Canonical Transformations

Variable changes preserving Hamiltonian form. Theory, formulas, worked examples and applications.

Conceptual context

Why can Hamiltonian mechanics change coordinates and momenta almost freely, while Lagrangian mechanics can only change coordinates? The answer is that Hamiltonian phase space has an extra structure — the symplectic structure, encoded in Poisson brackets — that transformations must preserve, and that extra freedom (mixing positions and momenta) is exactly what makes the Hamiltonian formalism so powerful: sometimes a clever choice of new variables turns a hard problem into a trivial one.

This leaf covers canonical transformations generated by a type-2 generating function, $F_2$, depending on the old coordinate $q_i$ and the new momentum $P_i$. Its partial derivatives give the old momentum $p_i$ and the new coordinate $Q_i$; the transformed Hamiltonian $K$ may differ from the original $H$ if the generator depends explicitly on time, via $\partial F_2/\partial t$; and the condition $C$ checks that the proposed transformation actually preserves the symplectic structure.

🟢 Essential level

The essential idea is that a change of variables (q,p)→(Q,P) is not automatically valid in Hamiltonian mechanics, even if it seems reasonable. It is only "canonical" — it only preserves the form of Hamilton's equations in the new variables — if it keeps intact a certain algebraic structure between the variables, the fundamental Poisson bracket.

The most practical way to guarantee that preservation, without having to check it by hand every time, is to generate the whole transformation from a single function, the generating function. Any transformation correctly derived from a generating function is automatically canonical; that is why this leaf focuses on how to use that function, not on verifying the condition from scratch each time.

A celebrated example illustrating the power of this idea: there is a canonical transformation turning the simple harmonic oscillator, with its quadratic Hamiltonian in position and momentum, into a system where the new coordinate is simply the phase angle advancing at a constant rate. The problem, which in the original variables requires solving a differential equation, becomes trivial in the new variables.

🔵 Formal level

Basic formulas

This first formal layer presents the reference relation used to recognize the model and read its variables before entering more abstract derivations. Identify what each term represents, which assumptions make it valid, and how the chosen coordinates connect to the physical motion that the model is intended to describe.

The formal level rests on four relations:

Type-2 generating relation (old momentum)

New coordinate from the generator

Transformed Hamiltonian

Real student questions

"Why isn't it enough to check that the transformation is invertible, as with any change of variables?" Because invertibility is a much weaker condition than canonicity: there are infinitely many invertible transformations of (q,p) that do not preserve the symplectic structure and are therefore not canonical.

"How do I choose which type of generating function to use (F₁ to F₄)?" It depends on which pair of variables one wants to fix as independent: type 2 (used in this leaf) fixes the old coordinate and the new momentum; the other three types fix different combinations.

Final synthesis

By the end of this leaf, the student should be able to derive a complete transformation from a type-2 generating function, verify it is canonical via the Poisson brackets, and build the transformed Hamiltonian including the temporal correction when appropriate.

The relevant final check is one of consistency: verifying that Hamilton's equations written with $K$ in the new variables (Q,P) produce exactly the same physics (the same trajectories) as the original equations with $H$ in (q,p).