Canonical Transformation and Symplectic Conditions

Given a system with $n$ generalized coordinates $q$, $n$ conjugated momenta $p$ and Hamiltonian $H$, the corresponding Hamilton's equations of motion are

$$\dot q_{i} = {\partial H \over \partial p_{i}}$$

$$\dot p_{i} = - {\partial H \over \partial q_{i}} ~~~~~ i = 1,2,.... n$$ (2.1)

These equations can be written in a more compact form by defining a column matrix with $2n$ elements such that

$${\bf x} = \left ( \begin{array}{r} {\bf q } \\ {\bf p } \end{array}\right ).$$ (2.2)

In this notation the Hamilton's equations (2.1) can be compactly written as

$$\dot {\bf x} = {\bf J} {\partial H \over \partial \bf x} ~~~~~ {\bf J} = \le... ...{rr} {\bf0} & {\bf 1} \\ {\bf -1} & {\bf0} \end{array}\right )$$ (2.3)

where ${\bf J}$ is a $2n\times 2n$ matrix, ${\bf 1}$ is an $n\times n$ identity matrix and ${\bf0}$ is a $n\times n$ matrix of zeroes. Eq. (2.3) is the so-called symplectic notation for the Hamilton's equations.^2.1

Using the same notation we now may define a transformation of variables from ${\bf x} \equiv \{ q,p \}$ to ${\bf y} \equiv \{ Q,P \}$ as

$${\bf y} = {\bf y}({\bf x})$$ (2.4)

For a restricted canonical transformation [72,73] we know that the function $H({\bf x})$ expressed in the new coordinates ${\bf y}$ serves as the Hamiltonian function for the new coordinates ${\bf y}$, that is the Hamilton's equations of motion in the ${\bf y}$ basis have exactly the same form as in Eq. (2.3):

$$\dot {\bf y} = {\bf J} {\partial H \over \partial \bf y}$$ (2.5)

If we now take the time derivative of Eq. (2.4), use the chain rule relating ${\bf x}$ and ${\bf y}$ derivatives and use Eq. (2.5), we arrive at

$$\dot {\bf y} = {\bf M} {\bf J} {\bf M}^t {\partial H \over \partial \bf y}.$$ (2.6)

Here ${\bf M}$ is the Jacobian matrix with elements

$$M_{ij} = \partial y_{i} /\partial x_{i},$$ (2.7)

and ${\bf M}^{t}$ is its transpose. By comparing Eqs. (2.5) and (2.6), we arrive at the conclusion that a transformation is canonical if, and only if, the Jacobian matrix ${\bf M}$ of the transformation Eq. 2.4 satisfies the condition

$${\bf M}{\bf J}{\bf M}^{t} = {\bf J}.$$ (2.8)

Eq. (2.8) is known as the symplectic condition for canonical transformations and represents an effective tool to test whether a generic transformation is canonical. Canonical transformations play a key role in Hamiltonian dynamics. For example, consider transformation $\phi$

$${\bf z} (t) = \phi (t,{\bf z}(0))$$ (2.9)

where $\{p_{0} q_{0} \}\equiv {\bf z}(0)$ and $\{P,Q \}\equiv {\bf z}(t)$, i.e. one writes the coordinates and momenta at time $t$, obtained from the solution of the Hamiltonian equation of motion, as a function of the coordinates and momenta at the initial time zero. This transformation, which depends on the scalar parameter $t$, is trivially canonical since both $\{p_{0} q_{0}\}$ and $\{P,Q \}$ satisfies the Hamilton equations of motion. Hence the above transformation defines the $t$-flow mapping of the systems and, being canonical, its Jacobian matrix obeys the symplectic condition (2.8). An important consequence of the symplectic condition, is the invariance under canonical (or symplectic) transformations of many properties of the phase space. These invariant properties are known as ``Poincare invariants'' or canonical invariants. For example transformations or $t$-flow's mapping obeying Eq. (2.8) preserve the phase space volume. This is easy to see, since the infinitesimal volume elements in the ${\bf y}$ and ${\bf x}$ bases are related by

$$d {\bf y} = \vert\det {\bf M}\vert d{\bf x}$$ (2.10)

where $\vert\det {\bf M}\vert$ is the Jacobian of the transformation. Taking the determinant of the symplectic condition Eq. (2.8) we see that $\vert\det {\bf M}\vert = 1$ and therefore

$$d {\bf y} = d {\bf x}.$$ (2.11)

For a canonical or symplectic $t$-flow mapping this means that the phase total space volume is invariant and therefore Liouville theorem is automatically satisfied.

A step-wise numerical integration scheme defines a $\Delta t$-flow mapping or equivalently a coordinates transformation, that is

$$\begin{array}{r} Q(\Delta t) = Q(q(0),p(0),\Delta t) \\ P(\Delta t) = P(q(0),p(0),\Delta t) \end{array} ~~~ {\bf y}(\Delta t) = {\bf y}( {\bf x}(0)).$$ (2.12)

We have seen that exact solution of the Hamilton equations has $t$-flow mapping satisfying the symplectic conditions (2.8). If the Jacobian matrix of the transformation (2.12) satisfies the symplectic condition then the integrator is termed to be symplectic. The resulting integrator, therefore, exhibits properties identical to those of the exact solution, in particular it satisfies Eq. (2.11). Symplectic algorithms have also been proved to be robust, i.e resistant to time step increase, and generate stable long time trajectory, i.e. they do not show drifts of the total energy. Popular MD algorithms like Verlet, leap frog and velocity Verlet are all symplectic and their robustness is now understood to be due in part to this property. [70,20,23,71]