Hamilton's principle/Explainers/Action from the endpoints

Technical version: Action as a function of its endpoints

This page explains Hamilton's own idea, which is not the one usually taught first. It uses no mathematics. The technical page it goes with is named above.

Turning the question round

Action on the true motion as a function of its endpoints.

Everything so far has used the total cost in one way: work it out for many different paths, and find the path where it stops changing.

Hamilton's idea was the reverse.

Work out the cost for the true path only. Then ask how that number depends on where the journey started and finished.

That is a completely different object. It is no longer a rule about paths. It is an ordinary quantity that depends on two places — a number attached to each pair of start and end points.

Hamilton called it the characteristic function.

What it knows

Lambert's theorem: chord and sum of radii.

The surprise is how much that single number contains.

Ask how the number changes when you move the finishing point slightly. The answer is the momentum the object has when it arrives.

Ask how it changes when you move the starting point slightly. The answer is the momentum it had when it set off, with the opposite sign.

So the function does not merely record a cost. Differentiating it gives you the motion — arrival speed, departure speed, and with a little more work the timing.

Hamilton's claim was that the whole of dynamics for a system reduces to finding this one function and differentiating it.

Why he thought this was better

Hamilton was explicit about what his method does and does not offer, and he is worth taking at his word.

He did not claim it computes faster. He said in the essay that it might well bring no practical advantage.

What he claimed is a difference of kind. The older methods produce the equations of motion, which then have to be solved. His method produces their solution directly, once the function is known.

He was blunt about the older principle. Applied to finding the motion, he said, it only gives you the differential equations, which you could get another way — and he noted that Lagrange, Laplace and Poisson had for that reason thought it of little use.

So the distinction to hold on to is: varying the path gives you the equations. Varying the endpoints gives you their solution.

Where the idea came from

Not from mechanics. Hamilton built this for optics first, in his work on systems of light rays, and then transcribed it.

He says so in the opening paragraph of the dynamics essay: the dynamical principle, he writes, is only another form of an idea already applied to optics.

He adds a remark whose reach he could not have known. The method, he suggests, is probably not confined to optics and dynamics, and might one day become a separate branch of analysis.

He was right in a way he had no means of anticipating. See Hamilton's principle/Explainers/Light and particles.

A warning for anyone reading Hamilton

There is a trap here, and it catches nearly everyone.

Modern books define the action as one particular quantity. Hamilton's characteristic function is a different quantity.

Both are called the action. Both principles built on them are called the principle of least action. They are not the same, and the difference matters.

Anyone who opens Hamilton's 1834 essay expecting the modern quantity will misread every equation in it. Hamilton introduced the modern one a year later, in his second essay, as a technical improvement.

The two are explained side by side at Hamilton's principle/Explainers/Two different actions.

The place it breaks

The characteristic function has a limitation, and it is one this volume has met several times now.

It assumes there is one motion connecting the start and the finish. Only then is the cost a single well-defined number.

Sometimes there is more than one. Between the poles of a sphere there are infinitely many shortest routes, all the same length. The function has no single value there, and differentiating it is meaningless.

Those are the conjugate points of Hamilton's principle/Explainers/Where the minimum fails again, arriving in a third guise: places where minimality fails, where the wave picture folds, and where this function loses its meaning.

Three descriptions, one phenomenon. That recurrence is one of the reasons the subject hangs together as well as it does.

What Hamilton actually demonstrated

The 1834 essay does not stop at general claims. Hamilton works the two-body problem — a planet round the sun — in detail with his method.

Out come the shape of the orbit, the classification into ellipse, parabola and hyperbola, and the relation between position and time.

And out comes something less obvious: the cost of the journey depends only on three quantities — the straight-line distance between the two positions, the sum of their distances from the sun, and the size of the orbit. It does not depend on where in the sky they are, or on the shape of the arc between them.

That fact is used in orbit determination to this day. It was known before Hamilton, but his method delivers it as a remark about which quantities his function contains, rather than as a separate calculation.

That is his case for the method: not speed, but making structural facts visible.

Where to go next

The volume now turns to why conserved quantities exist at all, at Hamilton's principle/Explainers/Symmetry and conservation.