Lagrange bracket

Lagrange brackets, with respect to variables and

A sum of the form

(*)

where and are certain functions of and .

If and are canonical variables and , are canonical transformations, then the Lagrange bracket is an invariant of this transformation:

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

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 , , induce a diffeomorphism , then the matrices formed from the elements and 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 denotes the mapping: , then the Lagrange bracket is equal to the product of the vectors and with respect to the canonical symplectic form (cf. Symplectic manifold) on the phase space. More generally, if is a symplectic form on a smooth manifold and is a smooth mapping from a surface to , then is an area form on . If is a standard area form on , then the function on could be called the Lagrange brackets of . 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)