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Susskind Lecture 8 - Hamiltonian Mechanics

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Susskind Lecture 8 - Hamiltonian Mechanics

Lecture 8 introduces the Hamiltonian as both energy and the engine of phase-space motion. It also completes the symmetry story for energy: time-translation invariance produces energy conservation.

Time-translation invariance

A system is time-translation invariant when the same experiment, begun later with the same initial conditions, gives the same outcome.

In the Lagrangian language this means:

L has no explicit t dependence

The value of L may still change as q(t) and q_dot(t) change. The point is that the formula for L does not itself depend on the clock time.

Hamiltonian definition

For generalized coordinates q_i and conjugate momenta p_i:

p_i = ∂L/∂q_dot_i
H = Σ_i p_i q_dot_i - L

After computing H, rewrite it as a function of q_i and p_i, not q_dot_i.

For ordinary systems with L = T - V, the Hamiltonian becomes:

H = T + V

So in the usual case, the Hamiltonian is the total energy.

Energy conservation

The general identity is:

dH/dt = -∂L/∂t

Therefore, if L has no explicit time dependence:

dH/dt = 0

Energy conservation is the conservation law associated with time-translation symmetry.

If a subsystem has a time-dependent external parameter, its Hamiltonian need not be conserved. The missing energy is in the larger system that controls that parameter.

Hamilton’s equations

Hamiltonian mechanics treats q_i and p_i as coordinates of phase space. The equations are first-order:

q_dot_i = ∂H/∂p_i
p_dot_i = -∂H/∂q_i

Knowing one point in phase space is enough to determine the immediate future.

Harmonic oscillator in phase space

For a simple oscillator, the Hamiltonian can be written in a symmetric form like:

H = (ω/2)(p^2 + q^2)

Hamilton’s equations become rotation in the q-p plane. The phase-space trajectory is a circle of constant energy. In ordinary configuration space the oscillator moves back and forth; in phase space it circulates.

This is a key mental upgrade: phase space shows both position and momentum at once.

Common pitfalls

  • Thinking conservation of energy means conservation of the Lagrangian. L is usually not conserved.
  • Forgetting to eliminate q_dot when forming H(q, p).
  • Assuming H always equals naive T + V. It does for many ordinary systems, but the definition is the Legendre-transform expression.
  • Missing the word explicit in “explicit time dependence.” q(t) dependence alone does not break time-translation invariance.
  • Treating Hamilton’s equations as more equations than Lagrange’s. Two first-order equations replace one second-order equation per degree of freedom.

See also

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