Learning Objectives:
- Understand the concept and purpose of canonical transformations in Hamiltonian mechanics.
- Learn how generating functions facilitate canonical transformations.
- Explore the role of infinitesimal generators in describing symmetries and conserved quantities.
- Establish a foundation for understanding Hamilton-Jacobi theory.
Key Concepts / Definitions
- Canonical Transformation: A transformation from old variables $(q, p)$ to new variables $(Q, P)$ that preserves the form of Hamilton’s equations.
- Generating Function: A function that defines a canonical transformation by connecting old and new variables.
- Infinitesimal Generator: A function that produces infinitesimal canonical transformations; often linked to symmetries and conserved quantities.
Canonical Transformations
Canonical transformations simplify problems in Hamiltonian mechanics by transforming to new variables $(Q, P)$ that preserve the structure of Hamilton’s equations:
\[\dot{q} = \frac{\partial H}{\partial p}, \quad \dot{p} = -\frac{\partial H}{\partial q} \quad \Rightarrow \quad \dot{Q} = \frac{\partial K}{\partial P}, \quad \dot{P} = -\frac{\partial K}{\partial Q}\]Here, $K(Q, P, t)$ is the new Hamiltonian in terms of transformed variables.
Generating Functions of Canonical Transformations
A generating function $F$ allows us to define a canonical transformation in such a way that the new coordinates $(Q, P)$ are derived systematically from the old ones $(q, p)$.
The key identity is:
\[p \, dq - P \, dQ = dF\]This implies the transformation is symplectic, meaning it preserves the area in phase space:
\[\int_C p \, dq = \int_{C'} P \, dQ\]Using different choices of independent variables in $F$, we define four standard types of generating functions.
In Hamiltonian mechanics, generating functions define canonical transformations and can be written in different forms depending on the choice of independent variables. The four standard types — $F_1$, $F_2$, $F_3$, and $F_4$ — are interrelated via Legendre transformations, which exchange variables in a controlled manner.
We begin with Type I: $F_1(q, Q)$ and obtain the others by performing Legendre transformations with respect to $q$, $Q$, or both.
Type I: $F_1(q, Q)$
This is the fundamental generating function from which others can be derived. It depends on the old coordinate $q$ and the new coordinate $Q$.
From the differential identity:
\[dF_1 = p \, dq - P \, dQ\]We read off:
\[p = \frac{\partial F_1}{\partial q}, \quad P = -\frac{\partial F_1}{\partial Q}\]Type II: $F_2(q, P)$
To eliminate the new coordinate $Q$ and introduce the new momentum $P$, we perform a Legendre transformation of $F_1$ with respect to $Q$:
\[F_2(q, P) = F_1(q, Q) + P Q\]Differentiating:
\[dF_2 = dF_1 + P \, dQ + Q \, dP = p \, dq + Q \, dP\]Therefore:
\[p = \frac{\partial F_2}{\partial q}, \quad Q = \frac{\partial F_2}{\partial P}\]Type III: $F_3(p, Q)$
To express the generating function in terms of the old momentum $p$ and new coordinate $Q$, we Legendre transform $F_1$ with respect to $q$:
\[F_3(p, Q) = F_1(q, Q) - p q\]Differentiating:
\[dF_3 = dF_1 - p \, dq - q \, dp = -P \, dQ - q \, dp\]So we obtain:
\[q = -\frac{\partial F_3}{\partial p}, \quad P = -\frac{\partial F_3}{\partial Q}\]Type IV: $F_4(p, P)$
This form uses both momenta, old and new. It is obtained by Legendre transforming $F_1$ with respect to both $q$ and $Q$:
\[F_4(p, P) = F_1(q, Q) - p q + P Q\]Differentiating:
\[dF_4 = dF_1 - p \, dq - q \, dp + P \, dQ + Q \, dP = -q \, dp + Q \, dP\]Hence:
\[q = -\frac{\partial F_4}{\partial p}, \quad Q = \frac{\partial F_4}{\partial P}\]🔁 Summary Table
| Type | Generating Function | Relations |
|---|---|---|
| I | $F_1(q, Q)$ | $p = \frac{\partial F_1}{\partial q}, \quad P = -\frac{\partial F_1}{\partial Q}$ |
| II | $F_2(q, P)$ | $p = \frac{\partial F_2}{\partial q}, \quad Q = \frac{\partial F_2}{\partial P}$ |
| III | $F_3(p, Q)$ | $q = -\frac{\partial F_3}{\partial p}, \quad P = -\frac{\partial F_3}{\partial Q}$ |
| IV | $F_4(p, P)$ | $q = -\frac{\partial F_4}{\partial p}, \quad Q = \frac{\partial F_4}{\partial P}$ |
Infinitesimal Canonical Transformations
For small transformations generated by a function $G(q, p)$:
\[\delta q = \epsilon \{q, G\} = \epsilon \frac{\partial G}{\partial p}, \quad \delta p = \epsilon \{p, G\} = -\epsilon \frac{\partial G}{\partial q}\]Here, $\epsilon$ is a small parameter. If $\{G, H\} = 0$, then $G$ is conserved and the transformation is a symmetry of the system.
🔗 Relation Between Hamiltonian and Generating Function
In classical mechanics, a generating function defines a canonical transformation, which maps one set of canonical variables $(q, p)$ to another $(Q, P)$, preserving the form of Hamilton’s equations.
We now explore how the Hamiltonian is related to the generating function, starting from the action integral.
🧭 The Action Integral
The action in Hamiltonian mechanics is:
\[S = \int_{t_1}^{t_2} \left( p \, \dot{q} - H(q, p, t) \right) dt\]If we perform a canonical transformation from $(q, p)$ to $(Q, P)$, the action becomes:
\[S' = \int_{t_1}^{t_2} \left( P \, \dot{Q} - K(Q, P, t) \right) dt\]Here, $K$ is the new Hamiltonian in the transformed variables.
🔄 Inserting the Generating Function
Let the transformation be generated by a function $F_1(q, Q, t)$ of type I. The total differential of $F_1$ is:
\[dF_1 = \frac{\partial F_1}{\partial q} dq + \frac{\partial F_1}{\partial Q} dQ + \frac{\partial F_1}{\partial t} dt\]We want to preserve the action up to a total derivative:
\[\int \left( p \, \dot{q} - H \right) dt \quad \longrightarrow \quad \int \left( P \, \dot{Q} - K \right) dt\]To ensure the equations of motion remain invariant, the two Lagrangian forms should differ by an exact differential:
\[p \, dq - H \, dt = P \, dQ - K \, dt + dF_1\]Here, $F_1$ is some function of the canonical variables (possibly also of time), whose total differential $dF_1$ adjusts for the change of variables.
📌 Matching Terms
From the differential identity:
\[p \, dq - H \, dt = P \, dQ - K \, dt + \frac{\partial F_1}{\partial q} dq + \frac{\partial F_1}{\partial Q} dQ + \frac{\partial F_1}{\partial t} dt\]Group terms:
- Coefficients of $dq$: $p = \frac{\partial F_1}{\partial q}$
- Coefficients of $dQ$: $P = -\frac{\partial F_1}{\partial Q}$
-
Coefficients of $dt$:
\[-H = -K + \frac{\partial F_1}{\partial t} \quad \Rightarrow \quad K = H + \frac{\partial F_1}{\partial t}\]
Bridge to Hamilton-Jacobi Theory
By appropriately choosing a generating function, we can transform a complicated Hamiltonian system into a simpler one—ideally into a system where the new Hamiltonian $K$ is zero or depends only on momenta, which allows direct integration.
This motivates the transition to Hamilton-Jacobi Theory.
The Hamilton-Jacobi theory arises from seeking a generating function (typically of Type II) that completely solves the equations of motion.
Let us consider a generating function of Type II: $F_2(q, P, t)$, and define it as Hamilton’s Principal Function:
\[S(q, P, t) = F_2(q, P, t)\]From this function, the transformation equations are:
\[p_i = \frac{\partial S}{\partial q_i}, \quad Q_i = \frac{\partial S}{\partial P_i}\]If we desire the new Hamiltonian $K(Q, P, t) = 0$, then from the relation:
\[K = H(q, p, t) + \frac{\partial S}{\partial t}\]and substituting $p_i = \frac{\partial S}{\partial q_i}$, we obtain the Hamilton-Jacobi Equation:
\[H\left(q_1, \dots, q_n, \frac{\partial S}{\partial q_1}, \dots, \frac{\partial S}{\partial q_n}, t \right) + \frac{\partial S}{\partial t} = 0\]Solving this partial differential equation gives us the principal function $S(q, P, t)$, which contains complete information about the system’s dynamics.
Solved Examples
-
Example 1:
Show that the transformation defined by $F_2(q, P) = \frac{1}{2}mq^2 \cot P$ is canonical.
Solution:
\(p = \frac{\partial F_2}{\partial q} = mq \cot P, \quad Q = \frac{\partial F_2}{\partial P} = -\frac{1}{2}mq^2 \csc^2 P\)
The transformation is canonical as it preserves Poisson brackets. -
Example 2:
Use an infinitesimal generator $G = q$ to find the transformation of $q$ and $p$. \(\delta q = \epsilon \{q, q\} = 0, \quad \delta p = \epsilon \{p, q\} = -\epsilon\)
Important Points / Summary
- Canonical transformations preserve the form of Hamilton’s equations.
- Generating functions provide a practical way to define canonical transformations.
- Infinitesimal generators correspond to conserved quantities and symmetries.
- The Hamilton-Jacobi theory uses generating functions to reduce dynamics to solving a PDE.
Practice Questions
Short Answer:
- Define a canonical transformation with an example.
- Explain the role of generating functions in canonical transformations.
- How is the Hamilton-Jacobi equation related to generating functions?
Numerical:
- Show that $F_1(q, Q) = qQ$ defines a canonical transformation and compute $p$, $P$.
- Let $G = pq$ be an infinitesimal generator. Find $\delta q$ and $\delta p$.
MCQs:
- Which of the following is not a valid type of generating function?
- (a) $F_1(q, Q)$
- (b) $F_2(q, P)$
- (c) $F_5(q, p)$
- (d) $F_4(p, P)$
- An infinitesimal generator $G$ leads to a conserved quantity if:
- (a) ${G, H} = 0$
- (b) $G$ is a function of time only
- (c) ${G, G} = 1$
- (d) $G$ commutes with all coordinates