Non‐Leibniz Hamiltonian and Lagrangian formalisms for certain class of dissipative systems

Abstract

The non‐Leibniz Hamiltonian and Lagrangian formalism for the certain class of dissipative systems is introduced in this article. The formalism is based on the generalized differentiation operator (κ‐operator) with a nonzero Leibniz defect. The Leibniz defect of the introduced operator linearly depends on one scaling parameter. In a special case, if the Leibniz defect vanishes, the generalized differentiation operator reduces to the common differentiation operator. The κ‐operator allows the formulation of the variational principles and corresponding equations of Lagrange and Hamiltonian types for dissipative systems. The solutions of some generalized dynamical equations are provided closed form.

Dissipation in Lagrangian Formalism

“…At this point, some kind of generalization is necessary if one wishes to obtain a differential equation that cannot be derived from a Lagrangian. One way is to generalize the functional derivative to a fractional order [ 1 , 2 , 3 , 4 , 5 , 6 ]. Another way is to increase the degrees of freedom, which is the basis of many different methods.…”

On the Lagrangians and potentials of a two coupled damped Duffing oscillators system and their application on three-node motif networks

“…In this work, we have shown that it is possible to construct Lagrangians [34,35] for two coupled damped Duffing oscillators both directionally and bi-directionally. In general it is not possible to obtain a potential for dissipative systems.…”