In my last post, I outlined how solving Newton’s for a single object moving in a static force field could be restated as flowing along a vector field in the tangent bundle (=cotangent bundle). However, if I had written out the details of what flowing along the vector field meant, it would have been clear that it was just a fancy way of saying the particle accelerates in the direction of the force. Is this just elaborate computational legerdemain, where I have used big words and new concepts to misdirect you from the fact that I am just moving the difficulty around?
The aim of this post is to demonstrate that the symplectic perspective has its own merits, by describing the technique of ‘symplectic reduction’. The principle here is a simple and beautiful one: if the mechanical system under consideration has a (good) symmetry, then there is some quantity one can assign to the possible states of the system which is conserved for all time. This meta-theorem is called Noether’s Theorem, though we will not be using it in full generality. Once a ‘conserved quantity’ is found (think of something like angular momentum), together with the symmetry it came from, the mechanical system can be reduced to one on a lower dimensional space.
The starting point is the following observation:
Observation. A function on a symplectic manifold is preserved by the flowing along its symplectic gradient. That is, if is the the flow after time , then .
Proof. A point flows to in time . The difference in at these two points is , the differential of along . However, by the definition of symplectic gradient, this is , which is zero by skew-symmetry. Hence, is constant on the images of after all time. End Proof.
This implies what I mentioned last time, that the total energy function is conserved by time evolution. It is also worth noting that the symplectic form itself is preserved under all symplectic gradient flows.
Ok, so lets see how to use this to our advantage in mechanics. We have a symplectic manifold (which is the tangent/cotangent bundle to space ), and we have a total energy function on , and we want to flow along .
Suppose that we have a 1-dimensional symmetry of this setup; that is, an action of on that fixes and the symplectic form. If we act by a very small number , points in move a very small amount away from themselves, and so in the limit as , we get a vector field on . This vector field generates the action of , in the sense that the action of a number is the same as flowing along for time .
In general, it is hard to say anything, but suppose further that the vector field is the symplectic gradient of some function on . Then the action of is just the flow along this vector field. Since the action of fixes , the infinitesmal action of also fixes , so
Hence, the flow of fixes . Thus, the value of for a particle is conserved as it moves through time; we call such a function a conserved quantity.
Since its value is conserved, we no longer have to think about the whole space when trying to see where a particle starting at point will go; we can restrict our attention to for . We have a problem, though, in that the space is not a symplectic manifold, since it is an odd dimensional manifold (for basic reasons, symplectic manifolds must be even dimensional).
However, we still have one piece of information we haven’t used yet: the symmetry itself. Since the action of preserves by the Observation, acts on . We can quotient by the action of , and the resulting manifold will inherit a well-defined symplectic form. This is is denoted and called the symplectic reduction at . The special case of is called the symplectic quotient of by .
Because the action of preserved , and descend to the quotient to give a function and its symplectic gradient . Now, if we can figure out where a point in goes when we flow along , we can lift this flow to a solution in . Hence, we’ve reduced the mechanical system to a (hopefully) easier problem.
(Note: the following example didn’t come out very well; you might want to skip it)
Lets see how this works for the example I mentioned last time. We had a planet moving in a two dimensional space around a fixed sun . If we write the coordinates of as , the total energy function looks like:
We need a symmetry to work with; the obvious one is rotational symmetry:
To find the infinitesmal action, we find the derivative with respect to at :
Finding a function whose symplectic gradient is this vector field is most easily done with guess and check, so I will leave the computation to the reader. The resulting is:
The reader might recognize this as the angular momentum of the planet around the sun (without the mass term). Therefore, we have discovered that (without some unaccounted for force acting on it), a planet’s angular momentum around its sun will be constant. Woo!
Ok, now lets assume we know our planet’s starting value for is . The first step in symplectic reduction is to restrict our attention to the subset of given by .
Next, we need to quotient out by the action of rotation. This is best done by changing coordinates. The two variables and get replaced by the radius , which is the only quantity that can be discerned after the quotient. Similarly, and are replaced by and . In these new coordinates, the restriction that becomes:
Thus, , and so the only free variables are and . Thus, the symplectic reduction is isomorphic to .
Now, if we were doing things by the book, we would see how the vector field descends to the reduction, and solve the flow there. However, I have a lazier idea. We know that the value of will be preserved by the flow, and the reduction is a two dimensional space. Thus, we know the flow will stay in the one dimensional level sets of , but then there’s no room for anything interesting to happen! The particle will move around these level sets.
The function with the change of coordinates is:
We then add the fact that ,
Declaring that the total energy of the planet is , we get:
This complicated looking formula nevertheless carves out a path in the heavens that the planet must take.
Notice that the symplectic reduction is a new symplectic manifold that isn’t necessarily the cotangent space of any regular manifold. This means we should care about symplectic gradient flows on arbitrary symplectic manifolds (since they might arise as symplectic reductions). We are starting to see that symplectic manifolds are inherently interesting guys, at least as far as classical mechanics is concerned.
Another advantage is that we can potentially find another symplectic reduction. This would amount to finding another symmetry of the system; however, in order for this new symmetry to descend to the symplectic reduction by an old symmetry, the two symmetries must commute. This is equivalent to having a symmetry.
In fact, for symmetries of our system, we don’t have to reduce successive times. We can upgrade our techniques so that we can just do one big reduction. Say, for instance, that we have functions such that the flows along :
1) preserves the energy function , and
2) commute with each other.
The functions are said to be ‘in involution with each other and ‘. What we want to do is come up with a way to specify the fixed values of all of the at once.
If we have such functions, we can cook up the following function :
This map is called a moment map of the system (it is not unique since the ‘s aren’t unique, they can each vary by a constant).
If we pick an element in , then the set is exactly the intersection of the level sets of each of the . This is just a fancy way of picking all the fixed values. By the same logic as above, acts on , and we can define the symplectic reduction at to be the quotient of this action. The function and its gradient descend to this new space, which is likely a much simpler space.
Phew, that was a fair amount of talking, and we didn’t even say anything about the case of a non-abelian symmetry group. Suffice it to say, there is a version of moment maps, conserved quantities and symplectic reduction that works here too and is quite beautiful. I might say more about this stuff in a later post, or I might share with you some of the fun facts about torus actions.