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 $q_i$, $i=1,\dots ,n$ and the corresponding moment $p_i$ where the Hamiltonian is $H(p,q,t)$ and $p=(p_1,\dots ,p_n)$, $=(q_1,\dots ,q_n)$. 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

$$\delta \int pdq-Hdt=\delta \int PdQ-Kdt=0$$\delta \int p d q - H d t = \delta \int P d Q - K d t = 0

vanish. Here we of course write $pdq:={\sum }_{i=1}^n{p}_{i}d{q}_i$. 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.

$$pdq-Hdt=PdQ-Kdt+dF$$p d q - H d t = P d Q - K d t + d F

$$dF=pdq-PdQ+(K-H)dt$$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 $PQ$, use the Leibniz rule $d(PQ)=PdQ+QdP$ and we see that for the generating function of the “second kind”, $\Phi (q,P,t)=F(q,P,t)+QP$ the total differential

$$d\Phi =d(F+PQ)=pdq+QdP+(K-H)dt$$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

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