Susskind Classical Mechanics - Phase space
Phase space is the arena of complete classical states. Configuration space tells you where the system is; phase space tells you where it is and how it is moving.
Core definition
For generalized coordinates q_i and conjugate momenta p_i:
phase-space point = (q_1, ..., q_N, p_1, ..., p_N)For N particles in three-dimensional space, there are 3N position coordinates and 3N momentum coordinates, so phase space is 6N-dimensional.
Why position space is not enough
A ball at the same height can be moving upward, falling downward, or momentarily at rest. Same position, different momentum, different future. This is why Susskind Lecture 1 - Classical Mechanics and Susskind Lecture 4 - Systems of More Than One Particle both insist on complete states.
Newtonian and Hamiltonian forms
Newtonian mechanics can be rewritten as first-order phase-space flow:
q_dot_i = p_i / m_i
p_dot_i = F_i(q)Hamiltonian mechanics generalizes this:
q_dot_i = ∂H/∂p_i
p_dot_i = -∂H/∂q_iThis is why Susskind Lecture 8 - Hamiltonian Mechanics feels like a return to Lecture 1: one point, one future arrow, but now in a continuous state space.
Geometry to remember
- A trajectory in ordinary space is q(t).
- A trajectory in phase space is (q(t), p(t)).
- A conserved quantity labels surfaces in phase space.
- Hamiltonian flow carries points along those surfaces.
- Susskind Classical Mechanics - Liouville theorem says Hamiltonian flow preserves phase-space volume.
Common pitfalls
- Calling configuration space “phase space.” Configuration space has q only; phase space has q and p.
- Using velocity when the formalism requires conjugate momentum. They coincide only for simple Lagrangians.
- Trying to visualize high-dimensional phase space literally. Use lower-dimensional examples as maps, not as exact pictures.