Susskind Classical Mechanics - Hamiltonian and Hamilton equations

Hamiltonian mechanics rewrites classical dynamics as first-order flow in phase space. It is the formulation that most directly foreshadows quantum mechanics.

Build H from L

Start with the Lagrangian and conjugate momenta:

p_i = ∂L/∂q_dot_i

Then define:

H = Σ_i p_i q_dot_i - L

Finally express H in terms of q_i and p_i, not q_dot_i.

Hamilton’s equations

The equations of motion are:

q_dot_i = ∂H/∂p_i
p_dot_i = -∂H/∂q_i

Each phase-space coordinate gets one first-order equation.

Energy meaning

For ordinary L = T - V systems:

H = T + V

More generally, H is the conserved energy associated with time-translation invariance when the Lagrangian has no explicit time dependence.

dH/dt = -∂L/∂t

Harmonic oscillator picture

For a simple oscillator, the Hamiltonian can look like a radius squared in phase space:

H = (ω/2)(p^2 + q^2)

Constant energy means constant radius. The oscillator is circular motion in phase space and back-and-forth motion in configuration space.

Why this formulation matters

• It makes the complete state explicit as (q, p).
• It turns second-order equations into pairs of first-order equations.
• It makes Susskind Classical Mechanics - Poisson brackets natural.
• It makes Susskind Classical Mechanics - Liouville theorem almost automatic.

Common pitfalls

• Computing H but leaving velocities in the answer.
• Assuming H is conserved even when L explicitly depends on time.
• Treating q and p as independent in Lagrangian mechanics; they become phase-space coordinates in Hamiltonian mechanics.

See also

• Susskind Classical Mechanics MOC
• Susskind Lecture 8 - Hamiltonian Mechanics
• Susskind Lecture 9 - Phase Space Fluid and Liouville Theorem
• Susskind Lecture 10 - Poisson Brackets and Angular Momentum
• Energy