Canonical transformations and Hamilton–Jacobi theory

Hamiltonian mechanics already buys you a lot of freedom in how you describe a system, but not every change of position and momentum variables is allowed. A canonical transformation is one that keeps Hamilton's equations looking exactly like Hamilton's equations in the new coordinates — it preserves the underlying structure (the Poisson brackets), even though the new coordinates might look nothing like position and momentum in any familiar sense.

That freedom turns out to be enormous. Among all the canonical transformations available for a given system, there is (locally) one that makes the new Hamiltonian identically zero everywhere. If the new Hamiltonian is zero, Hamilton's equations say the new coordinates don't change in time at all — they're constants, fixed by the system's initial conditions. Finding the generating function for that specific transformation is the content of the Hamilton–Jacobi equation, a single first-order partial differential equation in the old coordinates and time.

This reframes the entire problem. Instead of solving a system of coupled ordinary differential equations for how position and momentum evolve, you solve one partial differential equation for a single function, and the motion falls out algebraically from its derivatives — no further integration needed. It's rarely the easiest path for a generic problem, but for systems with enough symmetry it separates beautifully into one ordinary equation per coordinate, and it's the route that carries most directly into the WKB approximation and, from there, into quantum mechanics.

The one thing to hold onto: making the Hamiltonian zero is a specific, maximally ambitious goal for one particular use of canonical transformations. The transformations themselves are used far more broadly — simplifying coordinates, or moving to variables suited to periodic motion — without needing to drive the Hamiltonian all the way to zero every time.