Symplectic Geometry

We have already seen that canonical transformations can rotate positions and momenta into each other. These labels, therefore, have no invariant meaning, and so it is helpful to adopt a more covariant notation: let $ \mathsf{a} $ take values in $ \left\lbrace 1, \cdots , 2N \right\rbrace $, such that

\[y ^{\mathsf{a} } = \begin{cases} q _ {a} \ , & \text{if} \quad 1 \leq \mathsf{a} \leq N \\ p ^{a} \ , & \text{if} \quad N+1 \leq \mathsf{a} \leq 2N \ . \end{cases}\]

Similarly, let $ \partial _ {\mathsf{a} } = \partial /\partial y ^{\mathsf{a} } $. In terms of these phase space coordinates $ y ^{\mathsf{a} } $, the fundamental Poisson brackets can now be written as \begin{equation} \label{eq:fund-pb} \left\lbrace y ^{\mathsf{a} }, y ^{\mathsf{b} } \right\rbrace = \epsilon ^{\mathsf{ab} } \ , \end{equation} where $ \epsilon $ is the constant antisymmetric $ 2N \times 2N $ matrix, written in terms of $ N \times N $ blocks as

\[\epsilon ^{\textsf{ab} } = \begin{pmatrix} 0 & \mathbf{1} \\ -\mathbf{1} & 0 \end{pmatrix} \ .\]

This choice of coordinates is sometimes referred to as the Darboux coordinates. Hamilton’s equations can be rewritten as \begin{equation} \dot{y} ^{\mathsf{a} } = \left\lbrace y ^{\mathsf{a} }, H \right\rbrace = \epsilon ^{\mathsf{ab} } \, \partial _ {\mathsf{b} }H \ , \end{equation} and the Poisson brackets of any two functions can be written as \begin{equation} \left\lbrace A,B \right\rbrace = \epsilon ^{\mathsf{ab} }\,\partial _ {\mathsf{a} }A \,\partial _ {\mathsf{b} }B \ . \end{equation}

More generally, we will find it convenient to depart from strictly canonical coordinates on phase space that obey \eqref{eq:fund-pb} and instead work with coordinates that satisfy \begin{equation} \left\lbrace y ^{\mathsf{a} }, y ^{\mathsf{b} } \right\rbrace = \omega ^{\mathsf{ab} }(y) \ . \end{equation} The matrix $ \omega ^{\mathsf{ab} } $ is now not necessarily a constant, and can depend on the point in phase space where this Poisson bracket is being computed. From the antisymmetry of the Poisson bracket, we have that \begin{equation} \omega ^{\mathsf{ab} } = - \omega ^{\mathsf{ba} } \ . \end{equation} Further, we will insist on $ \omega ^{\mathsf{ab} } $ being invertible, and denote its inverse with lowered indices such that \begin{equation} \omega ^{\mathsf{ab} } \omega _ {\mathsf{bc} } = \delta ^{\mathsf{a} }_ {\mathsf{c} } \ . \end{equation} Finally, from the Jacobi identity we derive the following condition:

\[\begin{align} \left\lbrace y ^{a}, \left\lbrace y ^{b}, y ^{c} \right\rbrace \right\rbrace + \left\lbrace y ^{c}, \left\lbrace y ^{a}, y ^{b} \right\rbrace \right\rbrace + \left\lbrace y ^{b}, \left\lbrace y ^{c}, y ^{a} \right\rbrace \right\rbrace &= 0 \\ \Rightarrow \partial _ {\mathsf{a} } \omega _ {\mathsf{bc} } + \partial _ {\mathsf{c} } \omega _ {\mathsf{ab} } + \partial _ {\mathsf{b} } \omega _ {\mathsf{ca} } &= 0 \ . \end{align}\]

Taken together, we conclude that phase space comes equipped with a second-rank antisymmetric tensor $ \omega _ {\mathsf{ab} } $ that is everywhere invertible and satisfies Bianchi’s identity. This is called a symplectic manifold. The tensor $ \omega _ {\mathsf{ab} } $ may be thought of as analogous to the metric tensor on a Riemannian manifold, just as canonical transformations are analogous to coordinate transformations.