Susskind Classical Mechanics - Liouville theorem

Liouville theorem says Hamiltonian flow is incompressible in phase space. It is the continuous mechanics version of “classical reversible dynamics does not erase information.”

Phase-space fluid picture

Imagine a blob of possible initial states in phase space. Every point in the blob evolves by Hamilton’s equations.

The blob may stretch, shear, fold, and become filamented. But its total phase-space volume does not change.

Divergence calculation

Hamilton’s equations define the phase-space velocity field:

v_qi = q_dot_i = ∂H/∂p_i
v_pi = p_dot_i = -∂H/∂q_i

The divergence is:

Σ_i [∂v_qi/∂q_i + ∂v_pi/∂p_i]

Substitute the Hamiltonian velocities:

Σ_i [∂²H/∂q_i∂p_i - ∂²H/∂p_i∂q_i] = 0

The mixed partial derivatives cancel.

What it means

• Hamiltonian flow preserves phase-space volume.
• A set of possible states cannot be crushed into a smaller volume.
• This is tied to reversibility and information preservation.
• It is not the same as conservation of energy, though both are phase-space statements.

Common pitfalls

• Thinking Liouville theorem says physical-space density is conserved. It is about phase space.
• Thinking the blob keeps its shape. Only volume is protected.
• Ignoring momenta. A projected blob in position space can expand or contract.

See also

• Susskind Classical Mechanics MOC
• Susskind Lecture 9 - Phase Space Fluid and Liouville Theorem
• Susskind Classical Mechanics - Phase space
• Susskind Classical Mechanics - Hamiltonian and Hamilton equations