Data for: On Lagrangian treatment for dissipative systems

With a positive Leibniz defect the amplitude of free vibration remains constant with time with the fading frequency. The negative Leibniz defect leads the opposite behavior, demonstrating the growing frequency (“blue shift”). Notably that the Hamiltonian remains constant in time in both cases.

Thus the introduction of non-zero Leibniz defect leads to an alternative mathematical description of the conservative systems with some uncommon physical properties. Consider an ensemble of states that occupies a particular volume of phase space in the initial moment. The evolution of the volume of phase space is governed by Hamilton’s equations.

In the present lecture an alternative way is investigated. Instead of the modification of the Hamilton’s function, we examine the alteration of the derivatives. The formalism is based on the generalized differentiation operator (kappa-operator) with a non-zero Leibniz defect.

The Leibniz defect of the generalized differentiation operator linearly depends on one scaling parameter. In a special case, if the scaling parameter turns to one, the Leibniz defect vanishes and generalized differentiation operator reduces to the common differentiation operator. The generalized differentiation operator allows the formulation of the variational principles and corresponding Lagrange and Hamiltonian equations.

The developed formalism allows the representation of the mechanical systems with the Lie group methods. The analytical solution of the equations of non-Leibniz oscillator is found. As an example of the partial differential equation with the generalized derivatives the wave is studied.

Lecture delivered of 89th GAMM Meeting, 22/03/2018, Technical Universitz of Munich