Susskind Lecture 9 - Phase Space Fluid and Liouville Theorem
Lecture 9 asks Andrew to stop following one trajectory and imagine all possible trajectories at once. The result is the phase-space fluid picture.
Phase-space fluid
Picture phase space filled with infinitely many dots, one for every possible initial condition. Each dot moves according to Hamilton’s equations:
q_dot_i = ∂H/∂p_i
p_dot_i = -∂H/∂q_iThe collection of moving dots behaves like a fluid. The velocity field of this fluid is the Hamiltonian vector field.
Energy surfaces
If H has no explicit time dependence, each trajectory stays on a surface of constant energy:
H(q, p) = EFor one harmonic oscillator, these energy surfaces are circles in the q-p plane. For larger systems, they are high-dimensional surfaces that are impossible to draw but still mathematically precise.
Divergence and incompressibility
For ordinary fluid flow, zero divergence means volume is neither created nor destroyed locally. Susskind imports that idea into phase space.
The phase-space divergence of Hamiltonian flow is:
Σ_i [∂q_dot_i/∂q_i + ∂p_dot_i/∂p_i]Substitute Hamilton’s equations:
Σ_i [∂/∂q_i(∂H/∂p_i) - ∂/∂p_i(∂H/∂q_i)] = 0The mixed partial derivatives cancel. That is Liouville’s theorem.
Liouville theorem
Hamiltonian flow preserves phase-space volume.
A blob of possible states can stretch, twist, and become filamented, but its total phase-space volume stays the same. For the harmonic oscillator the blob merely rotates. For more general systems it can distort severely without changing volume.
This is the continuous analog of the reversible update arrows from Susskind Lecture 1 - Classical Mechanics: Hamiltonian mechanics does not crush many states into one.
Poisson bracket preview
Lecture 9 also introduces the Poisson bracket as a compact way to write time evolution. For a phase-space function F(q, p):
dF/dt = {F, H}That idea becomes central in Susskind Lecture 10 - Poisson Brackets and Angular Momentum.
Common pitfalls
- Thinking Liouville theorem says ordinary spatial volume is conserved. It is phase-space volume.
- Thinking blobs keep their shape. Only volume is guaranteed.
- Thinking energy conservation and Liouville theorem are the same. Energy conservation keeps a trajectory on a surface; Liouville preserves volume of a set of states.
- Forgetting that the theorem relies on Hamiltonian flow.
- Interpreting a phase-space dot as a physical particle. It represents a possible state.