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Susskind Lecture 10 - Poisson Brackets and Angular Momentum

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Susskind Lecture 10 - Poisson Brackets and Angular Momentum

Lecture 10 turns Poisson brackets into a working language for mechanics. They compute time evolution, encode symmetries, and reveal why angular momentum generates rotations.

Poisson bracket definition

For phase-space functions F(q, p) and G(q, p):

{F, G} = Σ_i [(∂F/∂q_i)(∂G/∂p_i) - (∂F/∂p_i)(∂G/∂q_i)]

Core properties:

{F, G} = -{G, F}
{F, F} = 0
{q_i, q_j} = 0
{p_i, p_j} = 0
{q_i, p_j} = δ_ij

The product rule also holds, so Poisson brackets behave like a derivative in each argument.

Time evolution

The time derivative of any phase-space function is:

dF/dt = {F, H}

Choosing F = q_i or F = p_i recovers Hamilton’s equations:

q_dot_i = {q_i, H}
p_dot_i = {p_i, H}

This is why Poisson brackets are not ornamental notation. They summarize the dynamics.

Angular momentum in three dimensions

For a particle with position r and momentum p:

L = r × p
L_x = y p_z - z p_y
L_y = z p_x - x p_z
L_z = x p_y - y p_x

If the Hamiltonian is rotationally invariant, angular momentum is conserved.

Angular momentum generates rotations

Taking a Poisson bracket with an angular momentum component gives the infinitesimal rotation generated by that component. For example, rotations about z are generated by L_z.

The angular momentum components satisfy:

{L_i, L_j} = ε_ijk L_k

This algebra is a classical preview of commutators in quantum mechanics.

Generators and conservation

Any phase-space function G can generate an infinitesimal transformation:

δF = ε {F, G}

If the transformation generated by G leaves the Hamiltonian unchanged, then:

{H, G} = 0

By antisymmetry, this is equivalent to:

dG/dt = {G, H} = 0

So symmetry generator and conserved quantity are the same object viewed from two angles.

Rotor/precession lesson

Susskind’s spinning rotor in a magnetic field shows the power of the bracket algebra. If the Hamiltonian is proportional to L_z, then L_z is constant and L_x, L_y rotate into each other. The angular momentum precesses around the field instead of simply falling toward it.

Common pitfalls

  • Forgetting the sign in the Poisson bracket definition.
  • Treating {F, H} as optional notation. It is the time derivative of F along the Hamiltonian flow.
  • Assuming angular momentum is one number. In 3D it is a vector with nontrivial bracket relations.
  • Confusing a symmetry transformation with time evolution. Both can be generated by brackets, but by different generators.
  • Missing the classical-to-quantum bridge: Poisson brackets become a template for commutators.

See also

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