Suppose there are ‘n’ particles inhabiting 3 dimensional space. Let’s describe the positions of the particles as a single point in 3n dimensional position coordinate space
(Note: We’re working in Cartesian coordinates unless otherwise noted.)
Now imagine an arbitrary vector field in this coordinate space ( is just the coordinate space position variable)
If our particles are in some arbitrary motion, will be a function of time, and the coordinate space point will trace out a path through the vector field . The particular vector at point is
We can take this a step further by assuming that , is a velocity field and that is constrained to move along the streamlines of the field. We can then relate the velocity field with the particle trajectory like this:
This is a form of Burgers’ Equation. Now, an interesting question that we might ask: “Is it possible to find a such that will trace out paths that match those traced out by physical systems? In particular, can we find a velocity field whose streamlines match Newtonian Mechanics?”
Let’s start by letting the masses of the particles equal and define a new set of variables for convenience (let’s call them “extended mass variables”)
The total acceleration vector of our phase space point is going to be:
We are going to concentrate on the component of this vector for simplicity. Newton’s 2nd law, equation (5), and letting V be a Potential Energy function will lead to
Switching over to momentum coordinates by letting , we get
One problem we might encounter with this equation is that streamlines may run into each other and cause “shock” (the vector field will become multi-valued at the point of intersection). To avoid this, we are going to restrict our velocity field to being Conservative, meaning that it is the Gradient of a Scalar Potential (remember, the Gradient of a Scalar function is a vector that points in the direction of steepest ascent). This is going to wind up ensuring that our streamlines never run into each other. We’ll call our Scalar function ‘S’ and let
In other words,
A neat trick this allows us to do is swap indices on the partial derivatives of .
This is a standard theorem from Calculus III; in 3-dimensions it means that the Curl of is zero if and only if is Conservative (See here for more info). This will allow us to transform equation (6) to
A bit more manipulation is going get us to an interesting place:
This is the Hamilton Jacobi Equation (HJE), in some ways the most powerful equation in Classical Mechanics. We will sometimes use an alternate notation for the HJE. Let’s define
as the spatial coordinates for the particle and let
be the Gradient with respect to the particle. Then the HJE looks like this:
To recap, if you solve this for S, you will have a function whose Gradient gives streamlines in phase space that agree with Newton’s Laws. This can be quite difficult to visualize, especially for multi-particle systems. One way to think about it is that we have a very large number (so large that it appears as a continuous fluid in phase space) of sets of ‘n’ particles. Each particle interacts with the other particles in it’s own set, but doesn’t know about the existence of the other sets of particles. The examples in the next section should help with visualizing solutions of the HJE.
By the way, the HJE is usually the last stop (or close to it) in most Classical Mechanics Texts. As such, the typical path to it goes through: Calculus of Variations, Lagrangian Mechanics, Hamiltonian Mechanics, Canonical Transformations, (and possibly other stuff). That is a long way to go and makes it seem more obscure than it actually is. The derivation shown above is the simplest I know of.
Note that since is the momentum of the particle, equation (18) can be rewritten as
where ‘T’ is the Kinetic Energy at a given point in phase space. Evidently, equals -1 times the total Energy at each point. This turns out to always be true if the Potential is Conservative. In that case with -E and worry only about the spatial part of S.
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