Equations of motion
In mathematical physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.
Contents
Description
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.
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
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.
Kinematics
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.
Translations, rotations, oscillations
In all cases, the main types of motion are translations, rotations, oscillations, or any combinations of these.
Diffential equations
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.
See also
- Computational physics
- Differential equation
- Euler method
- Function (mathematics)
- Mathematical physics
- Numerical analysis
- Verlet integration
External links
- Equations of motion @ Wikipedia
