nLab
generating function in classical mechanics

Local picture

Local picture is explained in

  • Landau-Lifschitz, Mechanics, vol I. of Course of theoretical physics, chapter 23, Canonical transformations

Suppose we choose a Lagrangean submanifold, what gives the splitting between submanifold coordinates q iq_i, i=1,,ni = 1,\ldots, n and the corresponding moment p ip_i where the Hamiltonian is H(p,q,t)H(p,q,t) and p=(p 1,,p n)p = (p_1,\ldots, p_n), =(q 1,,q n) = (q_1,\ldots, q_n). The canonical transformation by definition preserves the equations of motion; let the new coordinates be Q jQ_j and momenta P jP_j and the new Hamiltonian is KK. Thus both variations

δpdqHdt=δPdQKdt=0 \delta \int p d q - H d t = \delta \int P d Q - K d t = 0

vanish. Here we of course write pdq:= i=1 np idq ip d q := \sum_{i = 1}^n p_i d q_i. Subtracting the left and right hand side variations we get that the difference must be a (variation of the integral of the) total differential of a function, which can be chosen as a function of some set of old and new coordinates in a consistent way. For example in 1 dimension, we have 4 possibilities of one new and one old generalized coordinate.

pdqHdt=PdQKdt+dF p d q - H d t = P d Q - K d t + d F
dF=pdqPdQ+(KH)dt d F = p d q - P d Q + (K - H) d t

therefore if F=F(q,Q,t)F = F (q,Q, t) then Fq=p\frac{\partial F}{\partial q} = p, FQ=P\frac{\partial F}{\partial Q} = -P, K=Ft+HK = \frac{\partial F}{\partial t} + H, what gives the relation between the old and new coordinates and momenta and the new Hamiltonian KK, which must be expressed in terms of P,Q,tP, Q, t.

If F=F(q,P,t)F = F(q,P,t) then we add PQP Q, use the Leibniz rule d(PQ)=PdQ+QdPd (P Q) = P d Q + Q d P and we see that for the generating function of the “second kind”, Φ(q,P,t)=F(q,P,t)+QP\Phi(q,P,t) = F(q,P,t) + Q P the total differential

dΦ=d(F+PQ)=pdq+QdP+(KH)dt d \Phi = d (F + P Q) = p d q + Q d P + (K - H) d t

and Φq=p\frac{\partial \Phi}{\partial q} = p, FP=Q\frac{\partial F}{\partial P} = Q, K=Φt+HK = \frac{\partial \Phi}{\partial t} + H. A similar analysis can be done straightfowardly for other possibilities of the choice of arguments.

Global picture

An invariant picture of generating functions on symplectic manifolds is in

  • C. Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann. 292 (1992), 685–710, MR1157321, doi

  • L. Traynor, Symplectic homology via generating function, Geom. Funct. Anal. 4 (1994) 718-748, MR1302337, doi

An adaption of generating functions to the setup of symplectic micromorphisms is in

  • Alberto S. Cattaneo, Benoit Dherin, Alan Weinstein, Symplectic microgeometry II: generating functions, arxiv/1103.0672

Revised on December 11, 2013 13:30:50 by Zoran Škoda (88.207.224.62)