Canonical transformation

In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates $(\mathbf {q} ,\mathbf {p} )\rightarrow (\mathbf {Q} ,\mathbf {P} )$ that preserves the form of Hamilton's equations, although it might not preserve the Hamiltonian itself.

For clarity, we restrict the presentation here to calculus and classical mechanics. Readers familiar with more advanced mathematics such as cotangent bundles, exterior derivatives and symplectic manifolds should read the related symplectomorphism article. (Canonical transformations are a special case of a symplectomorphism.)

Generating function approach

The functional form of Hamilton's equations is

${\frac {d}{dt}}\mathbf {p} \equiv {\dot {\mathbf {p} }}=-{\frac {\partial H}{\partial \mathbf {q} }}$

${\frac {d}{dt}}\mathbf {q} \equiv {\dot {\mathbf {q} }}=~~{\frac {\partial H}{\partial \mathbf {p} }}$

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 $K(\mathbf {Q} ,\mathbf {P} )$ is a new Hamiltonian that must be determined.

To derive the relation between $(\mathbf {q} ,\mathbf {p} ,H)$ and $(\mathbf {Q} ,\mathbf {P} ,K)$, we use an indirect generating function approach. Both sets of variables must obey Hamilton's principle

$\delta \int _{t_{1}}^{t_{2}}\left[\mathbf {p} \cdot {\dot {\mathbf {q} }}-H(\mathbf {q} ,\mathbf {p} ,t)\right]dt=0$

$\delta \int _{t_{1}}^{t_{2}}\left[\mathbf {P} \cdot {\dot {\mathbf {Q} }}-K(\mathbf {Q} ,\mathbf {P} ,t)\right]dt=0$

To satisfy both variational integrals, we must have

$\lambda \left[\mathbf {p} \cdot {\dot {\mathbf {q} }}-H(\mathbf {q} ,\mathbf {p} ,t)\right]=\mathbf {P} \cdot {\dot {\mathbf {Q} }}-K(\mathbf {Q} ,\mathbf {P} ,t)+{\frac {dG}{dt}}$

In general, $\lambda$ is set equal to one; such canonical transformations are called restricted canonical transformations.

Here $G$ is a generating function of one old canonical variable ($\mathbf {q}$ or $\mathbf {p}$), one new canonical coordinate ($\mathbf {Q}$ or $\mathbf {P}$) and (possibly) the time $t$. Thus, there are four general types of generating functions: $G_{1}(\mathbf {q} ,\mathbf {Q} ,t)$, $G_{2}(\mathbf {q} ,\mathbf {P} ,t)$, $G_{3}(\mathbf {p} ,\mathbf {Q} ,t)$ and $G_{4}(\mathbf {p} ,\mathbf {P} ,t)$.