Canonical Momenta and Constants of Motion
The conjugate momentum pk for a generalized coordinate qk is defined by the equation
- .
An important special case of the Euler–Lagrange equation occurs when L does not contain a generalized coordinate qk explicitly,
that is, the conjugate momentum is a constant of the motion.
In such cases, the coordinate qk is called a cyclic coordinate. For example, if we use polar coordinates t, r, θ to describe the planar motion of a particle, and if L does not depend on θ, the conjugate momentum is the conserved angular momentum.
Read more about this topic: Hamilton's Principle, Mathematical Formulation
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