Hamilton's Principle and the Action
The paths a system can actually follow between fixed endpoints and times are those that make the action stationary, and the Euler-Lagrange equations are the condition for that. Stating mechanics as a claim about whole paths rather than about instants is what carries over to fields and to quantum theory.
What a learner can do afterwards
- States Hamilton's principle and writes the action of a simple system
- Shows that making the action stationary returns the equation of motion
- Explains what varying a path means and which endpoints stay fixed
1 · Read
The action S is one number for a whole path: S = ∫_{t1}^{t2} L dt, with L the Lagrangian along the path. Hamilton's principle states that the path a system follows between fixed start and end positions at fixed times t1 and t2 makes S stationary: every small bend keeping those endpoints fixed changes S by nothing to first order. Stationary means the first variation of S vanishes for all such admissible bends, not that S is a minimum; a saddle can qualify.
Worked example: one coordinate x with L = ½mẋ² − V(x), and endpoints x(t1) = x1, x(t2) = x2 fixed. Bend the path: x(t) + η(t), with η small and η(t1) = η(t2) = 0. The first change in S is ∫(∂L/∂x η + ∂L/∂ẋ η̇) dt. Integrate the second term by parts: the boundary term (∂L/∂ẋ η) at t1 and t2 vanishes because η is zero there. What remains is ∫(∂L/∂x − d/dt(∂L/∂ẋ)) η dt. Since η is arbitrary, the bracket vanishes: the Euler-Lagrange equation. Free particle check: V = 0 gives mẍ = 0, a straight line at constant speed.
Varying a path means bending it while pinning both endpoints: same start position and time, same end position and time. The bends are all smooth paths with those pinned ends, and stationarity demands zero first variation under every one of them, not under a single chosen nudge. This fixed-endpoint version is the one that returns the interior Euler-Lagrange equation directly. Leaving an end or end time free adds boundary terms; those become extra conditions on the free end, while the interior equation still follows from bends that vanish near the ends.
Stationary does not mean least: for ordinary positive kinetic energy the action can sit at a saddle, and several different paths can be stationary at once, so fixed endpoints need not pick a unique trajectory.
Mechanics becomes a claim about whole paths rather than instants, and that pattern travels: fields vary whole histories, and quantum theory weighs all paths at once. The fixed endpoint times are part of the setup, not an optimization over shorter times. Learn the stationary habit once and reuse it everywhere.
Pin positions and times at both ends, demand zero first variation under all bends, and the equation of motion drops out.
2 · Watch
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Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.