Hamiltonian mechanics

Lagrangian mechanics describes a system with position and velocity, producing one second-order differential equation per coordinate — an equation involving acceleration, which means solving it requires two pieces of initial information (position and velocity) per coordinate. Hamiltonian mechanics performs a change of variables, trading velocity for momentum, and the payoff is that each single second-order equation splits into two first-order equations: one saying how position changes with time, one saying how momentum does. Nothing physical is different — a trajectory computed either way is identical — but the mathematical structure becomes more symmetric and, for many purposes, more tractable.

The momentum used here, called canonical or generalised momentum, is defined directly from the Lagrangian rather than assumed in advance, and it does not always reduce to the familiar mass times velocity. For a charged particle in a magnetic field, for instance, the canonical momentum includes a contribution from the field itself, not just from motion — a distinction that matters enormously once quantum mechanics is built on this same foundation.

The central object, the Hamiltonian H, replaces the Lagrangian and for most ordinary systems equals the total energy, kinetic plus potential. That identification is not decoration: Hamilton's equations show that H is literally the generator of time evolution — the function whose derivatives tell every coordinate and momentum how to change next. This reframing, energy as the engine driving the system forward in time rather than merely a conserved bystander quantity, is what makes Hamiltonian mechanics the formulation that carries over most directly into quantum mechanics, where the Hamiltonian plays exactly the same generating role in the Schrödinger equation.