There’s a fascinating analogy between classical mechanics and thermodynamics, which I last talked about in 2012:
I’ve figured out more about it, and today I’m giving a talk about it in the physics colloquium at the University of British Columbia. It’s a colloquium talk that’s supposed to be accessible for upper-level undergraduates, so I’ll spend a lot of time reviewing the basics… which is good, I think.
I don’t know if the talk will be recorded, but you can see my slides here, and I’ll base this blog article on them.
Hamilton’s equations versus the Maxwell relations
Why do Hamilton’s equations in classical mechanics:
look so much like the Maxwell relations in thermodynamics?
William Rowan Hamilton discovered his equations describing classical mechanics in terms of energy around 1827. By 1834 he had also introduced Hamilton’s principal function, which I’ll explain later.
James Clerk Maxwell is most famous for his equations describing electromagnetism, perfected in 1865. But he also worked on thermodynamics, and discovered the ‘Maxwell relations’ in 1871.
Hamilton’s equations describe how the position and momentum of a particle on a line change with time if we know the energy or Hamiltonian :
Two of the Maxwell relations connect the volume , entropy , pressure and temperature of a system in thermodynamic equilibrium:
Using this change of variables:
become these relations:
These are almost like two of the Maxwell relations! But in thermodynamics we always use partial derivatives:
and we say which variables are held constant:
If we write Hamilton’s equations in the same style as the Maxwell relations, they look funny:
Can this possibly be right?
Yes! When we work out the analogy between classical mechanics and thermodynamics we’ll see why.
We can get Maxwell’s relations starting from this: the internal energy of a system in equilibrium depends on its entropy and volume
Temperature and pressure are derivatives of
Maxwell’s relations follow from the fact that mixed partial derivatives commute! For example:
To get Hamilton’s equations the same way, we need a function of the particle’s position and time such that
Then we’ll get Hamilton’s equations from the fact that mixed partial derivatives commute!
The trick is to let be ‘Hamilton’s principal function’. So let’s define that. First, the action of a particle’s path is
where is the Lagrangian:
The particle always takes a path from to that’s a critical point of the action. We can derive Hamilton’s equations from this fact.
Let’s assume this critical point is a minimum. Then the least action for any path from to is called Hamilton’s principal function
A beautiful fact: if we differentiate Hamilton’s principal function, we get back the energy and momentum :
You can prove these equations using
which implies that
where we integrate along the minimizing path. (It’s not as trivial as it may look, but you can do it.)
Now let’s fix a starting-point for our particle, and say its path ends at any old point . Think of Hamilton’s principal function as a function of just :
Then the particle’s momentum and energy when it reaches are:
This is just what we wanted. Hamilton’s equations now follow from the fact that mixed partial derivatives commute!
So, we have this analogy between classical mechanics and thermodynamics:
What’s really going on in this analogy? It’s not really the matchup of variables that matters most—it’s something a bit more abstract. Let’s dig deeper.
I said we could get Maxwell’s relations from the fact that mixed partials commute, and gave one example:
But to get the other Maxwell relations we need to differentiate other functions—and there are four of them!
• : internal energy
• : Helmholtz free energy
• : enthalpy
• : Gibbs free energy
They’re important, but memorizing all the facts about them has annoyed students of thermodynamics for over a century. Is there some other way to get the Maxwell relations? Yes!
Integrate around a loop :
This says the heat added to a system equals the work it does in this cycle
Green’s theorem implies that if a loop encloses a region
But we know these are equal!
So, we get
for any region enclosed by a loop. And this in turn implies
In fact, all of Maxwell’s relations are hidden in this one equation!
Mathematicians call something like a 2-form and write it as . It’s an ‘oriented area element’, so
Now, starting from
We can choose any coordinates and get
(Yes, this is mathematically allowed!)
If we take we get
We can actually cancel some factors and get one of the Maxwell relations:
(Yes, this is mathematically justified!)
Let’s try another one. If we take we get
Cancelling some factors here we get another of the Maxwell relations:
Other choices of give the other two Maxwell relations.
In short, Maxwell’s relations all follow from one simple equation:
Similarly, Hamilton’s equations follow from this equation:
All calculations work in exactly the same way!
By the way, we can get these equations efficiently using the identity and the product rule for :
Now let’s change viewpoint slightly and temporarily treat and as independent from and So, let’s start with with coordinates . Then this 2-form on :
is called a symplectic structure.
Choosing the internal energy function , we get this 2-dimensional surface of equilibrium states:
for any region in the surface , since on this surface and our old argument applies.
This fact encodes the Maxwell relations! Physically it says: for any cycle on the surface of equilibrium states, the heat flow in equals the work done.
Similarly, in classical mechanics we can start with with coordinates , treating and as independent from and This 2-form on :
is a symplectic structure. Hamilton’s principal function defines a 2d surface
We have for any region in this surface And this fact encodes Hamilton’s equations!
In thermodynamics, any 2d region in the surface of equilibrium states has
This is equivalent to the Maxwell relations.
In classical mechanics, any 2d region in the surface of allowed 4-tuples for particle trajectories through a single point has
This is equivalent to Hamilton’s equations.
These facts generalize when we add extra degrees of freedom, e.g. the particle number in thermodynamics:
or more dimensions of space in classical mechanics:
We get a vector space with a 2-form on it, and a Lagrangian submanifold : that is, a n-dimensional submanifold such that
for any 2d region
This is more evidence for Alan Weinstein’s “symplectic creed”:
As a spinoff, we get two extra Hamilton’s equations for a point particle on a line! They look weird, but I’m sure they’re correct for trajectories that go through a specific arbitrary spacetime point