Action

Action is a number assigned to an entire possible history of a system, not just to one instant. In the Susskind-style route, mechanics becomes: list possible paths, compute the action for each path, and the physical path is the one for which the action is stationary.

The usual definition is

$$S[q]={\int }_{t_1}^{t_2}L(q,\dot{q},t)dt,$$

where $L$ is the Lagrangian. For many beginner systems,

$$L=T-V,$$

kinetic energy minus Potential energy. The principle is often called least action, but the safer phrase is stationary action:

$$\delta S=0.$$

That means a tiny change to the path makes no first-order change in $S$. The action may be a minimum, maximum, or saddle.

Why it matters

Newton’s Laws of Motion says what forces do at each instant. The action principle says which whole history fits the constraints. This becomes powerful when coordinates are awkward, constraints are present, or symmetries are hidden.

Links forward

• Varying $S$ gives the Euler-Lagrange equations used in Lagrangian mechanics.
• The Legendre transform of the Lagrangian leads to Hamiltonian mechanics.
• Symmetries of the action are the cleanest route to Noether theorem.

Common pitfalls

• Do not read “least” too literally; use stationary action.
• The path endpoints are normally held fixed during the variation.
• $S$ has units of energy times time, the same dimensions as Planck’s constant.

Source trail: Susskind The Theoretical Minimum index; reference book: Classical Mechanics: The Theoretical Minimum by Leonard Susskind and George Hrabovsky.