The book is devoted to mathematical description of damped oscillations and their relation with the Euler equation. This equation has been studied by Leonard Euler starting from 1740 and also is known as the Cauchy-Euler equidimensional equation. To develop the Lagrangian and the Hamiltonian formalism extension of analytical mechanics to pseudo-Riemannian metrics is introduced. Depending on self-adjoint properties of variational functional, two different descriptions are presented. One of them, called the dual Bateman system, supply a damped system with its time reversal image. The second one imitates damping by the time dependent mass of the system. Quantization of damped oscillator and wide class of exactly solvable nonlinear quantum Euler problems are treated in details. Intriguing relations with Chern-Simons topological quantum mechanics and soliton theory are indicated. The book is self contained and pedagogically written as an introduction to the subject for students at advanced undergraduate and graduate level from science and engineering specialties. Also it could be a valuable source for researchers in the field.

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