Integral of Motion
A constant of motion may be defined in a given force field as any function of phase-space coordinates (position and velocity, or position and momentum) and time that is constant throughout a trajectory. A subset of the constants of motion are the integrals of motion, defined as any functions of only the phase-space coordinates that are constant along an orbit. Every integral of motion is a constant of motion, but the converse is not true because a constant of motion may depend on time. Examples of integrals of motion are the angular momentum vector, or a Hamiltonian without time dependence, such as . An example of a function that is a constant of motion but not an integral of motion would be the function for an object moving at a constant speed in one dimension.
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