Lagrange bracket

Lagrange brackets, with respect to variables $ u $ and $ v $

A sum of the form

$$ \tag{* } \sum_{i=1}^n \left ( \frac{\partial q _ {i} }{\partial u } \frac{\partial p _ {i} }{\partial v } - \frac{\partial q _ {i} }{\partial v } \frac{\partial p _ {i} }{\partial u } \right ) \equiv [ u , v ] _ {p , q } , $$

where $ q = ( q _ {1} \dots q _ {n} ) $ and $ p = ( p _ {1} \dots p _ {n} ) $ are certain functions of $ u $ and $ v $.

If $ q = ( q _ {1} \dots q _ {n} ) $ and $ p = ( p _ {1} \dots p _ {n} ) $ are canonical variables and $ Q = Q ( q , p ) $, $ P = P ( q , p ) $ are canonical transformations, then the Lagrange bracket is an invariant of this transformation:

$$ [ u , v ] _ {q , p } = \ [ u , v ] _ {Q , P } . $$

For this reason the indices $ q , p $ on the right-hand side of (*) are often omitted. The Lagrange bracket is said to be fundamental when the variables $ u $ and $ v $ coincide with some pair of the $ 2n $ variables $ q , p $. From them one can form three matrices:

$$ [ p , p ] = \ \{ [ p _ {i} , p _ {j} ] \} _ {i , j = 1 } ^ {n} ,\ \ [ q , q ] ,\ [ q , p ] , $$

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. Namely, if the functions $ u _ {i} = u _ {i} ( q , p ) $, $ 1 \leq i \leq n $, induce a diffeomorphism $ \mathbf R ^ {2n} \rightarrow \mathbf R ^ {2n} $, then the matrices formed from the elements $ [ u _ {i} , u _ {j} ] $ and $ ( u _ {j} , u _ {i} ) $ are inverse to each other.

References

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

Comments

If $ \psi $ denotes the mapping: $ ( u , v) \mapsto ( q ( u , v), p ( u , v)) $, then the Lagrange bracket is equal to the product of the vectors $ {\partial \psi } / {\partial u } $ and $ {\partial \psi } / {\partial v } $ with respect to the canonical symplectic form (cf. Symplectic manifold) on the phase space. More generally, if $ \omega $ is a symplectic form on a smooth manifold $ M $ and $ \psi $ is a smooth mapping from a surface $ S $ to $ M $, then $ \psi ^ {*} \omega $ is an area form on $ S $. If $ ds $ is a standard area form on $ S $, then the function $ \psi ^ {*} \omega /ds $ on $ S $ could be called the Lagrange brackets of $ \psi $. See [a1], Chapt. 3.

References

[a1]	R. Abraham, J.E. Marsden, "Foundations of mechanics" , Benjamin/Cummings (1978)
[a2]	F.R. [F.R. Gantmakher] Gantmacher, "Lectures in analytical mechanics" , MIR (1975) (Translated from Russian)