Hamilton's principle/Mathematics/Action as a function of its endpoints

Plain-language version: Action from the endpoints

Every use of the action so far has treated it as a functional to be varied over paths, with the endpoints held fixed. Hamilton's own idea was the opposite: evaluate the action on the actual motion, so that it is no longer a functional at all but an ordinary function of where the motion starts and ends. Its derivatives then reproduce the whole of dynamics. He called it the characteristic function, and he called the result the law of varying action.

This page follows his construction. The 1834 essay is a verified source for this volume, so the equations below are quoted rather than reconstructed.

The characteristic function

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

Hamilton defines $V$ as the accumulated living force of the system, that is, twice the kinetic energy integrated over the motion:

$V=\int \Sigma .m(x^{\prime }dx+y^{\prime }dy+z^{\prime }dz)={\int }_0^{t}2Tdt\mathrm{(}\mathrm{1}\mathrm{)}$

Note what (1) is not. It is not $\int Ldt$. Hamilton's $V$ is the abbreviated action $\int 2Tdt$, and the distinction between it and $\int Ldt$ is the subject of Abbreviated action: Maupertuis versus Hamilton. Confusing the two is the most common error in reading Hamilton, and the volume flags it here as well as there.

$V$ is evaluated along the true motion. It therefore depends on the initial coordinates $a,b,c$, the final coordinates $x,y,z$, and the energy constant $H$ — in total $6n+1$ arguments for a system of $n$ points, and not on any path.

The law of varying action

Hamilton's central result is the expression for how $V$ changes when the endpoints and the energy are varied:

$\delta V=\Sigma .m(x^{\prime }\delta x+y^{\prime }\delta y+z^{\prime }\delta z)-\Sigma .m(a^{\prime }\delta a+b^{\prime }\delta b+c^{\prime }\delta c)+t\delta H\mathrm{(}\mathrm{2}\mathrm{)}$

Reading (2) coefficient by coefficient gives three groups of relations. First, the final momenta:

$\frac{\delta V}{\delta x_1}=m_1{x_1}^{\prime },\frac{\delta V}{\delta y_1}=m_1{y_1}^{\prime },\dots \mathrm{(}\mathrm{3}\mathrm{)}$

second, the initial momenta, with the opposite sign:

$\frac{\delta V}{\delta a_1}=-m_1{a_1}^{\prime },\frac{\delta V}{\delta b_1}=-m_1{b_1}^{\prime },\dots \mathrm{(}\mathrm{4}\mathrm{)}$

and third, the time:

$\frac{\delta V}{\delta H}=t\mathrm{(}\mathrm{5}\mathrm{)}$

In modern notation (3) and (4) read $\partial S/\partial q=p$ and $\partial S/\partial q_0=-p_0$: the gradient of the action with respect to the final position is the final momentum, and with respect to the initial position is minus the initial momentum.

Why this is the whole solution

Equations (3) are $3n$ relations connecting the final coordinates, their velocities, the initial coordinates and $H$. Eliminating $H$ between them and (5) gives the intermediate integrals. Equations (4) with (5) give the final integrals. Hamilton states the consequence plainly: the general problem of dynamics is reduced to the search and differentiation of a single function.

The value of that reduction is not computational. Hamilton is explicit that it may bring no practical facility, and rests the case elsewhere: on the intellectual pleasure of reducing all researches respecting the forces and motions of body to the study of one characteristic function.

The equations $V$ must satisfy

Combining (3) with conservation of energy $T=U+H$ gives a partial differential equation for $V$ in the final coordinates, and combining (4) with the initial form gives a second one in the initial coordinates. These are equations (F.) and (G.), quoted on the Hamilton-Jacobi equation. Hamilton describes them as of the first order and the second degree, and says they furnish the principal means of discovering the form of that function.

So the logical order is: the law of varying action comes first, and the Hamilton-Jacobi equation is derived from it. The modern presentation reverses this, deriving the equation from canonical transformation theory and mentioning the characteristic function afterwards. Both are correct; only Hamilton's shows where the idea came from.

What Hamilton says about least action

Hamilton is precise about the relation between his principle and the older one, and the passage is the source of this volume's terminology. He observes that Lagrange and others, from the vanishing of the variation of the action integral, deduced only the well known law of least action — and then corrects the name in passing, writing of this well known law of least, or as it might be better called, of stationary action.

He goes further, and it is worth stating exactly what he claims. The older principle, applied to determine the motion, serves only to form the differential equations of the second order, which can always be otherwise found — and he notes that Lagrange, Laplace and Poisson had for that reason spoken lightly of its utility. His own law of varying action, by contrast, expresses by means of a single function not the mere differential equations of motion, but their intermediate and their final integrals.

That is the distinction the volume's landing page must carry: varying the path with fixed endpoints gives the equations of motion; varying the endpoints of the actual motion gives their solution.

Relation to the modern $S$

The modern on-shell action is $S(q,t;q_0,t_0)={\int }_{t_0}^{t}Ldt$ evaluated along the true motion, and its differential is

$dS=\sum _i{p}_{i}d{q}_i-\sum _i{p}_{0i}d{q}_{0i}-Hdt+H_{0}d{t}_0\mathrm{(}\mathrm{6}\mathrm{)}$

Equation (6) is (2) with the energy variation traded for a time variation, via the Legendre transform of the Legendre transform. Hamilton's $V$ holds $H$ as an argument and produces $t$; the modern $S$ holds $t$ and produces $H$. They are Legendre transforms of one another in the pair $(t,H)$, which is why the same information appears in both and why the sign patterns match.

Hamilton introduced this second function himself in the 1835 second essay, writing $S$ for the part of the characteristic function into which the time enters explicitly.

Singularities

$V$ fails to be a well-defined function wherever more than one motion connects the given endpoints with the given energy. That is again the conjugate point of the Jacobi condition and conjugate points, and the derivatives (3) and (4) fail there. The great-circle example is the cleanest case: between antipodes, infinitely many motions of the same action exist, and the characteristic function has no gradient.

Next

Noether's first theorem returns to the functional picture and derives conservation laws from symmetry.