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Difference between revisions of "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|Lagrange function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027500/c0275001.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027500/c0275002.png" /> are generalized coordinates, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027500/c0275003.png" /> generalized velocities, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027500/c0275004.png" /> the time, does not contain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027500/c0275005.png" /> explicitly, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027500/c0275006.png" /> is a cyclic coordinate, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027500/c0275007.png" />-th Lagrange equation has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027500/c0275008.png" /> (cf. [[Lagrange equations (in mechanics)|Lagrange equations (in mechanics)]]), which at once gives an integral of motion
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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|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)|Lagrange equations (in mechanics)]]), which at once gives an integral of motion
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027500/c0275009.png" /></td> </tr></table>
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$$\frac{\partial L}{\partial\dot q_j}=\text{const}.$$
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027500/c02750010.png" /> such that the Hamiltonian has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027500/c02750011.png" />, i.e. does not contain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027500/c02750012.png" />. Then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027500/c02750013.png" /> are called the action coordinates and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027500/c02750014.png" /> the angle coordinates.
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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.

Latest revision as of 14:39, 28 August 2014

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

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

References

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


Comments

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: http://encyclopediaofmath.org/index.php?title=Cyclic_coordinates&oldid=13639
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article