Canonical Structures

The Euler-Lagrange equations we derived earlier are equivalent to to a system of first-order equations known as Hamilton’s equations, which we will now derive. Using the definition of the canonical momentum, one begins by defining the Hamiltonian function \begin{equation} \label{eq:hamiltonian} H = p ^{a} \dot{q} _ {a} - L(q _ {a}, \dot{q} _ {a}) \ . \end{equation} We have in the above expression a term with repeated indices; such terms are to be understood as follows: \begin{equation} p ^{a} \dot{q} _ {a} \equiv \sum_ {a=1}^{N} p ^{a} \dot{q} _ {a} \ . \end{equation} In general, unless explicitly mentioned otherwise, all repeated indices — one raised and one lowered — in an expression imply a sum over the range of values the index may take. This is called the Einstein summation convention and will be employed throughout these lectures.

It is instructive to consider the total derivative of the Hamiltonian:

\[\begin{align} \mathrm{d} H &= \dot{q} _ {a} \mathrm{d} p ^{a} + p ^{a} \mathrm{d} \dot{q} _ {a} - \mathrm{d} L(q _ {a},\dot{q} _ {a}) \ , \\ \label{eq:d-hamiltonian} &= - \frac{\partial L}{\partial q _ {a}} \mathrm{d} q _ {a} + \dot{q} _ {a} \mathrm{d} p ^{a} \ . \end{align}\]

In going from the first line to the second, we have used the definition of the canonical momentum. We can conclude, then, that the Hamiltonian is only a function of generalised coordinates and momenta, i.e. it is a function on phase space. We could, then, just as well write \begin{equation} \label{eq:d-hamiltonian-2} \mathrm{d} H (q _ {a}, p ^{a}) = \frac{\partial H}{\partial q _ {a}} \mathrm{d} q _ {a} + \frac{\partial H}{\partial p ^{a}} \mathrm{d} p ^{a} \ , \end{equation} and on comparing the coefficients of $ \mathrm{d} p ^{a} $ and $ \mathrm{d} q _ {a} $ in \eqref{eq:d-hamiltonian} and \eqref{eq:d-hamiltonian-2} and using the Euler-Lagrange equations, we arrive at: \begin{equation} \label{eq:hamilton} \dot{q} _ {a} = \frac{\partial H}{\partial p ^{a}} \quad \mathrm{and} \quad \dot{p} ^{a} = - \frac{\partial H}{\partial q _ {a}} \ . \end{equation} Together, \eqref{eq:hamilton} are called Hamilton’s equations, a sent of $ 2N $ first-order differential equations that are completely equivalent to the Euler-Lagrange equations, and therefore Newton’s laws of motion.

They may be derived using a phase-space version of the principle of least action; defining the action associated to a trajectory $ (q _ {a}(t), p ^{a}(t)) $ in phase space as \begin{equation} \label{eq:phase-space-action} S[q _ {a}(t), p ^{a} (t)] = \int _ {t _ {i}} ^{t _ {f}} \mathrm{d}^{}t \, \left[ p ^{a} \dot{q} _ {a} - H(q _ {a}, p ^{a}) \right] \ , \end{equation} as is easily verified.

Poisson Brackets

There exists a particularly elegant formulation of Hamilton’s equations due to Poisson. For any two functions $ A $ and $ B $ on phase space, their Poisson bracket is defined as \begin{equation} \label{eq:poisson-bracket-def} \left\lbrace A,B \right\rbrace = \frac{\partial A}{\partial q _ {a}} \frac{\partial B}{\partial p ^{a}} - \frac{\partial A}{\partial p ^{a}} \frac{\partial B}{\partial q _ {a}} \ . \end{equation} In terms of this bracket, Hamilton’s equations can be written as \begin{equation} \dot{q} _ {a} = \left\lbrace q _ {a}, H \right\rbrace \quad \mathrm{and} \quad \dot{p} ^{a} = \left\lbrace p ^{a}, H \right\rbrace \ . \end{equation}

Indeed, for any function $ f\left[q _ {a}(t), p ^{a}(t), t\right]$ on phase space, we can write its time derivative in terms of the Poisson bracket as

\[\begin{align} \frac{\mathrm{d} f}{\mathrm{d} t} &= \frac{\partial f}{\partial t} + \frac{\partial f}{\partial q _ {a}} \dot{q} _ {a} + \frac{\partial f}{\partial p ^{a}} \dot{p} ^{a} \ , \\ &= \frac{\partial f}{\partial t} + \frac{\partial f}{\partial q _ {a}} \frac{\partial H}{\partial p ^{a}} - \frac{\partial f}{\partial p ^{a}} \frac{\partial H}{\partial q _ {a}} \ , \\ &= \frac{\partial f}{\partial t} + \left\lbrace f,H \right\rbrace \ . \end{align}\]

It follows, then, that any function that depends on time only implicitly through the phase space coordinates and that “Poisson commutes”” with the Hamiltonian — i.e. satisfies $ \left\lbrace f, H \right\rbrace = 0 $ — is a constant of motion.

The Poisson brackets have two properties that follow from \eqref{eq:poisson-bracket-def} and are of particular note. The first is antisymmetry: \begin{equation} \label{eq:pb-antisymmetry} \left\lbrace A, B \right\rbrace = - \left\lbrace B, A \right\rbrace \ . \end{equation} The second is the Jacobi identity: \begin{equation} \label{eq:pb-jacobi} \left\lbrace A, \left\lbrace B, C \right\rbrace \right\rbrace + \left\lbrace B, \left\lbrace C, A \right\rbrace \right\rbrace + \left\lbrace C, \left\lbrace A, B \right\rbrace \right\rbrace = 0 \ . \end{equation}

It is useful to note that the only non-trivial Poisson brackets of the canonical (or phase space) variables is \begin{equation} \label{eq:fundamental-Poisson-brackets} \left\lbrace q _ {a}, p ^{b} \right\rbrace = \delta _ {a} {}^{b} \ . \end{equation}

Canonical Transformations

A transformation $ \left( q _ {a}, p ^{a} \right) \rightarrow \left( Q _ {a}, P ^{a} \right) $ is said to be canonical if there exists a new Hamiltonian $ H’ $ such that the structure of Hamilton’s equations is unchanged, i.e. that \begin{equation} \dot{Q} _ {a} = \frac{\partial H’}{\partial P ^{a}} \quad \mathrm{and} \quad \dot{P} ^{a} = - \frac{\partial H’}{\partial Q _ {a}} \ . \end{equation} Since we have already established the action principle from which Hamilton’s equations arise, it follows that for a transformation to be canonical, the old and new actions in \eqref{eq:phase-space-action} differ by a total derivative, or \begin{equation} \label{eq:canonical-transformation-def} p ^{a} \dot{q} _ {a} - H = P ^{a} \dot{Q} _ {a} - H’ + \frac{\mathrm{d} G}{\mathrm{d} t} \ . \end{equation} The function $ G $ is called the generating function of the canonical transformation, and may be chosen to be a function of both the old and new phase space variables. For example, the choice $ G = P ^{a} \left( q _ {a} -Q _ {a} \right) $ is easily seen to be the identity transformation.

Consider now canonical transformations that differ infinitesimally from the identity transformation, i.e. \begin{equation} G = P ^{a} \left( q _ {a} - Q _ {a} \right) + \epsilon F \left( q _ {a}, p ^{a} \right) + \mathcal{O}\left( \epsilon ^{2} \right) \ . \end{equation} The function $ F $ is called a generator of a canonical transformation. Using \eqref{eq:canonical-transformation-def}, we find \begin{equation} \delta q _ {a} = \epsilon \left\lbrace q _ {a}, F \right\rbrace \quad \mathrm{and} \quad \delta p ^{a} = \epsilon \left\lbrace p ^{a}, F \right\rbrace \ . \end{equation} Time evolution, generated by the Hamiltonian thusly, is a canonical transformation. More generally, any constant of motion is a generator of a symmetry of the Hamiltonian.