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Lagrange Bracket

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

\begin{displaymath}[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].
\end{displaymath} (1)

The Lagrange bracket satisfies
\end{displaymath} (2)

\begin{displaymath}[[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].
\end{displaymath} (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


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

© 1996-9 Eric W. Weisstein