ISEGORIA / MATH ENCYCLOPEDIA
Hamiltonian mechanics: phase space and its geometry
Liouville’s theorem, symplectic integrators, and Noether’s link between symmetry and conservation.
Before you begin: Classical mechanics and calculus of variations
Predict, manipulate, then check your reasoning against the example and question. Graphs illustrate the mathematics; they do not replace a proof.
1. Liouville: the flow keeps area
Each point of phase space is a complete state (angle, momentum) of a pendulum. Drag the disc of initial states anywhere and advance time. The disc shears into a thin filament because swings of different amplitude have different periods, but its area never changes.
Worked example. The velocity field \((\partial_pH,-\partial_qH)\) has zero divergence, so by the divergence theorem the enclosed area cannot grow or shrink.
Watch out. Area is kept exactly by the true flow. The readout measures the polygon through 720 boundary points, so after long times a very thin filament is slightly under-resolved.
What does the thick curve through (±π, 0) separate?
The separatrix divides swinging motion (closed curves) from motion that goes over the top (open curves).
2. Integrators that respect the geometry
Integrate a Kepler orbit with three methods at the same step size. Explicit Euler spirals outward and RK4 slowly loses energy. Leapfrog is only second order, yet its energy error stays bounded over very long runs, because each of its steps is itself an exact symplectic map.
Worked example. Leapfrog almost exactly conserves a modified energy \(\tilde H=H+O(h^2)\) for exponentially long times, so the true energy only oscillates. The orbit precesses instead of decaying.
Watch out. Higher order is not always better over long times. RK4 is far more accurate per step, but its errors accumulate in one direction.
Why does Euler’s energy grow rather than shrink?
In the simplest case, a harmonic oscillator, each Euler step moves along the tangent to the circle of constant energy and so lands outside it: the energy is multiplied by exactly \(1+h^2\) per step.
3. Noether: symmetry gives conservation
A particle moves in a planar potential. With no ripple the potential is rotationally symmetric and angular momentum is conserved: the orbit is a clean rosette. Add an n-fold ripple and rotational symmetry breaks; angular momentum now wanders, while energy, protected by time-translation symmetry, stays fixed.
Worked example. At \(\varepsilon=0\), \(\partial V/\partial\phi=0\), so \(L=xp_y-yp_x\) is exactly constant.
Watch out. A discrete symmetry (rotation by \(2\pi/n\)) does not give a conserved quantity. Noether’s theorem needs a continuous family of symmetries.
Which symmetry protects the energy here?
Time translation: the potential does not depend on time.