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Susskind Lecture 6 - Principle of Least Action

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Susskind Lecture 6 - Principle of Least Action

Lecture 6 is the pivot from Newtonian mechanics to advanced mechanics. Instead of asking “what force acts right now?”, it asks “which whole path makes the action stationary?”

A safer phrase than “least action” is stationary action: the physical path makes the first-order change in the action vanish. It need not be an absolute minimum.

Two ways to pose the motion problem

Newtonian initial-value version:

given q(t0) and q_dot(t0), find q(t)

Action-principle boundary-value version:

given q(t0) and q(t1), find the path between them

The endpoints are fixed. The initial velocity is not specified directly; it is part of the path selected by the action principle.

Lagrangian and action

For many ordinary mechanical systems:

L(q, q_dot, t) = T - V

The action is the time integral of the Lagrangian:

S[q] = ∫ from t0 to t1 L(q, q_dot, t) dt

The bracket notation S[q] is a reminder that the action is a functional: it takes a whole path as input.

Euler-Lagrange equation

Stationary action gives the Euler-Lagrange equation:

d/dt(∂L/∂q_dot_i) - ∂L/∂q_i = 0

For L = (1/2)m x_dot^2 - V(x), this becomes:

m x_ddot = -dV/dx

So Newton’s law is recovered, but from a different organizing principle.

Generalized coordinates

The coordinates q_i do not have to be Cartesian positions. They can be angles, radii, pendulum coordinates, or any variables that uniquely describe the configuration.

This is one reason the Lagrangian method is powerful. If the natural coordinate is an angle, use an angle. The Euler-Lagrange equations still work.

Conjugate momentum

For any generalized coordinate q_i, define the conjugate momentum:

p_i = ∂L/∂q_dot_i

For Cartesian motion with ordinary L = T - V, this reduces to p = mv. In generalized coordinates, or in electromagnetic problems, p_i may be something else.

Cyclic coordinates

If a coordinate q_i does not appear in L, then ∂L/∂q_i = 0. The Euler-Lagrange equation becomes:

dp_i/dt = 0

So the conjugate momentum p_i is conserved. This is the first glimpse of the deep relation developed in Susskind Lecture 7 - Symmetries and Conservation Laws.

Coordinate transformations

Susskind uses moving and rotating frames to show why Lagrangians are practical. Instead of transforming Newton’s equations directly, rewrite L in the new coordinates and apply Euler-Lagrange again. Fictitious forces such as centrifugal and Coriolis terms then appear automatically.

Common pitfalls

  • Saying “least” when the correct local condition is stationary.
  • Letting the varied paths move the endpoints. The standard derivation fixes q(t0) and q(t1).
  • Treating the Lagrangian as energy. For ordinary systems L = T - V, while energy is T + V.
  • Assuming generalized momentum always equals mv.
  • Choosing too many coordinates for a constrained system. Generalized coordinates should be just enough to specify the configuration.

See also

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