Cyclic coordinates

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Generalized coordinates of a certain physical system that do not occur explicitly in the expression of the characteristic function of this system. When one uses the corresponding equations of motion, one may obtain at once for every cyclic coordinate the integral of motion corresponding to it. For example, if the Lagrange function $L(q_i,\dot q_i,t)$, where the $q_i$ are generalized coordinates, the $\dot q_i$ generalized velocities, and $t$ the time, does not contain $q_j$ explicitly, then $q_j$ is a cyclic coordinate, and the $j$-th Lagrange equation has the form $(d/dt)(\partial L/\partial\dot q_j)=0$ (cf. Lagrange equations (in mechanics)), which at once gives an integral of motion

$$\frac{\partial L}{\partial\dot q_j}=\text{const}.$$


[1] L.D. Landau, E.M. Lifshits, "Mechanics" , Pergamon (1965) (Translated from Russian)


The notion of a cyclic coordinate (angle coordinate, angle variable) ties in with action-angle coordinates in the theory of completely-integrable Hamiltonian systems. Each such system (with finite degrees of freedom) can be transformed into one with coordinates $(y_k,x_k)$ such that the Hamiltonian has the form $H(y_1,\dots,y_n)$, i.e. does not contain $x_1,\dots,x_n$. Then the $y_k$ are called the action coordinates and the $x_k$ the angle coordinates.

How to Cite This Entry:
Cyclic coordinates. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article