Sergei Yakovenko's blog: on Math and Teaching

Monday, February 6, 2017

Lecture 13 (Jan 30, 2017)

Symplectic manifolds

In parallel with the Riemannian manifolds equipped with a positive definite (symmetric) scalar product on each tangent space, it is interesting to consider manifolds equipped with an antisymmetric scalar product on each tangent space, i.e., with a differential 2-form \omega\in\Omega^2(M). This form is called a symplectic structure, if \mathrm d\omega=0 and an additional nondegeneracy condition is met.

It turns out that this structure naturally arises on the cotangent bundle M=T^*N of an arbitrary smooth manifold N. Moreover, this structure is intimately related with the mechanics of frictionless systems: the Hamiltonian differential equations can be naturally described by vector fields X which satisfy the Hamiltonian condition \mathrm i_X\omega=\mathrm d H, where H is a function (Hamiltonian, or full energy) on the symplectic manifold. Thus each Hamiltonian vector field is “encoded” by a single function, rather than by a tuple of functions. The commutator of Hamiltonian vector fields is again Hamiltonian: this is the invariant definition of the Poisson bracket.

There are two instant ramifications from this point. One can discuss integrability of the Hamiltonian vector fields. Another, less physically motivated direction is to study the symplectic geometry, first locally, then globally. It is a surprising twisted counterpart of the Riemannian geometry, which has no intrinsic curvature but nevertheless is very rich globally.

The lecture notes will be available later.

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