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
Famous quotes containing the words canonical and/or motion:
“If God bestowed immortality on every man then when he made him, and he made many to whom he never purposed to give his saving grace, what did his Lordship think that God gave any man immortality with purpose only to make him capable of immortal torments? It is a hard saying, and I think cannot piously be believed. I am sure it can never be proved by the canonical Scripture.”
—Thomas Hobbes (1579–1688)
“Wine gives a man nothing. It neither gives him knowledge nor wit; it only animates a man, and enables him to bring out what a dread of the company has repressed. It only puts in motion what had been locked up in frost.”
—Samuel Johnson (1709–1784)