What is Equation of motion: Definition and 266 Discussions

In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behavior of a physical system as a set of mathematical functions in terms of dynamic variables. These variables are usually spatial coordinates and time, but may include momentum components. The most general choice are generalized coordinates which can be any convenient variables characteristic of the physical system. 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 for the differential equations describing the motion of the dynamics.
There are two main descriptions of motion: dynamics and kinematics. Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dynamics 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. 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,












{\displaystyle 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,







{\displaystyle \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.

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  1. B

    Equations of motion for Lagrangian of scalar QED

    Well, I started with the first equation of motion for the scalar field, but I'm really not sure if I'm doing it the right way. \begin{equation} \begin{split} \frac{\partial \mathcal{L}}{\partial \varphi} &= \frac{\partial}{\partial \varphi} [(\partial_\mu \varphi^* -...
  2. morpheus343

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  3. brotherbobby

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  4. T

    I Rewriting Equation of Motion in terms of Dual Fields (Chern-Simons)

    I am reading the following notes: https://arxiv.org/pdf/hep-th/9902115.pdf and am trying to make the connection between equations (22) and (24). Specifically, I do not understand how they were able to get (24) from (22) using the dual field prescription. I guess naively I'm not even sure where...
  5. MatinSAR

    Finding shortest distance between an observer and a moving object

    The rocket has constant velocity so we can write it's equation of motion as : $$\vec r = \vec r_0+t \vec v_0 $$ We can write it for components along each axis : $$x = x_0 + v_{0,x}t$$$$y = y_0 + v_{0,y}t$$$$z = z_0 + v_{0,z}t$$We put known values in above equations...
  6. C

    Finding the time for an object to start rolling without slipping

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  7. G

    Help me solve this integral for the equation of motion

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  8. M

    I Confused about applying the Euler–Lagrange equation

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  9. Rick16

    I Lagrangian approach for the inclined plane

    I want to use the Lagrangian approach to find the equation of motion for a mass sliding down a frictionless inclined plane. I call the length of the incline a and the angle that the incline makes with the horizontal b. Then the mass has kinetic energy 1/2m(da/dt)2 and the potential energy should...
  10. E

    I Solving Spherically Symmetric Static Star Equations of Motion

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  11. M

    Damping and friction in syringe equation of motion

    Hello Everyone I want to model forces affecting on syringe plunger , but I do not know how to calculate terms like friction and damping coefficient. What I imagine is that : F_driving = ma + cv + f ----------------(1) where: f: friction c: coefficient of viscous damping m: mass of plunger (is...
  12. D

    I Equation of motion for a simple mechanical system

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  13. D

    I Equation of motion: choice of generalized coordinates

    I am looking at a textbook solution to the following problem of finding the equation of motion of a half disk. In the solution, the author considers the half disk has a COM at the black dot, and to find the instantaneous translational velocity of the center of mass (he considers rotational...
  14. Pironman

    I How to find the equation of motion using Lagrange's equation?

    Good morning, I'm not a student but I'm curious about physics. I would like to calculate the equation of motion of a system using the Lagrangian mechanics. Suppose a particle subjected to some external forces. From Wikipedia, I found two method: 1. using kinetic energy and generalized forces...
  15. D

    Finding the differential equation of motion

    Summary:: Differential equation of motion, parabola Hi. I've tried resolve this problem but I have two doubts. The first is about the differential equation of motion because I can't simplify it to the form y" + a*y' + b*y = F(t). I'm not sure if what I got is right. My second doubt is that I...
  16. snoopies622

    I Seeking the Equation of Motion for a charged mass attached to a spring

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  17. rudransh verma

    B About Second equation of motion

    I was wondering how the second equation of motion produces negative displacements ##s= ut+\frac 12 at^2## . Is ##\frac12 at^2## kind of distance operator?
  18. curiousPep

    Engineering Stability Analysis of Equilibrium Solutions using Small Perturbations

    When I use Lagrange to get the equations of motion, in order to find the equilibrium conditions I set the parameters q as constants thus the derivatives to be zero and then calculate the q's that satisfy the equations of motion obtained. In ordert to check about stability I think I need to add...
  19. curiousPep

    I Lagrangian mechanics - generalised coordinates question

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  20. Istiak

    Find the equation of motion using the Lagrangian for this Atwood machine

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  21. Istiak

    Find equation of motion of an inclined plane when there's friction

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  22. M

    Engineering Equation of motion for the translation of a single rod

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  23. greg_rack

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  24. L

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  25. J

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  26. H

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  27. A

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  28. T

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  29. O

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  30. rad1um

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  31. M

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  32. VapeL

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  33. J

    Understanding this Equation of motion with a constant

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  34. JD_PM

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  35. P

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  36. JD_PM

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  37. JD_PM

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  38. V

    Finding the equation of motion for Born-Infeld lagrangian

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  39. sergiokapone

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  40. J

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  41. FahdEl

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    import numpy as np import matplotlib.pyplot as plt G=6.67408e-11 M=1.989e30 m=5.972e24 X0=-147095000000 Y0=0 VX0=0 VY0=-30300 T=365*24*60 def rk(ax,ay,x,y,vx,vy,h): t=0 n=int(T/h) A=[[t],[x],[y],[vx],[vy]] for i in range(1,n): k1x=vx k1y=vy q1x=ax(x,y)...
  42. L

    I Equation of motion Chern-Simons

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  43. D

    Equation of motion for oscillations about a stable orbit

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  44. N

    Using first principles, how to get the equation of motion?

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  45. M

    Derive the equations of motion

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  46. M

    Is it possible to integrate acceleration?

    Alright so I was just messing around with Lagrangian equation, I just learned about it, and I had gotten to this equation of motion: Mg*sin{α} - 1.5m*x(double dot)=0 I am trying to get velocity, and my first thought was to integrate with dt, but I didn't know how to. And I'm not even sure it's...
  47. S

    Converting a nonlinear eqn of motion to a state-space model

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  48. Abhishek11235

    Deriving the small-x approximation for an equation of motion

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  49. J

    MHB 2nd order differential equation - equation of motion

    Hi There is an example in my textbook worded as follows; A particle of mass 2kg moves along the positive x-axis under the action of a force directed towards the origin. At time t seconds, the displacement of P from O is x metres and P is moving away from O with a speed of v ms^-1. The force has...
  50. JD_PM

    Dimensional analysis of an equation of motion

    Homework Statement The evolution of the density in a system of attractive spheres can be described by the following dynamic equation. $$\frac{\partial}{\partial t} \rho (r,t) = D_o [\nabla^2 \rho (r,t) + \beta \nabla \rho (r,t) \int dr' [\nabla V (|r-r'|)] \rho (r',t) g(r,r',t)]$$ a)...