differential geometry - Formulation of an Action-Principle in a general Phase Space (No Cotangent-Bundle)

If we come from the physics side, the hamiltonian formalism usually is introduced via generalized coordinates (which are just a collection of numbers stuffed into a vector ($\vec{q}$), and the lagrangian formalism. Legendre transform of the Lagrangian yields the hamiltonian, and so on.

In this formulation, the canonical equations of motion \begin{align} \dot{\vec{q}} = \frac{\partial H}{\partial p} \\ \dot{\vec{p}} = -\frac{\partial H}{\partial q} \end{align} follow from an action principle. The action \begin{align} \int dt~\left( \dot{\vec{q}} \cdot\vec{p} - H(\vec{q}, \vec{p}, t)\right) \end{align} is stationary with respect to variations with $\vec{q}(t_2)$ and $\vec{q}(t_1)$ fixed. Now I've read up more mathematical formulations, that employ the concept of manifolds. In some, the phase space is introduced as cotant space over the configuration space, which means that $\vec{q}$ now is a set of coordinates for a point in the configuration space Q (which is a manifold), $\dot{\vec{q}}$ is an element of the tangential space $T_{\vec{q}}Q$, and $\vec{p}$ is an element of the dual space to that tangent space. Up to here it's no problem to write down the above mentioned action in the same manner. $\dot{\vec{q}}\vec{p}$ now isn't anymore a scalar product, but it's instead the application of $\vec{p}$ to \dot{\vec{q}}, which maps to a real number (and which perfectly makes sense).

My question is now: Can we keep up with the action formulation for cases in which the phase space (possibly) isn't a cotangent space anymore? For example, if it's just an even-dimensional space, equipped with a symplectic form, can we somehow still write down an action, that makes sense from a mathematical point of view?

Or is my question not necessary, because every notion of phase space that appears in physics makes use of the phase space as a cotangent space?

Answer

1. 
   

   Let there be given a $2n$-dimensional symplectic manifold $(M,\omega)$, $$\mathrm{d}\omega~=~0 \tag{1}$$ with globally defined Hamiltonian function $H: M \to \mathbb{R}$. (Let us for simplicity assume point mechanics with no explicit time dependence. The construction can be generalized to field theory.)

   
   


2. 
   

   Locally in a contractible open coordinate neighborhood $U\subseteq M$ there exists a symplectic potential 1-form $$\vartheta ~=~\sum_{I=1}^{2n}\vartheta_I~\mathrm{d}z^I ~\in~ \Gamma(T^{\ast}M|_U),\tag{2}$$ such that $$ \omega|_U ~=~\mathrm{d}\vartheta .\tag{3}$$

   
   


3. 
   

   Given a path $\gamma \subset U$. Define the local Hamiltonian action $$S_U[\gamma]~:=~\int_{\gamma} \left( \vartheta - H ~\mathrm{d}t\right) ~=~\int_{t_i}^{t_f}\! \mathrm{d}t\left( \sum_{I=1}^{2n}\vartheta_I~\dot{z}^I -H\right). \tag{4}$$ One may show that the corresponding Euler-Lagrange (EL) eqs. are precisely Hamilton's eqs. $$ \dot{z}^I~=~\{z^I,H\} .\tag{5}$$ Here the globally defined Poisson bi-vector is the inverse of the symplectic 2-form.

   
   
   


4. 
   

   It is possible to globalize the local action (4) into a so-called Wu-Yang action via a sheaf-theoretic construction on $M$. This is e.g. explained in Ref. 1.

   
   

References:

1. E.S. Fradkin & V.Ya. Linetsky, BFV approach to geometric quantization, Nucl. Phys. B431 (1994) 569; Section 3.3.