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 $$S_U[\gamma]~:=~\int_{\gamma} \left( \vartheta - H ~\mathrm{d}t\right) ~=~\int_{t_i}^{t_f}\! \mathrm{d}t\left( \sum_{I=1}^{2n}\vartheta_I~\dot{z}^I -H\right). \tag{4}$$ 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.


  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 :).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.