Symplectic geometry. What's this?

[matematikawan]

My training is in mathematics. But during my free time I also try to understand fundamental physics.

Recently I came across a material which said that the geometry of classical mechanics is symplectic. I'm not sure of the meaning. It was relating to the Hamiltonian which I'm also not familier.

All this while I though that the geometry for Newtonian mechanics is Euclidean, the geometry for quantum mechanic is Hilbert space, the geometry for special relativity is Minkowski and for general relativity ... (don't know which metric space).

To study mechanic via Hamitonian do we really need to understand symplectic geometry? By the way can't we just study mechanics as Issac Newton did without going through the Lagragian or Hamitonian? What good are Lagragian and Hamitonian?

 

[Ben Niehoff]

Symplectic geometry derives from Hamilton's equations. A system can be described at any point in time by a set of N generalized coordinates q_i, and their corresponding generalized momenta p_i (called the "conjugate momenta"). The Hamiltonian is a function H(q, p; t) that is, in most circumstances, equal to the total energy of the system T + U. The equations of motion for the system are then given by

$$\dot q_i = \frac{\partial H}{\partial p_i}$$
$$\dot p_i = - \frac{\partial H}{\partial q_i}$$

So there is a special kind of skew symmetry between q_i and p_i, which is what induces the symplectic geometry of the 2N-dimensional phase space of (q, p). From the Greek roots, "symplectic" means "folded together". Mathematically, it means that the vector space of (q, p) has a non-degenerate anti-symmetric form $\omega$. More explanation on this page:

http://en.wikipedia.org/wiki/Symplectic_vector_space

Using the symplectic form given on that page,

$$\omega = \left( \begin{array}{cc} 0 & I_n \\ -I_n & 0 \end{array} \right)$$

Hamilton's equations of motion can be written succinctly as

$$\vec \eta = \omega \left( \frac{\partial H}{\partial \vec \eta} \right)^T$$

where $\vec \eta$ is the 2N-dimensional column vector (q, p).

As for the question, "Why Hamiltonian or Lagrangian mechanics when they are equivalent to Newton's?", it is for two reasons. One is that it is often much faster to set up a problem correctly in Lagrangian mechanics; it is easy to write down an expression for the energy, and one may forget about vector calculus and just use whatever arbitrary coordinates are convenient.

Second is that Hamilton's and Lagrange's mechanics make use of the quantities of energy and action (action has units of energy*time, or momentum*length), which from a theoretical standpoint are arguably more fundamental than the usual quantities of position and velocity. It is, essentially, a more holistic perspective of the dynamics: specify how the energy behaves, for any arbitrary system, and the whole of its future motion is determined from there.

The Hamiltonian perspective is the basis for all statistical mechanics; i.e., the mechanics of large systems (~ 10^23 bodies), because it is a much simpler way to view such a system. The Hamiltonian also serves as an easy way to see the correspondence between quantum and classical mechanics (however, Feynman's path integral approach uses the Lagrangian instead, as do many other theoretical areas).

Also, in Minkowski space it becomes difficult to concretely define "force" as a useful concept. The Lagrangian method is more universal, and is used instead.

 

[Naty1]

Einstein formulated general relativity used Riemann mathematics...manifolds...geometry...

 

[matematikawan]

Thanks Ben Niehoff. The explanation shed some light on why we need the Lagrangian and Hamiltonian formulation. All this while I have been using a lot of vector mechanics. Much easier to calculate mechanical energy which is a scalar quantity. I agree that it will be difficult to define force in relativity theory.

Need more time to digest the note in wikipedia. It has only definitions. Similarly with the hamiltonian. I will come back later on these.

Symplectic geometry derives from Hamilton's equations.
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The Hamiltonian also serves as an easy way to see the correspondence between quantum and classical mechanics (however, Feynman's path integral approach uses the Lagrangian instead, as do many other theoretical areas).

...

I came across this web site
http://www.scottaaronson.com/democritus/lec9.html
which says that quantum mechanic is not really a physical theory. And it is possible to develope quantum mechanics via the generalization of probability theory rather than from the historical hamiltonian.