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

Famous quotes containing the words direct and/or approach:

    “Pleasure is the rock which most young people split upon; they launch out with crowded sails in quest of it, but without a compass to direct their course, or reason sufficient to steer the vessel; for want of which, pain and shame, instead of pleasure, are the returns of their voyage.”
    —Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)

    “I have watched ... many literary fashions shoot up and blossom, and then fade and drop.... Yet with the many that I have seen come and go, I have never yet encountered a mode of thinking that regarded itself as simply a changing fashion, and not as an infallible approach to the right culture.”
    —Ellen Glasgow (1873–1945)