Equation of motion Definition and 28 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. curiousPep

    Engineering Equilibrium and stability

    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...
  2. curiousPep

    I Lagrangian mechanics - generalised coordinates question

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  3. H

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  4. O

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  5. sergiokapone

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  6. M

    Derive the equations of motion

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  7. M

    Is it possible to integrate acceleration?

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  8. S

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  9. Abhishek11235

    Deriving the small-x approximation for an equation of motion

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  10. S

    I Multistage continuous Rocket Eqn

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  11. I

    A weird velocity/acceleration question

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  12. M

    Find the time dependence.... (Mechanics)

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  13. S

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  14. W

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  15. Avijit

    Does rotational motion affect the translational motion?

    A flying object is moving in 3D space having translational velocity and the object is also rotating. Consider a body frame (xb-yb-zb) attached to the C.G of the moving body. Hence the body attached frame is also translating and rotating (as the object is flying) with respect to a fixed inertial...
  16. O

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  17. jeisson botache

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  18. A

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    Consider the 1d motion of a body under the influence of the force given by F = -m*γ*vα. m is mass, γ is a constant of appropriate dimension, v is velocity and α is dimensionless constant. The value of α for which the motion will come to a stop in finite time is to be calculated. I solved the...
  19. I

    I Evolution style algorithm to determine EOM

    I had seen a documentary about an algorithm that uses notions of evolution to deduce the equation of motion of a system by sampling a variable connected with the system. For example, they used the double pendulum case where they sampled the position of the free end of the pendulum and arrived...
  20. L

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  21. S

    Equations of motion for 4 dof

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  22. N

    Help with Projectile motion- equation problem

    Homework Statement A projectile is launched horizontally with an initial velocity v0 from a height h. If it is assumed that there is no air resistance, which of the following expressions represents the vertical trajectory of the projectile? (A) h–gv0^2/2x^2 (B) h–gv0^2x^2 (C) h-gx^2/2v0^2 (D)...
  23. AntoineCompagnie

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  24. Alexiy

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    Homework Statement Homework Equations 3. The Attempt at a Solution [/B] Hello guys,I posted images since its easier to write equations.Please can someone help me check this, if this is correct so far, then i should be able to find the velocity at C, using kinetic energy?
  25. X

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  26. E

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  27. olgerm

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  28. sergiokapone

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