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:

    “One should never direct people towards happiness, because happiness too is an idol of the market-place. One should direct them towards mutual affection. A beast gnawing at its prey can be happy too, but only human beings can feel affection for each other, and this is the highest achievement they can aspire to.”
    —Alexander Solzhenitsyn (b. 1918)

    “... the ordinary is simply the universal observed from the surface, that the direct approach to reality is not without, but within. Touch life anywhere ... and you will touch universality wherever you touch the earth.”
    —Ellen Glasgow (1873–1945)