Variational Formulation of Linear Time-Dependent Invariants

TL;DR

This paper presents a variational approach to derive linear time-dependent invariants for multi-dimensional quadratic systems using classical trajectory-based transformations within Lagrangian and Hamiltonian frameworks.

Contribution

It introduces a novel method to obtain invariants by considering coordinate and momentum variations aligned with classical trajectories, linking symmetry transformations to invariants.

Findings

Derivation of invariants from classical trajectories

Connection between Noether symmetries and invariants

Applicable to multi-dimensional quadratic systems

Abstract

It is shown that linear time-dependent invariants for arbitrary multi\-dimensional quadratic systems can be obtained from the Lagrangian and Hamiltonian formulation procedures by considering a variation of coordinates and momenta that follows the classical trajectory and defines a noetherian symmetry transformation.