =m{\frac {d^{2}(l\theta )}{dt^{2}}}\quad \Rightarrow \quad {\frac {d^{2}\theta }{dt^{2}}}=-{\frac {g}{l}}\sin \theta \,,}

and a damped, sinusoidally driven harmonic oscillator,

F 0 sin  ( ω t ) = m ( d 2 x d t 2 + 2 ζ ω 0 d x d t + ω 0 2 x ) . {\displaystyle F_{0}\sin(\omega t)=m\left({\frac {d^{2}x}{dt^{2}}}+2\zeta \omega _{0}{\frac {dx}{dt}}+\omega _{0}^{2}x\right)\,.} F_{0}\sin(\omega t)=m\left({\frac {d^{2}x}{dt^{2}}}+2\zeta \omega _{0}{\frac {dx}{dt}}+\omega _{0}^{2}x\right)\,.

For describing the motion of masses due to gravity, Newton's law of gravity can be combined with Newton's second law. For two examples, a ball of mass m thrown in the air, in air currents (such as wind) described by a vector field of resistive forces R = R(r, t),

− G m M | r | 2 e ^ r + R = m d 2 r d t 2 + 0 ⇒ d 2 r d t 2 = − G M | r | 2 e ^ r + A {\displaystyle -{\frac {GmM}{|\mathbf {r} |^{2}}}\mathbf {\hat {e}} _{r}+\mathbf {R} =m{\frac {d^{2}\mathbf {r} }{dt^{2}}}+0\quad \Rightarrow \quad {\frac {d^{2}\mathbf {r} }{dt^{2}}}=-{\frac {GM}{|\mathbf {r} |^{2}}}\mathbf {\hat {e}} _{r}+\mathbf {A} \,\!} -{\frac {GmM}{|\mathbf {r} |^{2}}}\mathbf {\hat {e}} _{r}+\mathbf {R} =m{\frac {d^{2}\mathbf {r} }{dt^{2}}}+0\quad \Rightarrow \quad {\frac {d^{2}\mathbf {r} }{dt^{2}}}=-{\frac {GM}{|\mathbf {r} |^{2}}}\mathbf {\hat {e}} _{r}+\mathbf {A} \,\!

where G is the gravitational constant, M the mass of the Earth, and A = R/m is the acceleration of the projectile due to the air currents at position r and time t.

The classical N-body problem for N particles each interacting with each other due to gravity is a set of N nonlinear coupled second order ODEs,

d 2 r i d t 2 = G ∑ i ≠ j m i m j | r j − r i | 3 ( r j − r i ) {\displaystyle {\frac {d^{2}\mathbf {r} _{i}}{dt^{2}}}=G\sum _{i\neq j}{\frac {m_{i}m_{j}}{|\mathbf {r} _{j}-\mathbf {r} _{i}|^{3}}}(\mathbf {r} _{j}-\mathbf {r} _{i})} {\frac {d^{2}\mathbf {r} _{i}}{dt^{2}}}=G\sum _{i\neq j}{\frac {m_{i}m_{j}}{|\mathbf {r} _{j}-\mathbf {r} _{i}|^{3}}}(\mathbf {r} _{j}-\mathbf {r} _{i})

where i = 1, 2, …, N labels the quantities (mass, position, etc.) associated with each particle.

Analytical mechanics[edit]

Main articles: Analytical mechanics, Lagrangian mechanics and Hamiltonian mechanics
As the system evolves, q traces a path through configuration space (only some are shown). The path taken by the system (red) has a stationary action (δS = 0) under small changes in the configuration of the system (δq).[19]

Using all three coordinates of 3D space is unnecessary if there are constraints on the system. If the system has N degrees of freedom, then one can use a set of N generalized coordinates q(t) = [q1(t), q2(t) ... qN(t)], to define the configuration of the system. They can be in the form of arc lengths or angles. They are a considerable simplification to describe motion, since they take advantage of the intrinsic constraints that limit the system's motion, and the number of coordinates is reduced to a minimum. The time derivatives of the generalized coordinates are the generalized velocities

q ˙ = d q d t . {\displaystyle \mathbf {\dot {q}} ={\frac {d\mathbf {q} }{dt}}\,.} {\displaystyle \mathbf {\dot {q}} ={\frac {d\mathbf {q} }{dt}}\,.}

The Euler–Lagrange equations are[2][20]

d d t ( ∂ L ∂ q ˙ ) = ∂ L ∂ q , {\displaystyle {\frac {d}{dt}}\left({\frac {\partial L}{\partial \mathbf {\dot {q}} }}\right)={\frac {\partial L}{\partial \mathbf {q} }}\,,} {\frac {d}{dt}}\left({\frac {\partial L}{\partial \mathbf {\dot {q}} }}\right)={\frac {\partial L}{\partial \mathbf {q} }}\,,

where the Lagrangian is a function of the configuration q and its time rate of change dq/dt (and possibly time t)

L = L [ q ( t ) , q ˙ ( t ) , t ] . {\displaystyle L=L\left[\mathbf {q} (t),\mathbf {\dot {q}} (t),t\right]\,.} L=L\left[\mathbf {q} (t),\mathbf {\dot {q}} (t),t\right]\,.

Setting up the Lagrangian of the system, then substituting into the equations and evaluating the partial derivatives and simplifying, a set of coupled N second order ODEs in the coordinates are obtained.

Hamilton's equations are[2][20]

p ˙ = − ∂ H ∂ q , q ˙ = + ∂ H ∂ p , {\displaystyle \mathbf {\dot {p}} =-{\frac {\partial H}{\partial \mathbf {q} }}\,,\quad \mathbf {\dot {q}} =+{\frac {\partial H}{\partial \mathbf {p} }}\,,} \mathbf {\dot {p}} =-{\frac {\partial H}{\partial \mathbf {q} }}\,,\quad \mathbf {\dot {q}} =+{\frac {\partial H}{\partial \mathbf {p} }}\,,

where the Hamiltonian

H = H [ q ( t ) , p ( t ) , t ] , {\displaystyle H=H\left[\mathbf {q} (t),\mathbf {p} (t),t\right]\,,} H=H\left[\mathbf {q} (t),\mathbf {p} (t),t\right]\,,

is a function of the configuration q and conjugate "generalized" momenta

p = ∂ L ∂ q ˙ , {\displaystyle \mathbf {p} ={\frac {\partial L}{\partial \mathbf {\dot {q}} }}\,,} {\displaystyle \mathbf {p} ={\frac {\partial L}{\partial \mathbf {\dot {q}} }}\,,}

in which ∂/∂q = (∂/∂q1, ∂/∂q2, …, ∂/∂qN) is a shorthand notation for a vector of partial derivatives with respect to the indicated variables (see for example matrix calculus for this denominator notation), and possibly time t,

Setting up the Hamiltonian of the system, then substituting into the equations and evaluating the partial derivatives and simplifying, a set of coupled 2N first order ODEs in the coordinates qi and momenta pi are obtained.

The Hamilton–Jacobi equation is[2]

− ∂ S ( q , t ) ∂ t = H ( q , p , t ) . {\displaystyle -{\frac {\partial S(\mathbf {q} ,t)}{\partial t}}=H\left(\mathbf {q} ,\mathbf {p} ,t\right)\,.} -{\frac {\partial S(\mathbf {q} ,t)}{\partial t}}=H\left(\mathbf {q} ,\mathbf {p} ,t\right)\,.

where

S [ q , t ] = ∫ t 1 t 2 L ( q , q ˙ , t ) d t , {\displaystyle S[\mathbf {q} ,t]=\int _{t_{1}}^{t_{2}}L(\mathbf {q} ,\mathbf {\dot {q}} ,t)\,dt\,,} S[\mathbf {q} ,t]=\int _{t_{1}}^{t_{2}}L(\mathbf {q} ,\mathbf {\dot {q}} ,t)\,dt\,,

is Hamilton's principal function, also called the classical action is a functional of L. In this case, the momenta are given by

p = ∂ S ∂ q . {\displaystyle \mathbf {p} ={\frac {\partial S}{\partial \mathbf {q} }}\,.} {\displaystyle \mathbf {p} ={\frac {\partial S}{\partial \mathbf {q} }}\,.}

Although the equation has a simple general form, for a given Hamiltonian it is actually a single first order non-linear PDE, in N + 1 variables. The action S allows identification of conserved quantities for mechanical systems, even when the mechanical problem itself cannot be solved fully, because any differentiable symmetry of the action of a physical system has a corresponding conservation law, a theorem due to Emmy Noether.

All classical equations of motion can be derived from the variational principle known as Hamilton's principle of least action

δ S = 0 , {\displaystyle \delta S=0\,,} \delta S=0\,,

stating the path the system takes through the configuration space is the one with the least action S.

Electrodynamics[edit]

Lorentz force f on a charged particle (of charge q) in motion (instantaneous velocity v). The E field and B field vary in space and time.

In electrodynamics, the force on a charged particle of charge q is the Lorentz force:[21]

F = q ( E + v × B ) {\displaystyle \mathbf {F} =q\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)\,\!} \mathbf {F} =q\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)\,\!

Combining with Newton's second law gives a first order differential equation of motion, in terms of position of the particle:

m d 2 r d t 2 = q ( E + d r d t × B ) {\displaystyle m{\frac {d^{2}\mathbf {r} }{dt^{2}}}=q\left(\mathbf {E} +{\frac {d\mathbf {r} }{dt}}\times \mathbf {B} \right)\,\!} m{\frac {d^{2}\mathbf {r} }{dt^{2}}}=q\left(\mathbf {E} +{\frac {d\mathbf {r} }{dt}}\times \mathbf {B} \right)\,\!

or its momentum:

d p d t = q ( E + p × B m ) {\displaystyle {\frac {d\mathbf {p} }{dt}}=q\left(\mathbf {E} +{\frac {\mathbf {p} \times \mathbf {B} }{m}}\right)\,\!} {\frac {d\mathbf {p} }{dt}}=q\left(\mathbf {E} +{\frac {\mathbf {p} \times \mathbf {B} }{m}}\right)\,\!

The same equation can be obtained using the Lagrangian (and applying Lagrange's equations above) for a charged particle of mass m and charge q:[22]

L = 1 2 m r ˙ ⋅ r ˙ + q A ⋅ r ˙ − q ϕ {\displaystyle L={\tfrac {1}{2}}m\mathbf {\dot {r}} \cdot \mathbf {\dot {r}} +q\mathbf {A} \cdot {\dot {\mathbf {r} }}-q\phi } {\displaystyle L={\tfrac {1}{2}}m\mathbf {\dot {r}} \cdot \mathbf {\dot {r}} +q\mathbf {A} \cdot {\dot {\mathbf {r} }}-q\phi }

where A and ϕ are the electromagnetic scalar and vector potential fields. The Lagrangian indicates an additional detail: the canonical momentum in Lagrangian mechanics is given by:

P = ∂ L ∂ r ˙ = m r ˙ + q A {\displaystyle \mathbf {P} ={\frac {\partial L}{\partial {\dot {\mathbf {r} }}}}=m{\dot {\mathbf {r} }}+q\mathbf {A} } \mathbf {P} ={\frac {\partial L}{\partial {\dot {\mathbf {r} }}}}=m{\dot {\mathbf {r} }}+q\mathbf {A}

instead of just mv, implying the motion of a charged particle is fundamentally determined by the mass and charge of the particle. The Lagrangian expression was first used to derive the force equation.

Alternatively the Hamiltonian (and substituting into the equations):[20]

H = ( P − q A ) 2 2 m + q ϕ {\displaystyle H={\frac {\left(\mathbf {P} -q\mathbf {A} \right)^{2}}{2m}}+q\phi \,\!} H={\frac {\left(\mathbf {P} -q\mathbf {A} \right)^{2}}{2m}}+q\phi \,\!

can derive the Lorentz force equation.

General relativity[edit]

Geodesic equation of motion[edit]

Geodesics on a sphere are arcs of great circles (yellow curve). On a 2D–manifold (such as the sphere shown), the direction of the accelerating geodesic is uniquely fixed if the separation vector ξ is orthogonal to the "fiducial geodesic" (green curve). As the separation vector ξ0 changes to ξ after a distance s, the geodesics are not parallel (geodesic deviation).[23]
Main articles: Geodesics in general relativity and Geodesic equation

The above equations are valid in flat spacetime. In curved space spacetime, things become mathematically more complicated since there is no straight line; this is generalized and replaced by a geodesic of the curved spacetime (the shortest length of curve between two points). For curved manifolds with a metric tensor g, the metric provides the notion of arc length (see line element for details), the differential arc length is given by:[24]

d s = g α β d x α d x β {\displaystyle ds={\sqrt {g_{\alpha \beta }dx^{\alpha }dx^{\beta }}}} ds={\sqrt {g_{\alpha \beta }dx^{\alpha }dx^{\beta }}}

and the geodesic equation is a second-order differential equation in the coordinates, the general solution is a family of geodesics:[25]

d 2 x μ d s 2 = − Γ μ α β d x α d s d x β d s {\displaystyle {\frac {d^{2}x^{\mu }}{ds^{2}}}=-\Gamma ^{\mu }{}_{\alpha \beta }{\frac {dx^{\alpha }}{ds}}{\frac {dx^{\beta }}{ds}}} {\frac {d^{2}x^{\mu }}{ds^{2}}}=-\Gamma ^{\mu }{}_{\alpha \beta }{\frac {dx^{\alpha }}{ds}}{\frac {dx^{\beta }}{ds}}

where Γ μαβ is a Christoffel symbol of the second kind, which contains the metric (with respect to the coordinate system).

Given the mass-energy distribution provided by the stress–energy tensor T αβ, the Einstein field equations are a set of non-linear second-order partial differential equations in the metric, and imply the curvature of space time is equivalent to a gravitational field (see principle of equivalence). Mass falling in curved spacetime is equivalent to a mass falling in a gravitational field - because gravity is a fictitious force. The relative acceleration of one geodesic to another in curved spacetime is given by the geodesic deviation equation:

D 2 ξ α d s 2 = − R α β γ δ d x α d s ξ γ d x δ d s {\displaystyle {\frac {D^{2}\xi ^{\alpha }}{ds^{2}}}=-R^{\alpha }{}_{\beta \gamma \delta }{\frac {dx^{\alpha }}{ds}}\xi ^{\gamma }{\frac {dx^{\delta }}{ds}}} {\frac {D^{2}\xi ^{\alpha }}{ds^{2}}}=-R^{\alpha }{}_{\beta \gamma \delta }{\frac {dx^{\alpha }}{ds}}\xi ^{\gamma }{\frac {dx^{\delta }}{ds}}

where ξα = x2α − x1α is the separation vector between two geodesics, D/ds (not just d/ds) is the covariant derivative, and Rαβγδ is the Riemann curvature tensor, containing the Christoffel symbols. In other words, the geodesic deviation equation is the equation of motion for masses in curved spacetime, analogous to the Lorentz force equation for charges in an electromagnetic field.[26]

For flat spacetime, the metric is a constant tensor so the Christoffel symbols vanish, and the geodesic equation has the solutions of straight lines. This is also the limiting case when masses move according to Newton's law of gravity.

Spinning objects[edit]

In general relativity, rotational motion is described by the relativistic angular momentum tensor, including the spin tensor, which enter the equations of motion under covariant derivatives with respect to proper time. The Mathisson–Papapetrou–Dixon equations describe the motion of spinning objects moving in a gravitational field.

Analogues for waves and fields[edit]

Unlike the equations of motion for describing particle mechanics, which are systems of coupled ordinary differential equations, the analogous equations governing the dynamics of waves and fields are always partial differential equations, since the waves or fields are functions of space and time. For a particular solution, boundary conditions along with initial conditions need to be specified.

Sometimes in the following contexts, the wave or field equations are also called "equations of motion".

Field equations[edit]

Equations that describe the spatial dependence and time evolution of fields are called field equations. These include

  • Maxwell's equations for the electromagnetic field,
  • Poisson's equation for Newtonian gravitational or electrostatic field potentials,
  • the Einstein field equation for gravitation (Newton's law of gravity is a special case for weak gravitational fields and low velocities of particles).

This terminology is not universal: for example although the Navier–Stokes equations govern the velocity field of a fluid, they are not usually called "field equations", since in this context they represent the momentum of the fluid and are called the "momentum equations" instead.

Wave equations[edit]

Equations of wave motion are called wave equations. The solutions to a wave equation give the time-evolution and spatial dependence of the amplitude. Boundary conditions determine if the solutions describe traveling waves or standing waves.

From classical equations of motion and field equations; mechanical, gravitational wave, and electromagnetic wave equations can be derived. The general linear wave equation in 3D is:

1 v 2 ∂ 2 X ∂ t 2 = ∇ 2 X {\displaystyle {\frac {1}{v^{2}}}{\frac {\partial ^{2}X}{\partial t^{2}}}=\nabla ^{2}X} {\frac {1}{v^{2}}}{\frac {\partial ^{2}X}{\partial t^{2}}}=\nabla ^{2}X

where X = X(r, t) is any mechanical or electromagnetic field amplitude, say:[27]

  • the transverse or longitudinal displacement of a vibrating rod, wire, cable, membrane etc.,
  • the fluctuating pressure of a medium, sound pressure,
  • the electric fields E or D, or the magnetic fields B or H,
  • the voltage V or current I in an alternating current circuit,

and v is the phase velocity. Nonlinear equations model the dependence of phase velocity on amplitude, replacing v by v(X). There are other linear and nonlinear wave equations for very specific applications, see for example the Korteweg–de Vries equation.

Quantum theory[edit]

In quantum theory, the wave and field concepts both appear.

In quantum mechanics, in which particles also have wave-like properties according to wave–particle duality, the analogue of the classical equations of motion (Newton's law, Euler–Lagrange equation, Hamilton–Jacobi equation, etc.) is the Schrödinger equation in its most general form:

i ℏ ∂ Ψ ∂ t = H ^ Ψ , {\displaystyle i\hbar {\frac {\partial \Psi }{\partial t}}={\hat {H}}\Psi \,,} i\hbar {\frac {\partial \Psi }{\partial t}}={\hat {H}}\Psi \,,

where Ψ is the wavefunction of the system, Ĥ is the quantum Hamiltonian operator, rather than a function as in classical mechanics, and ħ is the Planck constant divided by 2π. Setting up the Hamiltonian and inserting it into the equation results in a wave equation, the solution is the wavefunction as a function of space and time. The Schrödinger equation itself reduces to the Hamilton–Jacobi equation when one considers the correspondence principle, in the limit that ħ becomes zero.

Throughout all aspects of quantum theory, relativistic or non-relativistic, there are various formulations alternative to the Schrödinger equation that govern the time evolution and behavior of a quantum system, for instance:

  • the Heisenberg equation of motion resembles the time evolution of classical observables as functions of position, momentum, and time, if one replaces dynamical observables by their quantum operators and the classical Poisson bracket by the commutator,
  • the phase space formulation closely follows classical Hamiltonian mechanics, placing position and momentum on equal footing,
  • the Feynman path integral formulation extends the principle of least action to quantum mechanics and field theory, placing emphasis on the use of a Lagrangians rather than Hamiltonians.

See also[edit]

  • Scalar (physics)
  • Vector
  • Distance
  • Displacement
  • Speed
  • Velocity
  • Acceleration
  • Angular displacement
  • Angular speed
  • Angular velocity
  • Angular acceleration
  • Equations for a falling body
  • Parabolic trajectory
  • Curvilinear coordinates
  • Orthogonal coordinates
  • Newton's laws of motion
  • Torricelli's equation
  • Euler–Lagrange equation
  • Generalized forces
  • Defining equation (physics)
  • Newton–Euler laws of motion for a rigid body

References[edit]

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  20. ^ Jump up to: a b c Classical Mechanics (second edition), T.W.B. Kibble, European Physics Series, 1973, ISBN 0-07-084018-0
  21. Jump up ^ Electromagnetism (second edition), I.S. Grant, W.R. Phillips, Manchester Physics Series, 2008 ISBN 0-471-92712-0
  22. Jump up ^ Classical Mechanics (second Edition), T.W.B. Kibble, European Physics Series, Mc Graw Hill (UK), 1973, ISBN 0-07-084018-0.
  23. Jump up ^ Misner, Thorne, Wheeler, Gravitation
  24. Jump up ^ C.B. Parker (1994). McGraw Hill Encyclopaedia of Physics (second ed.). p. 1199. ISBN 0-07-051400-3.
  25. Jump up ^ C.B. Parker (1994). McGraw Hill Encyclopaedia of Physics (second ed.). p. 1200. ISBN 0-07-051400-3.
  26. Jump up ^ J.A. Wheeler; C. Misner; K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. pp. 34–35. ISBN 0-7167-0344-0.
  27. Jump up ^ H.D. Young; R.A. Freedman (2008). University Physics (12th ed.). Addison-Wesley (Pearson International). ISBN 0-321-50130-6.
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Hickey (surname)

Hickey (surname)

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Hickey is a common surname of Irish origin. The original form is Ó hÍceadha, which is still used in Ireland. Notable people with the surname include:

Contents

 [hide]
  • 1 Arts and entertainment
  • 2 Sport
  • 3 Politics and law
  • 4 History and current affairs
  • 5 Religion
  • 6 Science and academia
  • 7 Fictional characters
  • 8 See also

Arts and entertainment[edit]

  • Cheryl Hickey, born 1976, entertainment reporter for the Global Television Network
  • Dale Hickey born 1937, Australian artist
  • Dave Hickey, American art critic, author of Air Guitar: Essays on Art & Democracy (1998)
  • Eddie Hickey, US sportsman
  • Emily Henrietta Hickey, Irish author and translator
  • Ersel Hickey, rockabilly singer
  • James Harden-Hickey, 19th-century American writer
  • Kenny Hickey, US musician
  • Michael Hickey, US screenwriter
  • Rich Hickey, Creator of the Clojure programming language
  • Tom Hickey, Irish actor, famous for his role in The Riordans
  • Thomas Hickey, 1741–1842, Irish painter
  • William Hickey (actor)
  • William Hickey (memoirist)
  • May Wynn, actress, born Donna Lee Hickey before taking the stage name "May Wynn," after playing a character by that name in The Caine Mutiny

Sport[edit]

  • Ambrose Hickey (1945–2016), Irish Gaelic footballer
  • Anthony Hickey, born 1992, American basketball player
  • Charlie Hickey, American college baseball coach
  • Colin Hickey, Australian speed skater
  • Denis Hickie, Irish international Rugby Union
  • Jack Hickey (rugby), Australian Dual internationalist Rugby Union, Rugby league
  • Jarrad Hickey, born 1985, Australian Rugby League player
  • Jim Hickey (American football)
  • Jim Hickey (baseball coach)
  • Miriam Hickey, US Soccer player
  • Noah Hickey, New Zealand football player
  • Noel Hickey, Irish sportsman
  • Pat Hickey, born 1953, Canadian Ice Hockey player
  • Pat Hickey (sports administrator), born 1945, Irish sports administrator
  • Red Hickey, 1917–2006, athlete, played American Football
  • Reg Hickey, player and coach for Geelong in Australian rules football
  • Thomas Hickey, professional ice hockey player, L.A. Kings

Politics and law[edit]

  • Bonnie Hickey, Canadian politician
  • Eileen Hickey (New York) (1945–1999), New York politician
  • Eileen M. Hickey (1886–1960), Northern Irish politician
  • James Hickey (Irish politician)
  • Jimmy Hickey Jr., American politician
  • John J. Hickey, American politician
  • Maggie Hickey, Australian politician
  • Theodore M. Hickey, American politician
  • Vivian Hickey (1916-2016), American politician and educator
  • William J. Hickey, New York politician and judge

History and current affairs[edit]

  • Colonel James Hickey (soldier), in charge of the US Special forces team which captured Saddam Hussein
  • Elizabeth Hickey historian and writer
  • Jim Hickey (weather presenter)
  • Thomas Hickey (soldier), executed for mutiny during the American Revolutionary War

Religion[edit]

  • Antony Hickey, born 1586, Irish Franciscan theologian
  • David Francis Hickey (1882–1973), American-born Catholic bishop in Belize
  • Rev. William Hickey (1787-1875), Irish priest, writer and philanthropist

Science and academia[edit]

  • Joseph Hickey (1907-1993), American ornithologist
  • Raymond Hickey, Irish linguist

Fictional characters[edit]

  • Earl Hickey, fictional character in the NBC show My Name Is Earl
  • Randy Hickey, fictional character in the NBC show My Name Is Earl

See also[edit]

  • Hickey (disambiguation)
  • Ó hÍceadha
  • Irish medical families
WPanthroponymy.svg This page lists people with the surname Hickey. If an internal link intending to refer to a specific person led you to this page, you may wish to change that link by adding the person's given name(s) to the link.
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  • This page was last edited on 4 December 2017, at 16:46.
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