Susskind Classical Mechanics - Action and Lagrangian
The Lagrangian formulation packages a mechanical system into one function and gets the equations of motion by asking for stationary action.
Lagrangian
For many ordinary systems:
L(q, q_dot, t) = T - VBut the deeper rule is not “always T - V.” The deeper rule is: write the correct L(q, q_dot, t), then apply the action principle.
Action
The action assigns a number to a whole path:
S[q] = ∫ L(q, q_dot, t) dtThat makes S a functional. It depends on the entire function q(t), not just on one value of q.
Stationary action
The physical path satisfies:
δS = 0with endpoints fixed. “Stationary” is more precise than “least,” because the stationary point can be a minimum, maximum, or saddle.
Euler-Lagrange equation
For each generalized coordinate q_i:
d/dt(∂L/∂q_dot_i) - ∂L/∂q_i = 0This is the main machine. Feed in a Lagrangian; get equations of motion.
Conjugate momentum
The momentum conjugate to q_i is:
p_i = ∂L/∂q_dot_iThis definition is broader than p = mv. It is essential for Susskind Lecture 8 - Hamiltonian Mechanics and for magnetic forces in Susskind Lecture 11 - Electric and Magnetic Forces.
Why Andrew should care
The Lagrangian method is often the easiest route when constraints or non-Cartesian coordinates are present. Pendulums, rotating frames, and particles on surfaces become coordinate-choice problems instead of force-component bookkeeping nightmares.
Common pitfalls
- Treating the action as a function of one coordinate rather than a functional of a path.
- Forgetting endpoint conditions in variational arguments.
- Assuming generalized coordinates must be x, y, z.
- Equating conjugate momentum with mechanical momentum without checking L.