Lagrange Bracket

Let $F$ and $G$ be infinitely differentiable functions of $x$, $u$, and $p$. Then the Lagrange bracket is defined by

$$[F,G]=\sum_{\nu=1}^n \left[{{\partial F\over\partial p_\nu}\l... ...rtial x_\nu}+p_\nu{\partial F\over\partial u}}\right)}\right].$$ (1)

The Lagrange bracket satisfies

$$[F,G]=-[G,F]$$ (2)

$$[[F,G],H]+[[G,H],F]+[[H,F],G] = {\partial F\over\partial u}[G... ...rtial G\over\partial u}[H,F]+{\partial H\over\partial u}[F,G].$$ (3)

If $F$ and $G$ are functions of $x$ and $p$ only, then the Lagrange bracket $[F,G]$ collapses the Poisson Bracket $(F,G)$.

See also Lie Bracket, Poisson Bracket

References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1004, 1980.