TOPICS
Search

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

 (u,v)=sum_(r=1)^n((partialu)/(partialq_r)(partialv)/(partialp_r)-(partialu)/(partialp_r)(partialv)/(partialq_r))
(1)

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,

 (u_l,u_m)=-(u_m,u_l)
(2)

(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

 sum_(t=1)^(2n)(u_t,u_r)[u_t,u_s]=delta_(rs),
(3)

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,

 [A,B]=AB-BA=ih(A,B),
(4)

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,

 [q,p]=qp-pq=ih(q,p)=ih,
(5)

where h is h-bar.


See also

Lagrange Bracket, Lie Bracket

Explore with Wolfram|Alpha

References

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.

Referenced on Wolfram|Alpha

Poisson Bracket

Cite this as:

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

Subject classifications