In the present thesis we give generalization of analytical mechanics to describe dynamical systems with dissipation. The Lagrangian function in this case is determined by nonstationary pseudo-Riemannian metric for the kinetic energy, and by general quadratic form, nondiagonal in the generalized coordinates and velocities. Skew symmetric nondiagonal terms in our approach play the role of dissipation coefficients. As an application we study in details the classical damped harmonic oscillator. We show that two known formulations of this oscillator, the Bateman dual and the Caldirola Kanai formulations are particular realizations of our general approach. The Hamiltonian formulation and quantization of the model in both representations are given. Moreover Ostrogradsky generalization of Lagrangian and Hamiltonian formalism for description of systems with higher order derivatives and its application to the constant coefficient equations of an arbitrary order are considered. We construct related with the last one the Euler differential equation of an arbitrary order and its Lagrangian and Hamiltonian structure. Quantum Euler systems are introduced and solved for the stationary Schrodinger picture. Nonstationary nonlinear quantum models corresponding to arbitrary Euler Hamiltonian are solved exactly in the Heisenberg picture.