Lagrange Bracket

Let (q_1,...,q_n,p_1,...,p_n) be any functions of two variables (u,v). Then the expression

is called a Lagrange bracket (Lagrange 1808; Whittaker 1944, p. 298).

The Lagrange brackets are anticommutative,

(Plummer 1960, p. 136).

If (q_1,...,q_n,p_1,...,p_n) are any functions of 2n variables (Q_1,...,Q_n,P_1,...,P_n), then

where the summation on the right-hand side is taken over all pairs of variables (u_k,u_l) in the set (Q_1,...,Q_n,P_1,...,P_n).

But if the transformation from (q_1,...,q_n,p_1,...,p_n) to (Q_1,...,Q_n,P_1,...,P_n) is a contact transformation, then

giving

Furthermore, these may be regarded as partial differential equations which must be satisfied by (q_1,...,q_n,p_1,...,p_n), considered as function of (Q_1,...,Q_n,P_1,...,P_n) in order that the transformation from one set of variables to the other may be a contact transformation.

Let (u_1,...,u_(2n)) be 2n independent functions of the variables (q_1,...,q_n,p_1,...,p_n). Then the Poisson bracket (u_r,u_s) is connected with the Lagrange bracket [u_r,u_s] by

where delta_(rs) is the Kronecker delta. But this is precisely the condition that the determinants formed from them are reciprocal (Whittaker 1944, p. 300; Plummer 1960, p. 137).

See also

Explore with Wolfram|Alpha

References

Referenced on Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "Lagrange Bracket." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LagrangeBracket.html

Subject classifications