Canonical Maps
Hamilton–Jacobi theory naturally extends to canonical transformations because the action function $S$ itself behaves like a generating function. This extension clarifies why Hamiltonian mechanics can be simplified by changing phase-space variables and how continuous symmetry transformations arise from infinitesimal generators.
Canonical transformations
In Hamiltonian mechanics, the state of a system is described by canonical coordinates and momenta $(q_i,p_i)$. A transformation to new variables $(Q_i,P_i)$ is called a canonical transformation if the new variables also satisfy Hamilton’s equations in canonical form. The importance of such transformations is that they preserve the structure of mechanics while possibly making the problem much simpler.
The differential form of Hamilton’s principal function is
\[dS=\sum_i p_i\,dq_i-H\,dt\]Suppose we transform to new variables $(Q_i,P_i)$ with a new Hamiltonian $K$. Then the corresponding canonical differential form is
\[dS'=\sum_i P_i\,dQ_i-K\,dt\]A transformation is canonical if the difference between the old and new one-forms is an exact differential. Thus one requires
\[\sum_i p_i\,dq_i-H\,dt-\left(\sum_i P_i\,dQ_i-K\,dt\right)=dF\]where $F$ is some function. This is the basic condition for a canonical transformation. It means that the equations of motion preserve their Hamiltonian form under the transformation.
Thus canonical transformations preserve the symplectic structure of phase space. They do not merely change coordinates; they preserve the dynamical framework itself.
- coefficient of $dq_i$:
- coefficient of $dQ_i$:
- coefficient of $dt$:
Now suppose we do not want the generating function in terms of $Q_i$. We want a new generating function in terms of $P_i$ instead. Since $P_i$ is related to $Q_i$ by $$ P_i=-\frac{\partial F_1}{\partial Q_i} $$ the natural way to replace the variable $Q_i$ by its conjugate quantity $P_i$ is by a Legendre transform. So define $$ F_2(q_i,P_i,t)=F_1(q_i,Q_i,t)+\sum_i Q_iP_i $$ This is exactly the Legendre transformation of $F_1$ with respect to $Q_i$ because:
- old independent variable: $Q_i$
- conjugate derivative variable: $-P_i=\dfrac{\partial F_1}{\partial Q_i}$
Generating functions
The function $F$ appearing above is called a generating function. Depending on which old and new variables are chosen as independent arguments, there are four standard types of generating functions.
If $F_1 = F_1(q,Q,t)$, then
If $F_2 = F_2(q,P,t)$, then
If $F_3 = F_3(p,Q,t)$, then
If $F_4 = F_4(p,P,t)$, then
Among these, the type-$2$ generating function is especially important in Hamilton–Jacobi theory. If one writes
\[F_2(q,P,t)=S(q,P,t)\]then
\[p_i=\frac{\partial S}{\partial q_i}, \qquad Q_i=\frac{\partial S}{\partial P_i}\]If the new Hamiltonian is chosen to vanish,
\[K=K(Q,P)=0\]then the condition
\[K=H+\frac{\partial S}{\partial t}=0\]becomes
\[\frac{\partial S}{\partial t}+H\left(q_i,\frac{\partial S}{\partial q_i},t\right)=0\]which is precisely the Hamilton–Jacobi equation. Thus Hamilton–Jacobi theory is the special case of a canonical transformation generated by $S$ that transforms the dynamics into a trivial one with constant new variables.
This also explains the origin of the constants in Hamilton–Jacobi theory. If $P_i=\alpha_i$, then
\[Q_i=\frac{\partial S}{\partial \alpha_i}=\beta_i\]and because $K=0$, Hamilton’s equations in the new variables give
\[\dot p_i=0, \qquad \dot Q_i=0\]Hence both $\alpha_i$ and $\beta_i$ are constants.
Infinitesimal Canonical Transformations
Take an infinitesimal canonical transformation as one which is very close to the identity transformation:
\[Q_i=q_i+\delta q_i,\qquad P_i=p_i+\delta p_i\]where $\delta q_i$ and $\delta p_i$ are very small quantities of first order in a small parameter $\varepsilon$.
Now begin with a type-2 generating function and choose it in the form
\[F_2(q,P,t)=\sum_i q_iP_i+\varepsilon G(q,P,t)\]Here:
- $\sum_i q_iP_i$ gives the identity transformation
- $\varepsilon G$ gives a small deviation from identity
- $G$ is called the infinitesimal generator
Why $\sum_i q_iP_i$ gives identity
For a type-2 generating function, the transformation equations are
\[p_i=\frac{\partial F_2}{\partial q_i},\qquad Q_i=\frac{\partial F_2}{\partial P_i}\]If
\[F_2=\sum_i q_iP_i\]then
\[p_i=P_i,\qquad Q_i=q_i\]So the old and new variables are the same. Hence this is the identity transformation.
Add a small correction
Now take
\[F_2(q,P,t)=\sum_i q_iP_i+\varepsilon G(q,P,t)\]Then
\[p_i=\frac{\partial F_2}{\partial q_i} = P_i+\varepsilon \frac{\partial G}{\partial q_i}\]and
\[Q_i=\frac{\partial F_2}{\partial P_i} = q_i+\varepsilon \frac{\partial G}{\partial P_i}\]Since the transformation is infinitesimal, the difference between $P_i$ and $p_i$ is already of order $\varepsilon$. Therefore, inside first-order terms, we may replace $P_i$ by $p_i$. Thus
\[Q_i=q_i+\varepsilon \frac{\partial G}{\partial p_i}\]Hence
\[\delta q_i=Q_i-q_i=\varepsilon \frac{\partial G}{\partial p_i}\]Now from
\[p_i=P_i+\varepsilon \frac{\partial G}{\partial q_i}\]we get
\[P_i=p_i-\varepsilon \frac{\partial G}{\partial q_i}\]Therefore
\[\delta p_i=P_i-p_i=-\varepsilon \frac{\partial G}{\partial q_i}\]So the infinitesimal canonical transformation is
\[\boxed{\delta q_i=\varepsilon \frac{\partial G}{\partial p_i},\qquad \delta p_i=-\varepsilon \frac{\partial G}{\partial q_i}}\]Why this looks like Hamilton’s equations
Hamilton’s equations are
\[\dot q_i=\frac{\partial H}{\partial p_i},\qquad \dot p_i=-\frac{\partial H}{\partial q_i}\]Now compare with
\[\delta q_i=\varepsilon \frac{\partial G}{\partial p_i},\qquad \delta p_i=-\varepsilon \frac{\partial G}{\partial q_i}\]Dividing by $\varepsilon$,
\[\frac{\delta q_i}{\varepsilon}=\frac{\partial G}{\partial p_i},\qquad \frac{\delta p_i}{\varepsilon}=-\frac{\partial G}{\partial q_i}\]This has exactly the same structure as Hamilton’s equations, except that:
- $H$ is replaced by $G$
- time $t$ is replaced by the transformation parameter $\varepsilon$
So $G$ generates motion in phase space with respect to $\varepsilon$, just as $H$ generates motion with respect to time.
Main idea
- $H$ generates time evolution
- $G$ generates a canonical transformation
- infinitesimal means the transformation is only a very small step away from identity
- a finite canonical transformation can be built by repeating many such infinitesimal steps
These relations have exactly the same form as Hamilton’s equations, except that the evolution parameter is now $\varepsilon$ instead of time.
Using Poisson brackets, these can be written compactly as
\[\delta q_i=\varepsilon \{q_i,G\}, \qquad \delta p_i=\varepsilon \{p_i,G\}\]and for any dynamical quantity $f(q,p,t)$,
\[\delta f=\varepsilon \{f,G\}\]Thus the infinitesimal generator produces canonical transformations through Poisson brackets. This is the phase-space analogue of the way a generator produces a continuous symmetry in other branches of physics.
Meaning of the generator
The generator $G$ determines the direction in which the phase-space point moves under the infinitesimal transformation. Different choices of $G$ produce different transformations.
If
\[G=p\]for one degree of freedom, then
\[\delta q=\varepsilon, \qquad \delta p=0\]which is a translation in coordinate.
If
\[G=-q\]then
\[\delta q=0, \qquad \delta p=\varepsilon\]which is a translation in momentum.
If the Hamiltonian itself acts as the generator, then
\[\delta q_i=\varepsilon \frac{\partial H}{\partial p_i}, \qquad \delta p_i=-\varepsilon \frac{\partial H}{\partial q_i}\]which is exactly the time development of the system for an interval $\varepsilon$. Therefore the Hamiltonian is the generator of time evolution.
Relation with Poisson brackets and conservation laws
For any quantity $f(q,p,t)$, the total time evolution is
\[\frac{df}{dt}=\{f,H\}+\frac{\partial f}{\partial t}\]If a quantity $G$ has no explicit time dependence and satisfies
\[\{G,H\}=0\]then
\[\frac{dG}{dt}=0\]so $G$ is conserved. Hence conserved quantities are deeply connected with generators. A symmetry generator that leaves the Hamiltonian unchanged gives a constant of motion.
This idea links canonical transformations with conservation laws and prepares the way for more advanced ideas such as action-angle variables, Lie transformations, and Noether-type symmetry analysis in Hamiltonian form.
Main points
- A canonical transformation preserves Hamilton’s equations in canonical form.
- The condition for canonicity is that the difference of the old and new phase-space one-forms is an exact differential.
- That exact differential defines a generating function.
- There are four standard types of generating functions.
- Hamilton–Jacobi theory is a special canonical transformation generated by $S$ with new Hamiltonian $K=0$.
- Infinitesimal canonical transformations are generated by a function $G$.
- The change of any quantity under an infinitesimal canonical transformation is given by its Poisson bracket with $G$.
- The Hamiltonian itself is the generator of time evolution.
Practice questions
- Show that a transformation is canonical if the difference between the old and new phase-space one-forms is an exact differential.
- Derive the relations for the type-$2$ generating function $F_2(q,P,t)$.
- Explain how Hamilton’s principal function $S$ acts as a generating function in Hamilton–Jacobi theory.
- Prove that an infinitesimal canonical transformation generated by $G$ preserves Poisson bracket structure.
- Show that the Hamiltonian generates time translations in phase space.