nLab generating function in classical mechanics

to be merged with canonical transformation

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 $q_i$, $i = 1,\ldots, n$ and the corresponding moment $p_i$ where the Hamiltonian is $H(p,q,t)$ and $p = (p_1,\ldots, p_n)$, $= (q_1,\ldots, q_n)$. The canonical transformation by definition preserves the equations of motion; let the new coordinates be $Q_j$ and momenta $P_j$ and the new Hamiltonian is $K$. Thus both variations

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

vanish. Here we of course write $p d q := \sum_{i = 1}^n p_i d q_i$. 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.

$$p d q - H d t = P d Q - K d t + d F$$

$$d F = p d q - P d Q + (K - H) d t$$

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

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

$$d \Phi = d (F + P Q) = p d q + Q d P + (K - H) d t$$

and $\frac{\partial \Phi}{\partial q} = p$, $\frac{\partial F}{\partial P} = Q$, $K = \frac{\partial \Phi}{\partial t} + H$. 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

• C. Viterbo, Generating functions, symplectic geometry, and applications, Proc. ICM 537-547 doi

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

• David Théret, A complete proof of Viterbo’s uniqueness theorem on generating functions, Topology and its Applications 96 (1999) 249–266 pdf

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