07 May 2026
Canonical Transformations and Generating Functions
Canonical transformations, generating functions, standard forms, and their link with Hamilton-Jacobi theory.
In Hamiltonian mechanics, the variables $(q,p)$ describe a point in phase space. One may change to new variables $(Q,P)$, but not every change is acceptable. A canonical transformation is a change of phase-space variables that preserves Hamilton’s form of equations.
The aim is not merely to rename variables. A good canonical transformation can reveal conserved quantities, simplify the Hamiltonian, or even reduce the problem to constant new momenta.
Canonical condition
Let
\[(q_i,p_i)\to(Q_i,P_i).\]The transformation is canonical if the old and new phase-space one-forms differ by an exact differential:
\[\boxed{ \sum_i p_i\,dq_i-H\,dt= \sum_i P_i\,dQ_i-K\,dt+dF. }\]Here $F$ is a generating function and $K$ is the new Hamiltonian.
This condition says that the variational structure of Hamiltonian mechanics is unchanged, apart from an exact differential. Since an exact differential changes only endpoint terms in the action, the equations of motion keep their form.
Type-1 generating function
If
\[F=F_1(q,Q,t),\]then comparing coefficients gives
\[\boxed{ p_i=\frac{\partial F_1}{\partial q_i}, \qquad P_i=-\frac{\partial F_1}{\partial Q_i}, \qquad K=H+\frac{\partial F_1}{\partial t}. }\]Four standard generating functions
The common choices are:
| Type | Function | Relations |
|---|---|---|
| $F_1$ | $F_1(q,Q,t)$ | $p_i=\partial F_1/\partial q_i$, $P_i=-\partial F_1/\partial Q_i$ |
| $F_2$ | $F_2(q,P,t)$ | $p_i=\partial F_2/\partial q_i$, $Q_i=\partial F_2/\partial P_i$ |
| $F_3$ | $F_3(p,Q,t)$ | $q_i=-\partial F_3/\partial p_i$, $P_i=-\partial F_3/\partial Q_i$ |
| $F_4$ | $F_4(p,P,t)$ | $q_i=-\partial F_4/\partial p_i$, $Q_i=\partial F_4/\partial P_i$ |
For all four,
\[K=H+\frac{\partial F}{\partial t}\]with the appropriate generating function.
Identity transformation
For type $F_2$,
\[F_2(q,P,t)=\sum_i q_iP_i\]gives
\[p_i=P_i, \qquad Q_i=q_i.\]Thus it generates the identity transformation.
Scale transformation
Consider the type-$2$ generating function
\[F_2(q,P)=aqP,\]where $a$ is a non-zero constant. The transformation equations are
\[p=\frac{\partial F_2}{\partial q}=aP,\]and
\[Q=\frac{\partial F_2}{\partial P}=aq.\]Therefore
\[Q=aq,\qquad P=\frac{p}{a}.\]This transformation is canonical because
\[\{Q,P\}=\left\{aq,\frac{p}{a}\right\}=\{q,p\}=1.\]It rescales coordinate and momentum in inverse ways, preserving phase-space area.
Hamilton-Jacobi connection
If a type-$2$ generating function is chosen as
\[F_2(q,P,t)=S(q,P,t),\]and the new Hamiltonian is set to zero,
\[K=0,\]then
\[H+\frac{\partial S}{\partial t}=0.\]Since
\[p_i=\frac{\partial S}{\partial q_i},\]this becomes the Hamilton-Jacobi equation:
\[\boxed{ H\left(q_i,\frac{\partial S}{\partial q_i},t\right) +\frac{\partial S}{\partial t}=0. }\]Main points
- Canonical transformations preserve Hamilton’s equations.
- Generating functions encode canonical transformations.
- Four standard forms are used depending on the chosen independent variables.
- $F_2=\sum_iq_iP_i$ gives the identity transformation.
- Hamilton-Jacobi theory is a special canonical transformation generated by $S$.
Practice questions
- State the canonical one-form condition.
- Derive the transformation equations for $F_1(q,Q,t)$.
- Write the four standard generating functions.
- Show that $F_2=\sum_iq_iP_i$ gives the identity transformation.
- Explain how the Hamilton-Jacobi equation follows from $F_2=S$ and $K=0$.
Discussion