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Lagrange Bracket

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

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

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

The Lagrange brackets are anticommutative,

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

(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

sum_(r=1)^n(dp_rdeltaq_r-deltap_rdq_r)=sum_(k,l)[u_k,u_l](du_ldeltau_k-deltau_ldu_k),
(3)

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

sum_(r=1)^n(dp_rdeltaq_r-deltap_rdq_r)=sum_(r=1)^n(dP_rdeltaQ_r-deltaP_rdQ_r),
(4)

giving

[P_i,P_k]=0 for i,k=1,2,...,n
(5)
[Q_i,Q_k]=0 for i,k=1,2,...,n
(6)
[Q_i,P_k]=0 for i,k=1,2,...,n,i!=k
(7)
[Q_i,P_i]=0 for i=1,2,...,n.
(8)

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

sum_(t=1)^(2n)(u_t,u_r)[u_t,u_s]=delta_(rs),
(9)

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

Lie Bracket, Poisson Bracket

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References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1004, 1980.Lagrange. Mém. de l'Institute de France, 1808. Reprinted in Oeuvres, Vol. 4. p. 713.Plummer, H. An Introductory Treatise of Dynamical Astronomy. New York: Dover, p. 136, 1960.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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Lagrange Bracket

Cite this as:

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

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