The goal for this post is to give a general outline of how to do some very basic Newtonian physics, using the language of symplectic topology (I prefer to say symplectic topology instead of symplectic geometry, since a symplectic manifold has no local invariants). First, we’ll outline the basic kind of problem we care about, that of a particle sitting in a manifold with some forces acting on it. Then, we’ll go over the basic symplectic techniques needed, and finally we’ll state how the law of physics looks like in this setting.
This is a post I have meant to write for a while, since several times now I have thought about writing a post which would assume it as background. I’m also hoping that writing about a topic so dear to my heart helps jog me out of my non-posting funk.
Here is the setup. We start with a Riemannian manifold which we will declare to be ‘space’ and we are interested in understanding how a particle living inside this manifold will move around with various forces acting on it. The forces in question will be as simple as possible:
1) They only depend on the position of the particle.
2) They are constant over time.
Thus, we can turn the forces into a single vector field on , which at each point is the total force felt by the particle. To make this vector field even simpler, we impose another assumption:
3) There is a function on such that the gradient of is the force vector field. The function is called the potential, or the ‘potential energy’.
Many physicists wouldn’t even bother listing this as an assumption, since force fields in nature are always ‘closed’, which means that, on contractible pieces of space, a potential always exists. Therefore, assumption 3 is just saying that the topology of isn’t a problem.
An example is due. Let , two dimensional space. Now imagine we have a very heavy Sun fixed at the origin of this space, and that we want to watch a lighter Planet whiz around this sun. High school physics tells us that the magnitude of the force attracting to is
where is constant, the ‘s are the masses of the celestial bodies and is the position of the Planet. The potential energy of this system is also well known:
It should be noted that the potential is only ever determined up to a constant, and so the potential is usually shifted so that zero is some significant quantity. Here, the potential is set so that the energy approaches zero as the Planet gets infinitely far from the Sun.
What next? We can’t yet determine the motion of our particle in , because in addition to knowing where the particle starts, we also need to know the initial velocity of our particle. In fact, we need to keep track of the velocity of the particle as it moves around , or else we won’t know how it will move forward in time. Therefore, we start thinking about the tangent bundle to , where tangent vectors will denote possible velocities a particle could have at that point. Now, instead of picking a starting position and velocity in , we can just pick a starting point in . Similarly, the only important information to keep track of as the particle moves along is the right point in . In this context, is called the phase space of .
Since where a particle goes depends only on its position in , we can look at how it moves over an infinitesmally small amount of time. This gives a vector pointing in the direction that the particle is starting to move. Doing this everywhere yields a vector field on that tells the whole story of how the particle moves over time: it just flows along this vector field. I feel compelled to give this vector field a name, so why don’t we call it the time evolution vector field.
The question then becomes: How do we construct this vector field from some given forces? We can derive it from , but this takes a bit of work and cleverness; it would be a whole other post (which has probability 1 of being posted in the limit of this blog at infinity). Instead, I will simply declare a law of nature that determines the time evolution vector field from the potential. However, we first will need some tools from symplectic topology.
Assorted Symplectic Topology Tools
This will be a real quick rundown of the relevant facts. First, a symplectic structure on a manifold is a closed, non-degenerate 2-form on . Explicitly, this means that for every point in , we have a function from pairs of tangent vectors to which has a bunch of properties:
1) It is bilinear, that is, linear in each argument.
2) It is skew-symmetric; .
3) It is non-degenerate; .
The meaning of the symplectic form being closed is not visible on the level of tangent spaces and is not relevant for the physics at hand.
One of the most important examples of a symplectic space is that of the cotangent bundle to any smooth manifold . To see this structure, consider the point , where is a point in and is a tangent vector at . The tangent space at decomposes into the sum of the tangent space to at , and the tangent space to the cotangent space at . However, the tangent space to a vector space is isomorphic to the original vector space, so we discover that the tangent space to at is .
The symplectic form on the cotangent bundle is then very simple: Given two tangent vectors to , we can express them as and by the above decomposition. Then . That this defines a bilinear, skew-symmetric, non-degenerate form is straight-forward.
The last tool we need is that of a ‘symplectic gradient’. Given a real-valued function on a manifold , we can take its exterior derivative . This is a 1-form on which takes in a vector field and spits out the directional derivative of along that vector field. However, we want to turn this 1-form into a vector field, perhaps because we hate trying to visualize 1-forms.
In the case of the usual (Riemannian) gradient, we can use the metric to turn into a vector field, by looking for a vector field such that , where is the inner product on tangent vectors coming from the Riemannian structure. The symplectic gradient works exactly the same way. If we have a symplectic form sitting around, we can turn into a vector field with the property that . We call the symplectic gradient of .
The reason we can always find such a vector field (in both cases) is the non-degeneracy of the bilinear form. In general, a non-degenerate bilinear form on a vector space always defines an isomorphism between and its dual space, , by .
The Symplectic Law of Nature
We are almost ready to figure out how to do physics. First, we need a symplectic manifold, or else the last few paragraphs will have been a rather absurd digression. To do this, let us use the fact that a Riemannian structure defines an isomorphism between and (use the just-stated fact that it induces an isomorphism above each point in ). This means that has a symplectic structure, since has one.
Finally, the important theorem:
Theorem. Let be a Riemannian manifold, let be the mass of the particle in question, and let be a potential on . Let be the real-valued function on given by
Then the time evolution vector field $latex v_t$ on of the system is the symplectic gradient of , divided by . That is,
The function in the theorem is called the ‘total energy’ or sometimes the ‘Hamiltonian’, and is called the ‘kinetic energy’.
It should be noted that the above theorem holds no content beyond the simple of Newton; however, this formulation has many advantages. First, it produces this curious quantity called the ‘total energy’ of a state, which you can show without much difficulty is preserved as the particle moves through time. Second, it makes it much easier to use continuous symmetries of the forces to reduce the complexity of solving the flow, by the technique of ‘symplectic quotients’, which I hope to write about soon. Third, the conceptual distance between quantum mechanics and classical mechanics is smaller here; if one takes the limit as , the position and momentum operators on state-space become position and momentum operators on the cotangent bundle, and the symplectic form is the ghost of their commutator.