where is an arbitrary parameter, not necessarily arclength. (Indeed, for particles moving at the speed of light the arclength is zero, so it’s no good as a parameter!)

You can see this done e.g. starting in Section 3.5.2 of this book:

• John Baez, Blair Smith and Derek Wise, *Lectures on Classical Mechanics*.

It’s just a draft but it’s fairly readable.

]]>I suppose the awkward bit would be finding an expression for the action that is also agnostic with respect to the choice of time coordinate. A physical theory with action does seem to give a special role to , while it’s much more elegant to parametrise by arc length as you did and never have to mention time at all.

I wonder if it would turn into a field theory at that point, though.

]]>
I think what I don’t understand is what role the HPF plays in this recipe for mechanics. Is it:

1. Use some minimisation procedure to find path

2. Evaluate HPF for that path

3. Decide which represents time, label as and hence derive Hamilton’s equations

?

Yes, that’s about right.

In which case I think I was looking for step 1 somewhere in your post, and getting confused.

I explained it here:

Suppose we have a particle on the line. Consider smooth paths where it starts at some fixed position at some fixed time and ends at the point at the time . Nature will choose a path with least action—or at least one that’s a stationary point of the action. Let’s assume there’s a unique such path, and that it depends smoothly on and . For this to be true, we may need to restrict and to a subset of the plane, but that’s okay: go ahead and pick such a subset.

Given and in this set, nature will pick the path that’s a stationary point of action; the action of this path is called

Hamilton’s principal functionand denoted (Beware: this is not the same as entropy!)

Of course this is just a special case of a more general procedure. For example we could consider a spacetime manifold and fix a point

Suppose we can define a quantity called the ‘action’ for any path . Suppose there is a unique action-minimizing path from to any point . Then given , let be the action of the action-minimizing path from to the point Then

is **Hamilton’s principal function**, and starting from here we can derive Hamilton’s equations as explained in my blog article.

We can work in arbitrary coordinates and write the coordinates of as We don’t need to choose one coordinate and call it ‘time’: the formalism doesn’t really require that. Hamilton’s equations are really just the trivial statement that

Here I’m assuming that everything is as differentiable as we want it to be, and just to be careful I’m assuming there’s a unique action-minimizing path. The second assumption is false when there are ‘caustics’.

]]>I think what I don’t understand is what role the HPF plays in this recipe for mechanics. Is it:

1. Use some minimisation procedure to find path

2. Evaluate HPF for that path

3. Decide which represents time, label as , and hence derive Hamilton’s equations

?

In which case I think I was looking for step 1 somewhere in your post, and getting confused.

Incidentally, I find methods like this that put time and space coordinates on an equal footing to be very attractive — as far as I’m concerned, they only differ because they show up in different places in a Lorentzian metric. However, looking it up it seems that this is called ‘non-autonomous mechanics’ and is in general fairly tricky, e.g. I found http://arxiv.org/pdf/math-ph/0604063.pdf

(Though the text you posted http://math.ucr.edu/home/baez/toby/Hamiltonian.pdf also seems to apply to this case, though it seems to assume you’ve already picked a hypersurface, and doesn’t contain any details of a minimisation procedure)

]]>Given that the integral of Hamilton’s principal function over a path…

I’m confused. Hamilton’s principal function is itself defined as an integral over a path; it’s not something we typically integrate over a path.

]]>The symplectic structure is even more interesting and more useful when applied to the modern (post-classical) versions. Hamilton-Jacobi theory is amusing, but it’s less useful than out-and-out QM, and not significantly simpler.

2) Stat mech is basically the analytic continuation of QM, continued in the direction of imaginary time. This point and its ramifications are discussed in e.g. Feynman and Hibbs **Quantum Mechanics and Path Integrals** (1965). The classical limit of QM is obtained by the method of stationary phase, whereas the classical limit of stat mech is obtained by the method of steepest descent … so the two subjects are very nearly but not quite identical.

Again: If we’re going to make connections, it is even more interesting and more useful to connect the modern (post-classical) versions.

FWIW note that Planck invented QM as an outgrowth from stat mech … not directly from classical mechanics. So the connections are there, and have been since Day One.

3) Many (but not all) of the familiar formulas of thermodynamics can usefully be translated to the language of differential forms. In many cases all that is required is a re-interpretation of the symbols, leaving the form of the formula unchanged; for instance we interpret dE = T dS – P dV as a *vector* equation.

I say “not all” formulas because more than a few of the formulas you see in typical thermodynamics books are nonsense. This includes (almost) all expressions involving “dQ” or “dW”. Such things simply do not exist (except in trivial cases). Daniel Schroeder in **An Introduction to Thermal Physics** (1999) rightly calls them a crime against the laws of mathematics. With a modicum of self-discipline it is straightforward to do thermodynamics without committing such crimes.

Differential forms make thermodynamics simpler and more visually intuitive … and simultaneously more sophisticated, more powerful, and more correct.

Although there are fat books on the subject of differential topology, only the tiniest fraction of that is necessary for present purposes. An introductory discussion (including pictures) can be found at https://www.av8n.com/physics/thermo-forms.htm and the application to thermodynamics is worked out in some detail at https://www.av8n.com/physics/thermo/.

]]>]]>It has been shown that contact geometry is the proper framework underlying classical thermodynamics and that thermodynamic fluctuations are captured by an additional metric structure related to Fisher’s Information Matrix. In this work we analyze several unaddressed aspects about the application of contact and metric geometry to thermodynamics. We consider here the Thermodynamic Phase Space and start by investigating the role of gauge transformations and Legendre symmetries for metric contact manifolds and their significance in thermodynamics. Then we present a novel mathematical characterization of first order phase transitions as equilibrium processes on the Thermodynamic Phase Space for which the Legendre symmetry is broken. Moreover, we use contact Hamiltonian dynamics to represent thermodynamic processes in a way that resembles the classical Hamiltonian formulation of conservative mechanics and we show that the relevant Hamiltonian coincides with the irreversible entropy production along thermodynamic processes. Therefore, we use such property to give a geometric definition of thermodynamically admissible fluctuations according to the Second Law of thermodynamics. Finally, we show that the length of a curve describing a thermodynamic process measures its entropy production.