Hamiltonian problems via wavelets

We present applications of wavelet analysis to polynomial approximations for a number of nonlinear problems. According to the orbit method and by using construction from the geometric quantization theory we construct the symplectic and Poisson structures associated with generalized wavelets by using metaplectic structure and corresponding polarization. We consider the generalization of our variational wavelet approach to the case of Hamiltonian systems for which we need to preserve underlying symplectic or Poissonian or quasicomplex structures in any type of calculations. We use our approach for the problem of explicit calculations of Arnold-Weinstein curves via Floer variational approach from symplectic topology. The loop solutions are parametrized by the solutions of reduced algebraical problem-matrix quadratic mirror filters equations. We give wavelet characterization of symplectic Hilbert scales of spaces. We consider wavelet approach to the calculations of Melnikov functions in the theory of homoclinic chaos in perturbed Hamiltonian systems and to computations in constrained dynamical problems.