The Hamiltonian, Phase Space and the Canonical Equations
Trading velocities for momenta turns one second-order equation per coordinate into two first-order ones, and makes phase space the natural arena. The Hamiltonian equals the conserved energy whenever it carries no explicit time dependence.
What a learner can do afterwards
- Constructs a Hamiltonian from a Lagrangian and identifies the conjugate momenta
- Writes and solves the canonical equations for a simple system
- Sketches a phase-space trajectory and reads the motion off it
1 · Read
You start from a Lagrangian L that depends on position x and velocity v. You define the conjugate momentum as p equals dL/dv, the partial derivative of L with respect to v. That one trade turns a single second order equation into two first order ones. The two rules that carry you forward are x prime equals dH/dp and p prime equals minus dH/dx.
Take a cart on a spring, where m times x double prime equals minus k times x. Its Lagrangian is one half m v squared minus one half k x squared, so the momentum is p equals m v. The replacement pair is x prime equals p over m and p prime equals minus k times x. You step x and p forward together, side by side.
You draw the motion in phase space, with x on the horizontal axis and p on the vertical axis. For the spring cart the Hamiltonian is p squared over 2m plus one half k x squared, and it holds one fixed value E. That fixed value pins the path to an ellipse, and a bigger loop means a larger energy. Where p is positive, x is growing, so the state point drifts rightward across the upper half.
Recall that p equals dL/dv, so for the spring Lagrangian you get p equals m v. Whenever H carries no explicit time dependence, its value stays constant along the motion, so H is the conserved energy. Build p first, then form H from x and p, and check that no stray t sits inside H. If H itself depends on t, the energy is free to drift.
Trade velocity for momentum, step two first order equations, and read the energy off the phase loop.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.