Complex Lagrangian dynamics

TL;DR

The paper extends classical mechanics by introducing a complex Lagrangian framework, defined as mathfrak L = L + i M, to describe non-stationary and dissipative dynamics and to connect with existing complex Hamiltonian formalisms. It develops the complex Euler-Lagrange equations, a complex action principle, and a geometric formulation, illustrating with simple models like inverted and non-stationary oscillators to show how the imaginary part M encodes non-conservative forces. A key result is the demonstrated compatibility between the complex Lagrangian and complex Hamiltonian descriptions, including an explicit mapping of variables and conserved-quantity considerations that generalize Noether-like statements. The approach offers a conceptually simple, unifying framework that holds potential for quantum-field extensions and a deeper treatment of dissipative classical systems.

Abstract

In this article one introduces a formalism of classical mechanics where complex Lagrangian functions are admitted. The results include complex versions of the Lagrangian function, of the Euler-Lagrange equation, of the Hamilton principle, a geometric formulation, and the relation to a previous complex Hamiltonian formalism. The framework is particularly suitable for non-stationary motion, and various pathways can be followed in future investigation.