# 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?

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 \begin{align} S_U[\gamma]~:=~&\int_{\gamma} \left( \vartheta - H ~\mathrm{d}t\right)\cr ~=~&\int_{t_i}^{t_f}\! \mathrm{d}t\left( \sum_{I=1}^{2n}\vartheta_I~\dot{z}^I -H\right). \end{align}\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.

Let's write it another way:

$$L = \theta - H,$$

where $\theta(p,q)$ is a 1-form such that $d\theta = \omega$, the symplectic form. This is called the Liouville form. For a particle on the line it's $\theta = pdq$, $d\theta = dp dq$. On a compact manifold, we cannot take $\theta$ to be globally defined, but it is still possible to make the action

$$S = \int L = \int \theta - \int H$$

well-defined if $\theta$ is a connection 1-form. Now one has to choose a gauge, meaning essentially a covering of your symplectic manifold by cotangent spaces, and keep track so that everything is gauge invariant. In fact, $L$ won't be gauge invariant but its holonomy $S$ will be, and so one can still formulate Euler-Lagrange equations, etc.

By the way, in quantum mechanics, $\theta$ is required to be a connection on a $U(1)$ bundle, which leads to quantization of the integrals of $\omega$ over closed surfaces.

You should check out this book by V. Arnold called Mathematical Methods of Classical Mechanics (and indeed all the rest of his books too :).