View source for Lagrange bracket

''Lagrange brackets, with respect to variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l0571301.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l0571302.png" />''

A sum of the form

<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/l/l057/l057130/l0571303.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l0571304.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l0571305.png" /> are certain functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l0571306.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l0571307.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l0571308.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l0571309.png" /> are canonical variables and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713011.png" /> are canonical transformations, then the Lagrange bracket is an invariant of this transformation:

<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/l/l057/l057130/l05713012.png" /></td> </tr></table>

For this reason the indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713013.png" /> on the right-hand side of (*) are often omitted. The Lagrange bracket is said to be fundamental when the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713015.png" /> coincide with some pair of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713016.png" /> variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713017.png" />. From them one can form three matrices:

<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/l/l057/l057130/l05713018.png" /></td> </tr></table>

the first two of which are the zero, and the last one is the unit matrix. There is a definite connection between Lagrange brackets and [[Poisson brackets|Poisson brackets]]. Namely, if the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713020.png" />, induce a diffeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713021.png" />, then the matrices formed from the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713023.png" /> are inverse to each other.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Lagrange,   "Oeuvres" , '''6''' , Gauthier-Villars  (1873)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.T. Whittaker,   "Analytical dynamics of particles and rigid bodies" , Dover, reprint  (1944)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.I. Lur'e,   "Analytical mechanics" , Moscow  (1961)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  H. Goldstein,   "Classical mechanics" , Addison-Wesley  (1957)</TD></TR></table>



====Comments====
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713024.png" /> denotes the mapping: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713025.png" />, then the Lagrange bracket is equal to the product of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713027.png" /> with respect to the canonical symplectic form (cf. [[Symplectic manifold|Symplectic manifold]]) on the phase space. More generally, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713028.png" /> is a symplectic form on a smooth manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713030.png" /> is a smooth mapping from a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713031.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713032.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713033.png" /> is an area form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713034.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713035.png" /> is a standard area form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713036.png" />, then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713037.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713038.png" /> could be called the Lagrange brackets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057130/l05713039.png" />. See [[#References|[a1]]], Chapt. 3.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Abraham,   J.E. Marsden,   "Foundations of mechanics" , Benjamin/Cummings  (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F.R. [F.R. Gantmakher] Gantmacher,   "Lectures in analytical mechanics" , MIR  (1975)  (Translated from Russian)</TD></TR></table>