Canonical Transformation - Direct Approach

Direct Approach

The functional form of Hamilton's equations is

\dot{\mathbf{p}} = -\frac{\partial H}{\partial \mathbf{q}}
\dot{\mathbf{q}} =~~\frac{\partial H}{\partial \mathbf{p}}

By definition, the transformed coordinates have analogous dynamics

\dot{\mathbf{P}} = -\frac{\partial K}{\partial \mathbf{Q}}
\dot{\mathbf{Q}} =~~\frac{\partial K}{\partial \mathbf{P}}

where K(Q,P) is a new Hamiltonian that must be determined.

In general, a transformation (q,p,t) → (Q,P,t) does not preserve the form of Hamilton's equations. For time independent transformations between (q,p) and (Q,P) we may check if the transformation is restricted canonical, as follows. Since restricted transformations have no explicit time dependence (by construction), the time derivative of a new generalized coordinate Qm is

\dot{Q}_{m} = \frac{\partial Q_{m}}{\partial \mathbf{q}} \cdot \dot{\mathbf{q}} + \frac{\partial Q_{m}}{\partial \mathbf{p}} \cdot \dot{\mathbf{p}} = \frac{\partial Q_{m}}{\partial \mathbf{q}} \cdot \frac{\partial H}{\partial \mathbf{p}} - \frac{\partial Q_{m}}{\partial \mathbf{p}} \cdot \frac{\partial H}{\partial \mathbf{q}} = \lbrace Q_m, H \rbrace

where is the Poisson bracket.

We also have the identity for the conjugate momentum Pm

\frac{\partial H}{\partial P_{m}} = \frac{\partial H}{\partial \mathbf{q}} \cdot \frac{\partial \mathbf{q}}{\partial P_{m}} + \frac{\partial H}{\partial \mathbf{p}} \cdot \frac{\partial \mathbf{p}}{\partial P_{m}}

If the transformation is canonical, these two must be equal, resulting in the equations

\left( \frac{\partial Q_{m}}{\partial p_{n}}\right)_{\mathbf{q}, \mathbf{p}} = -\left( \frac{\partial q_{n}}{\partial P_{m}}\right)_{\mathbf{Q}, \mathbf{P}}
\left( \frac{\partial Q_{m}}{\partial q_{n}}\right)_{\mathbf{q}, \mathbf{p}} = \left( \frac{\partial p_{n}}{\partial P_{m}}\right)_{\mathbf{Q}, \mathbf{P}}

The analogous argument for the generalized momenta Pm leads to two other sets of equations

\left( \frac{\partial P_{m}}{\partial p_{n}}\right)_{\mathbf{q}, \mathbf{p}} = \left( \frac{\partial q_{n}}{\partial Q_{m}}\right)_{\mathbf{Q}, \mathbf{P}}
\left( \frac{\partial P_{m}}{\partial q_{n}}\right)_{\mathbf{q}, \mathbf{p}} = -\left( \frac{\partial p_{n}}{\partial Q_{m}}\right)_{\mathbf{Q}, \mathbf{P}}

These are the direct conditions to check whether a given transformation is canonical.

Read more about this topic:  Canonical Transformation

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