Susskind Classical Mechanics Common pitfalls
A compact trap list for Susskind Classical Mechanics MOC. These are the places where the notation looks familiar but Susskind is using it to make a deeper structural point.
State and phase space
- Position is not a complete state. You need velocities or momenta too.
- Configuration space is not phase space. Configuration space is q; phase space is q plus p.
- A phase-space point is not a physical particle. It represents the whole state of the system.
Motion and dynamics
- Velocity is not speed. Velocity has direction.
- Acceleration is not “speeding up.” Circular motion has acceleration at constant speed.
- Newton’s law says force determines acceleration, not velocity.
- A second-order equation in q becomes two first-order equations in q and p.
Energy and potentials
- Force points down the potential-energy gradient: F_i = -∂V/∂x_i.
- Potential energy is not conserved by itself; T + V is conserved for closed conservative systems.
- Potential energy can be shifted by a constant without changing physics.
- Energy of a subsystem may change if the larger system supplies or removes energy.
Action and Lagrangians
- “Least action” usually means stationary action, not necessarily a minimum.
- The varied paths have fixed endpoints.
- The Lagrangian is not the energy. For ordinary systems L = T - V, while H = T + V.
- Generalized momentum p_i = ∂L/∂q_dot_i need not equal mv.
Symmetries and conservation
- A symmetry is an invariance of the dynamics, not just a visual pattern.
- Cyclic coordinate means absent from L, not circular.
- Momentum conservation comes from translation symmetry; angular momentum conservation comes from rotation symmetry; energy conservation comes from time-translation symmetry.
- External fields can hide the larger closed system where the conservation law actually lives.
Hamiltonian and Poisson-bracket mechanics
- After forming H, eliminate velocities in favor of canonical momenta.
- H is conserved only when the Lagrangian has no explicit time dependence.
- Poisson brackets are antisymmetric: swapping the arguments changes the sign.
- A generator G both produces a transformation and is conserved when that transformation is a symmetry of H.
Liouville and gauge issues
- Liouville theorem preserves phase-space volume, not necessarily shape or position-space volume.
- Magnetic vector potential A is not unique; B = ∇ × A is gauge invariant.
- Canonical momentum in a magnetic field is gauge dependent; mechanical momentum is not.
- A static magnetic field changes direction of velocity, not kinetic energy.