What is Generalized momentum, cyclic coordinate? — Physics

Definition 1.10 University Physics — Year 3 · Chapter 1 — Lagrangian Mechanics

The generalized momentum conjugate to the coordinate $q_i$ is

p_i = \frac{\partial L}{\partial\dot q_i} .

For a Cartesian coordinate it is the ordinary momentum $m\dot x$ ; for an angle it is an angular momentum. A coordinate that does not appear in $L$ (though its velocity does) is called cyclic.

Examples

Example 1.12 (Central force, solved by inspection)

A particle in a central potential $E_p(r)$ , in polar coordinates in its plane of motion:

L = \tfrac12 m(\dot r^2 + r^2\dot\varphi^2) - E_p(r) .

$\varphi$ is cyclic, so $p_\varphi = mr^2\dot\varphi$ — the angular momentum — is conserved: Kepler’s law of areas, which the Year 1 volume derived from the torque equation, here falls out before any equation is solved. The remaining radial equation is $m\ddot r = mr\dot\varphi^2 - E_p'(r)$ , i.e. the one-dimensional motion in the effective potential $E_p(r) + p_\varphi^2/2mr^2$ of that volume.

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