Classical Mechanics, The Symplectic Way!

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    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 M 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 M, 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 -V  on M such that the gradient of -V is the force vector field.  The function V 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 M isn’t a problem.

An example is due.  Let M=\mathbb{R}^2, two dimensional space.  Now imagine we have a very heavy Sun S fixed at the origin of this space, and that we want to watch a lighter Planet P whiz around this sun.  High school physics tells us that the magnitude of the force attracting P to S is

\frac{Gm_Pm_S}{r^2}=\frac{Gm_Pm_S}{x^2+y^2}

where G is constant, the m‘s are the masses of the celestial bodies and (x,y) is the position of the Planet.  The potential energy of this system is also well known:

V=-\frac{Gm_Pm_S}{r}=-\frac{Gm_Pm_S}{\sqrt{x^2+y^2}}.

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 M, 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 M, or else we won’t know how it will move forward in time.  Therefore, we start thinking about the tangent bundle TM to M, where tangent vectors will denote possible velocities a particle could have at that point.  Now, instead of picking a starting position and velocity in M, we can just pick a starting point in TM.  Similarly, the only important information to keep track of as the particle moves along is the right point in TM.  In this context, TM is called the phase space of M.

    Since where a particle goes depends only on its position in TM, 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 TM 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 F=ma, 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 N is a closed, non-degenerate 2-form on N.  Explicitly, this means that for every point in N, we have a function \{-,-\} from pairs of tangent vectors to \mathbb{R} which has a bunch of properties:

1) It is bilinear, that is, linear in each argument.

2) It is skew-symmetric; \{v,w\}=-\{w,v\}.

3) It is non-degenerate; \forall v, \exists w\; s.t.\; \{v,w\}\neq 0.

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 T^*M to any smooth manifold M.  To see this structure, consider the point (x,v), where x is a point in M and v is a tangent vector at x.  The tangent space at (x,v) decomposes into the sum of the tangent space to M at x, and the tangent space to the cotangent space at x.  However, the tangent space to a vector space is isomorphic to the original vector space, so we discover that the tangent space to T^*M at (x,v) is T_xM\oplus T^*_xM.

    The symplectic form on the cotangent bundle is then very simple: Given two tangent vectors to (x,v), we can express them as w \oplus \alpha and w' \oplus \alpha' by the above decomposition.  Then \{w\oplus \alpha, w'\oplus \alpha'\}=\alpha(w')-\alpha'(w).  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 f on a manifold M, we can take its exterior derivative df.  This is a 1-form on M which takes in a vector field and spits out the directional derivative of f 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 df into a vector field, by looking for a vector field grad(f) such that (grad(f),v)=df(v), 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 df into a vector field X_f with the property that \{X_f,v\}=df(v).  We call X_f the symplectic gradient of f.

    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 \omega on a vector space W always defines an isomorphism between W and its dual space, W^*, by v\in W\rightarrow \omega(v,-)\in W^*.

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 TM and T^*M (use the just-stated fact that it induces an isomorphism above each point in M).  This means that TM has a symplectic structure, since T^*M has one.

    Finally, the important theorem:

Theorem.  Let M be a Riemannian manifold, let m be the mass of the particle in question, and let V be a potential on M.  Let E be the real-valued function on TM given by

E(x,v)=V(x)+\frac{m|v|^2}{2}.

Then the time evolution vector field $latex v_t$ on TM of the system is the symplectic gradient X_{E} of E, divided by m.  That is,

mv_t=X_E.

The function E in the theorem is called the ‘total energy’ or sometimes the ‘Hamiltonian’, and \frac{m|v|^2}{2} is called the ‘kinetic energy’. 

    It should be noted that the above theorem holds no content beyond the simple F=ma 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 \hbar\rightarrow 0, 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.

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15 Responses to “Classical Mechanics, The Symplectic Way!”

  1. Aaron F. Says:

    Oooh, very cool! Is this just a rephrasing of Hamiltonian mechanics, or is it something different? I wish I had the time to comprehend it right now…

    That nonwithstanding, I do have the time to catch some possible typos. 🙂 In the interests of accuracy and flawlessness…

    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.

    I don’t think this is right. There are plenty of force fields in nature that can’t be written as the gradient of a scalar potential—the electric field in the vicinity of an AC power line comes to mind. Of course, this method seems to be geared towards problems in Newtonian gravity and electrostatics, where everything can be represented by scalar potentials… but I wouldn’t generalize that to all of physics!

    It is skew-symmetric; {v,w} = {w,v}

    Either that’s supposed to be {v,w} = -{w,v} or I’m having an extreme senior moment. Which is entirely possible. 😉

  2. John Armstrong Says:

    Aaron, yes, this is Hamiltonian mechanics. In fact, it’s not too far off to say that Hamiltonian mechanics and symplectic topology are the same thing.

    As such, pretty much everything in the world of Hamiltonian mechanics (at least as far as I’ve seen) does have a potential. I don’t know anyone who’d call electrodynamics part of “classical mechanics”, which probably handles your qualm.

  3. Greg Muller Says:

    Its true this is Hamiltonian mechanics, in terms of one equation rather than two. Also, I agree with John in that Hamiltonian mechanics is wholly subsumed by symplectic topology, but whether or not sympectic topology is subsumed by Hamiltonian mechanics is an interesting question. In my mind, Hamiltonian mechanics is the symplectic topology of
    1) cotangent bundles, and
    2) symplectic reductions of cotangent bundles.
    But are there symplectic manifolds that don’t arise this way? I would guess so, but I don’t know an example off hand.

    Yes, I am ruling out electrodynamics by assuming the existance of a potential, but I already ruled it out by assuming that force only depended on position. Still, an elegant framework for handling Maxwell’s equations is given by Yang-Mills theory, which has a similarly geometric flavor.

  4. No wai Says:

    The coadjoint orbits of a Lie Group are symplectic. Their geometric quantization gives interesting insight into the representation of the Lie Group via the Kirillov orbit method. As far as I am aware there is no general method to obtain them via Marsden-Weinstein reduction.

    (Of course, when you go to infinite dimensions there are many non-cotangent symplectic manifolds, such as the space of connections on Riemann surfaces).

    BTW by far and away the best introductory text to the subject is by de Silva, some universities might be able to get it free online from
    http://www.springerlink.com/content/hq3au3baggr3/

  5. John Armstrong Says:

    Greg, I should qualify that a bit. I mean that when people say “Hamiltonian mechanics” these days, they often mean “symplectic geometry, but with a textual cue that in theory this has something to do with physics”.

  6. Henry Cohn Says:

    By the way, I once wrote up as simple-minded an account as I could of where this comes from (motivationally):

    http://research.microsoft.com/~cohn/Thoughts/symplectic.html

    It could be a nice complement to your write-up.

  7. Aaron Bergman Says:

    E&M isn’t any different from the rest of mechanics. You define your field strengths away from the sources and everything is happy. In other words, let your spacetime be the complement of the sources and work there.

  8. Introduction to Symplectic Topology « Rigorous Trivialities Says:

    […] ideas, specifically in the area of symplectic topology. (It appears that the nice folks over at The Everything Seminar has beaten me to it, but I had this mostly written, so I decided to post it anyway.) The outline of […]

  9. jpivarski Says:

    Thanks, Greg! My ears perk up whenever I hear anything about metric geometry and physics (someday, I *will* understand Yang-Mills theory). I look forward to your follow-up!

    By the way, there are a lot of forces which can’t be described as a gradient of a scalar: magnetic force, for instance, is purely curl (divergence-free) and not time-dependent (unlike electric force near AC currents). And then there are the very puzzling forces from Physics I, like friction and normal forces and “ropes,” which adjust themselves to the other applied forces, depending on direction…

  10. Aaron F. Says:

    Yes, I am ruling out electrodynamics by assuming the existance of a potential, but I already ruled it out by assuming that force only depended on position.

    Ohhh, true! I didn’t even think about that.

    By the way, there are a lot of forces which can’t be described as a gradient of a scalar: magnetic force, for instance, is purely curl (divergence-free) and not time-dependent (unlike electric force near AC currents).

    But magnetic forces, of course, depend on velocity. In fact, every purely position-dependent force I can think of off the top of my head is irrotational, although I can’t think of any fundamental reason why that should be so.

    And then there are the very puzzling forces from Physics I, like friction and normal forces and “ropes,” which adjust themselves to the other applied forces, depending on direction…

    Yes, the four fundamental forces of the universe: gravity, normal force, tension, and friction. 🙂

  11. Maurice de Gosson Says:

    What makes the importance of the symplectic approach to classical mechanics is (besides its formal simplicity) the “principle of the symplectic camel” which is a considerable refinement of Liouville”s theorem: during a Hamiltonian motion, it is not just phase space volume that is preserved, but also the symplectic capacity. It is (I do not want to be too technical) as if quantum mechanics has left an imprint in the classical world (or the other way round!). Feel free to go to m y homepage and download a few papers I have written about these fascinating things.
    http://www.freewebs.com/cvdegosson
    Maurice

  12. Quijote Says:

    Explicitly, this means that for every point in N , we have a function {-,-} from pairs of tangent vectors to R which has a bunch of properties

    Don’t you mean “tangent vectors to N” instead of R?

    The tangent space at decomposes into the sum of the tangent space to M at x, and the tangent space to the cotangent space at x.

    Don’t we need a connection for this?

  13. Quijote Says:

    Sorry, the first is a simple misunderstanding. Of course R denotes he range of that function. But the second question is still holds.

  14. Anonymous Grad Says:

    To answer Quijote’s (second) question – the author isn’t being entirely precise here. The statement

    The tangent space at (x,v) decomposes into the sum of the tangent space to M at x, and the tangent space to the cotangent space at x.

    has a couple of things wrong with it (I imagine in an effort to gloss over some of the technicalities). Just think of it as a very local statement. For example, something like this is true over a coordinate patch on M.

    Also, in response to no wai’s comment, the coadjoint orbits of a Lie group G can be very naturally obtained as the symplectic reduction of the cotangent bundle T*G, where G acts on T*G via the lifted action. In fact, this example was discussed in the original paper by Marsden and Weinstein.

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