Hamilton's principle

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

Of all the ways a ball could fly from your hand to a window, it takes one. Of all the paths light could follow from a lamp to your eye, it takes one. For three centuries physics has had a single answer to why: along the true path, a quantity called the action is stationary.

This volume is about that statement — what the quantity is, what stationary means and why it is not the same as least, how far the idea reaches, and why it turned out to be more fundamental than the laws it was invented to reformulate.

What is stationary

A true path and several comparison paths between fixed endpoints.

Take a system with a starting configuration and an ending configuration, and a fixed time to get from one to the other. For every path between them — not just the physical one, but every geometrically imaginable curve — compute

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

the accumulated difference between kinetic and potential energy along that path. Each path gets a number. The true path is the one where that number stops changing to first order: nudge the path slightly and $S$ does not change, to the accuracy of the nudge.

That is Hamilton's principle. It is equivalent to Newton's laws for ordinary mechanical systems, and it is stated in one line rather than one line per particle.

The one-line version conceals a real strangeness. To evaluate $S$ for a path you need the whole path — where the ball goes at the end is part of the input. Newton's laws are local in time: given position and velocity now, they say what happens next. Hamilton's principle is a statement about a completed journey. The two give the same answer, and they do not sound like the same kind of claim.

Why "stationary" and not "least"

Chronology of the action principle.

The principle was born as the principle of least action, and the name has stuck for nearly three centuries. It is wrong, and this volume takes some care over how wrong.

Stationary means the first variation vanishes — the same condition that a function's derivative is zero at a maximum, a minimum, or a saddle. It does not say the action is smallest. Sometimes it is. Sometimes it is a maximum, and sometimes it is neither.

The clearest case needs no mathematics. On a sphere, the shortest route between two cities is the short way round the great circle through them. But the long way round the same great circle is also a valid stationary path: it satisfies the same equation, at every point, and no local test distinguishes it from the short one. It is stationary and it is emphatically not shortest.

Optics gives a case where a real physical path is a maximum. Light reflecting from a suitably curved mirror takes longer than nearby paths would, and it goes that way regardless. Fermat's principle, stated as least time, is simply false there.

Hamilton knew this and said so, in 1834, in a passing correction he made no fuss about — the law of least, or as it might be better called, of stationary action. The mathematics behind his correction took the rest of the century, and it is the subject of this volume's second mathematics track.

Where the distinction bites. It is not pedantry. The point past which a path stops being shortest is a conjugate point, and conjugate points are physical: they are the foci of a lens, the multiple images produced by a gravitational lens, and the places where the classical approximation to quantum mechanics breaks down. A theory that says least cannot describe any of them.

Why the idea matters

The two long gaps in the subject's history.

Four reasons, in increasing order of consequence.

It is economical. One scalar function replaces a vector equation per particle, and coordinates can be chosen freely — angles, distances, whatever suits the problem — because the recipe does not care. For a constrained system the constraint forces, which dominate a Newtonian treatment, never appear.

It explains conservation laws. Momentum, energy and angular momentum are conserved because the action does not change under translations, time shifts and rotations. That is Noether's theorem of 1918, and it converts conservation laws from experimental facts into consequences of symmetry. It also works in reverse: find a symmetry and you have found a conserved quantity.

It builds theories. Write the most general action consistent with the symmetries you require, and the equations of motion follow. Electromagnetism comes out of demanding that a phase can be chosen independently at each point. General relativity comes out of demanding invariance under arbitrary coordinate changes. In both cases the symmetry chose the theory and almost nothing else had to be assumed.

It is why classical mechanics is true. In quantum mechanics every path contributes, each carrying a phase equal to its action in units of $\hslash$. For anything larger than an atom those phases oscillate so violently that neighbouring paths cancel — except near a path where the action is stationary, where neighbours share a phase and reinforce. The classical path is not selected. It is the one that survives interference.

That last point turns the principle inside out. The question is not why nature extremises the action. Nature does not extremise anything: it sums over everything, and stationary phase is where a sum of oscillating terms fails to cancel. Hamilton's principle is a theorem about interference — and it explains, as a bonus, why the correct word is stationary. Constructive interference needs only stationary phase. A maximum reinforces exactly as a minimum does.

How to read this volume

Every page carrying hard mathematics has a plain-language counterpart, named at the top of the page and linking back. You can read either level throughout, or switch between them.

The history. From Fermat's least time in the 1650s to Feynman's sum over paths in 1948, by way of the 1744 priority dispute, Lagrange's reformulation of mechanics, Hamilton's two essays, Jacobi, and Noether. Hamilton 1834 and 1835 is the pivot; Noether 1918 is where the subject stops being a technique and becomes an explanation.

The mathematics, in three tracks. The machinery builds the tools: functionals, the first variation, the fundamental lemma, and the Euler-Lagrange equation, starting at Functionals and the space of paths. Constraints and sufficiency asks when a stationary path is actually a minimum, and answers with worked counterexamples rather than assertions, starting at Holonomic constraints and reaching the central case at the Jacobi condition and conjugate points. Structure reorganises everything — as a transform, a flow in phase space, a wave equation, and a function of endpoints — starting at the Legendre transform.

The applications, from the brachistochrone to the path integral at the Feynman path integral, each worked rather than described.

The apparatus. Notation is worth reading before any primary source: three symbol collisions in this subject are severe enough to make Hamilton's own essays unreadable if they are not declared. Sources states exactly which text in this volume is quoted, which is editorial, and what remains unverified.

A note on sources

This volume quotes verbatim only from sources it has opened and checked. Where a source could not be verified, it says so, attributes no equation to it, and names what access would close the gap. No equation anywhere in this volume is a reconstruction.

That discipline is deliberate. A companion volume on Maxwell's On Physical Lines of Force discovered, after it was written, that every reachable electronic copy of that paper had lost its mathematics — the prose survived and read as complete, and the equations had to be rebuilt from secondary literature. This volume checked its sources before it was written rather than after.

Where to start

If you want the idea: read on at Comparing paths.

If you want the mathematics: Functionals and the space of paths.

If you want the one thing this volume most wants to convince you of: the Jacobi condition and conjugate points, where a great-circle arc is stationary and is not shortest, and the word least stops being tenable.