Hamilton's principle/Explainers/Two different actions

Technical version: Abbreviated action: Maupertuis versus Hamilton

Hamilton's principle: map
Reading the volume How to read this volume Notation Glossary Equation index Figure gallery
Reading by level Start here: Comparing paths The mathematics: Functionals and the space of paths Stationary is not least: The Jacobi condition and conjugate points The applications: The brachistochrone The history: Hamilton 1834 and 1835
Sources Sources Chronology
Related Noether's theorems On Physical Lines of Force

This page separates two things that are both called the action. It uses no mathematics. The technical page it goes with is named above.

A confusion worth clearing up

What each action principle holds fixed.

There are two different quantities in this subject.

Both are called the action. Both have a principle built on them. Both principles are called the principle of least action.

They are not the same, and treating them as one is the most common error in the whole area. This page keeps them apart.

What each one holds fixed

The difference is not in the formulas. It is in what you are allowed to vary when you compare paths.

Hamilton's version. Fix where the journey starts, where it ends, and how long it takes. Compare all paths that get from one to the other in that time. Their energies may differ.

Maupertuis' version. Fix where the journey starts, where it ends, and the energy. Compare all paths with that energy. How long they take may differ.

One holds the clock and lets the energy float. The other holds the energy and lets the clock float.

These are different competitions over the same set of curves. The winner turns out to be the same curve, which is why the two get confused, but the questions being asked are not the same question.

What each one is good for

Hamilton's version is the general one.

It needs no conservation of energy. It works for systems where the energy changes, where things are pushed and driven. It extends to fields, and hence to electromagnetism and gravity. Everything later in this volume uses it.

Maupertuis' version is the specialised one.

It requires energy to be conserved, so it applies to a narrower class of problems. In exchange, it removes time from the question entirely.

That trade is sometimes exactly what you want. If you care about the shape of a planet's orbit and not about where it will be next Tuesday, the version without time in it is simpler. You get the ellipse and you never mention the clock.

Recovering the timing afterwards is possible, and is extra work.

Turning motion into geometry

The specialised version has a consequence worth stating on its own.

With time removed, the problem becomes purely about shape. And it turns out to be exactly the problem of finding the shortest path — not in ordinary space, but in a space that has been stretched and distorted by the forces acting.

So a particle moving through a landscape of hills and valleys is travelling as straight as it can, in a warped space whose warping the landscape supplies.

That is a genuine reformulation, not a metaphor. And it is the same statement as the optical one on Light and particles: light bends in glass because glass warps the effective geometry, and a particle bends near a mass for the same kind of reason.

The idea that gravity is geometry, rather than being a force acting within it, has this as its ancestor.

Which one Hamilton used

Here is the trap for anyone reading the original sources.

Hamilton's famous 1834 essay uses the specialised quantity — the one usually attributed to Maupertuis — throughout. His central function is that one.

The quantity that modern textbooks call Hamilton's action appears in his second essay, a year later, introduced as a technical convenience for avoiding awkward steps.

So the thing named after Hamilton is not the thing he built his first paper on.

Anyone who opens the 1834 essay carrying the modern definition will misread every equation in it. This volume flags the point at each place a reader could trip over it, and the symbols are tabulated at Notation.

Neither is "least"

The word least is wrong for both, and this volume has now shown that several times over.

Maupertuis in 1744 meant least seriously. His argument for the principle was that nature is economical, and economy was the point of it.

Hamilton corrected the name in 1834, calling it the law of least, or as it might be better called, of stationary action. He did it in passing, without emphasis.

The mathematical justification for his correction took the rest of the century, and this volume works through it in the pages from Is it really a minimum onwards.

The quantum picture at Sum over histories then supplies an independent reason: what matters is that the quantity barely changes, and a maximum does that just as well as a minimum.

How to keep them apart

Three habits to hold on to, and they cover almost every case.

Ask what is held fixed. Time fixed means Hamilton's. Energy fixed means Maupertuis'.

Ask whether time appears in the answer. If the result is a shape with no schedule, it is the specialised version.

Distrust the word "action" without qualification. Especially in older sources, and especially in Hamilton's own.

Where to go next

The optical version of the same correspondence, and what came of it, is at Light and particles.