Poisson Bracket

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


is called a Poisson bracket (Poisson 1809; Whittaker 1944, p. 299). Plummer (1960, p. 136) uses the alternate notation {u,v}.

The Poisson brackets are anticommutative,


(Plummer 1960, p. 136).

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).

If A and B are physically measurable quantities (observables) such as position, momentum, angular momentum, or energy, then they are represented as non-commuting quantum mechanical operators in accordance with Heisenberg's formulation of quantum mechanics. In this case,


where [A,B] is the commutator and (A,B) is the Poisson bracket. Thus, for example, for a single particle moving in one dimension with position q and momentum p,


where h is h-bar.

See also

Lagrange Bracket, Lie Bracket

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Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1004, 1980.Plummer, H. An Introductory Treatise of Dynamical Astronomy. New York: Dover, pp. 136-137, 1960.Poisson. J. de l'École Polytech. 8, p. 266, 1809.Whittaker, E. T. A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: With an Introduction to the Problem of Three Bodies. New York: Dover, 1944.

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Poisson Bracket

Cite this as:

Weisstein, Eric W. "Poisson Bracket." From MathWorld--A Wolfram Web Resource.

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