# Action functional in the formalism of symplectic manifolds with Hamiltonian

Call Hamiltonian a symplectic manifold $$(M, \omega)$$ equipped with a distinguished Hamiltonian $$h \in \mathcal C^\infty(\mathbb R \times M)$$.

Wikipedia 'Tautological form' page has a section about defining the action functional for the phase space Hamiltonian manifold $$(T^*X, -d \theta, h)$$, where $$\theta$$ is the tautological 1-form.

It defines it as $$S = \theta(X_h)$$ where $$X_h$$ is the Hamiltonian flow (or better, the associated vector field). This is already a bit confusing, because usually $$S$$ depends on a given trajectory, but I guess one recovers the usual functional by integration: $$\mathcal S[q] = \int_0^1 S(q(t)) \,dt$$ where $$q : \mathbb R \to X$$ is a given trajectory.

Q: How do I generalize this for a general Hamiltonian manifold?

In the above definition, we made essential use of the fact that $$\omega$$ is not just closed, but also exact, i.e. it's of the form $$d\theta$$ for some 1-form $$\theta$$. A paper I found circumvents this problem by assuming $$\pi_2(M) = 0$$, but the setup is somewhat confusing to me.

• Jun 15 at 18:49