The book is devoted to a mathematical presentation of the analogies that exist between hydrodynamics, geometric optics and mechanics. It turns out that the study of families of trajectories of Hamiltonian systems essentially boils down to the problems of multidimensional hydrodynamics of an ideal fluid. In particular, the well-known Hamilton-Jacobi method corresponds to the case of potential currents. Some applications of this approach are described, in particular, the vortex method of precise integration of differential equations of dynamics.