Hamilton's principle/Mathematics/Abbreviated action: Maupertuis versus Hamilton

Plain-language version: Two different actions

Two distinct quantities are called the action, two distinct statements are called the principle of least action, and the two are routinely conflated. This page separates them. The distinction is not pedantic: the principles hold different things fixed, are stationary over different classes of paths, and one of them cannot even be stated without conservation of energy.

The two quantities

What each action principle holds fixed.

Hamilton's action is

$S={\int }_{t_0}^{t_1}Ldt={\int }_{t_0}^{t_1}(T-V)dt\mathrm{(}\mathrm{1}\mathrm{)}$

The abbreviated action, also called Maupertuis' action, is

$W=\int \sum _i{p}_{i}d{q}_i=\int 2Tdt\mathrm{(}\mathrm{2}\mathrm{)}$

the second equality holding when $T$ is a homogeneous quadratic in the velocities. They are related by the Legendre transform in the $(t,H)$ pair:

$W=S+\int Hdt=S+E(t_1-t_0)\mathrm{(}\mathrm{3}\mathrm{)}$

the last form holding when energy is conserved.

Hamilton's own $V$ is $W$, not $S$. His characteristic function on Action as a function of its endpoints is defined as the accumulated living force, ${\int }_0^{t}2Tdt$, which is (2). Readers who assume Hamilton's action means $\int Ldt$ misread every equation in the 1834 essay. He introduced the function of the other kind, with time entering explicitly, only in the 1835 second essay, and named it $S$.

The two principles

	Hamilton's principle	Maupertuis' principle
quantity	$S=\int Ldt$	$W=\int \sum pdq$
held fixed	endpoints and both times	endpoints and the energy
free to vary	the path, at fixed duration	the path and the duration
energy	need not be conserved	conservation is a hypothesis
applies to	any Lagrangian system	conservative systems only
determines	the path and its timing	the path in space only

The row that matters is the third. Hamilton's principle compares paths traversed in the same time and lets the energy differ. Maupertuis' principle compares paths of the same energy and lets the transit time differ. These are different comparison classes over the same set of curves, so they are different variational problems that happen to share their solutions.

Deriving Maupertuis from Hamilton

Start from (3) and vary, holding the endpoints and $E$ fixed but allowing $t_1$ to vary:

$\delta W=\delta S+E\delta (t_1-t_0)\mathrm{(}\mathrm{4}\mathrm{)}$

By the law of varying action, quoted on Action as a function of its endpoints, the variation of $S$ at fixed endpoints with variable end time is $\delta S=-H\delta t_1=-E\delta t_1$. Substituting into (4), the two terms cancel and

$\delta W=0\mathrm{(}\mathrm{5}\mathrm{)}$

So $W$ is stationary among energy-conserving paths joining the fixed endpoints. That is Jacobi's form of the principle of least action.

The cancellation in (4) is the whole relationship between the two principles, and it is why they select the same curves while making different statements about them.

Jacobi's form

For $T=\frac{1}{2}\sum g_{ik}{\dot{q}}_i{\dot{q}}_k$ and potential $V$, energy conservation gives $T=E-V$, and (2) becomes a purely geometric integral:

$W=\int \sqrt{2(E-V(q))}ds,d{s}^2=\sum _{i,k}{g}_{ik}d{q}_{i}d{q}_k\mathrm{(}\mathrm{6}\mathrm{)}$

No time appears. The trajectory is a geodesic of the conformally rescaled metric

${\tilde{g}}_{ik}=2(E-V(q))g_{ik}\mathrm{(}\mathrm{7}\mathrm{)}$

This is a genuinely different way of seeing mechanics: a particle in a potential travels in a straight line, in a curved space whose curvature the potential supplies. It also makes the optical analogy exact — compare (6) with Fermat's integral $\int nds$ and the correspondence $n\leftrightarrow \sqrt{2(E-V)}$ of the Hamilton-Jacobi equation is read off directly.

The timing is lost and must be recovered afterwards from $dt=ds/\sqrt{2(E-V)}$. That loss is the price of the geometric form, and it is the practical difference between the two principles.

Least, and the failure of least

Neither principle asserts a minimum, and this is where the two histories diverge.

Maupertuis in 1744 argued from a metaphysical premise — that nature acts with economy — and the word least was load-bearing in his argument. Euler gave the mathematical statement in the same year and treated it as a variational condition. The subsequent priority dispute is treated on the Maupertuis-Koenig affair.

Hamilton, ninety years later, states the correction plainly, describing the older result as the law of least, or as it might be better called, of stationary action.

The failure is exhibited, not asserted, on the Jacobi condition and conjugate points: past a conjugate point the action is stationary and is not minimal. For Maupertuis' principle in Jacobi form (6) that statement is geometrically transparent — the long way round a great circle is a geodesic and is not the shortest path — and it is the same counterexample, because for a free particle on a sphere (6) is arc length.

What each is good for

Hamilton's principle is the general one. It requires no conservation law, extends to dissipative and time-dependent systems in modified forms, and generalises to field theory, where the abbreviated action has no natural analogue. Every application in this volume beyond Central force and the Kepler problem uses it.

Maupertuis' principle is the specialised one. Where it applies it removes time from the problem entirely, turning dynamics into geometry, and it is the right tool for orbit shapes, for the semiclassical quantisation conditions, and for anything where the trajectory matters and the schedule does not.

The confusion, named

Three habits produce the conflation, and all three appear in the literature:

• calling both quantities $S$;
• calling both principles the principle of least action;
• citing Hamilton for a statement about $\int Ldt$ while quoting an equation from 1834, where $V$ is $\int 2Tdt$.

This volume writes $S$ for (1) and $W$ for (2) throughout, reserves Hamilton's principle for the first statement and Maupertuis' principle for the second, and uses stationary rather than least for both. The concordance on Notation records the collision with Hamilton's own usage, in which $V$ means (2).

O7 closes here

The structure track has taken the machinery of O5 and O6 and reorganised it four times: as a transform, as a flow in phase space, as a wave equation, and as a function of endpoints. Noether's two theorems explain why the first integrals found along the way existed. This page separates the two principles that the whole subject is named after.

Next

the brachistochrone opens the applications track.