# A Theoretical Framework for Lagrangian Descriptors

## TL;DR

This paper establishes a rigorous theoretical foundation for Lagrangian Descriptors (LDs), proving their effectiveness in detecting invariant manifolds and other phase space structures in various dynamical systems.

## Contribution

It introduces an alternative definition for LDs and rigorously proves their ability to identify stable and unstable manifolds in specific 2D hyperbolic saddle cases, extending to invariant sets like n-tori.

## Key findings

- LDs reliably detect invariant manifolds in 2D hyperbolic saddle points.
- LDs can highlight additional invariant sets such as n-tori.
- Application to classical examples demonstrates the method's effectiveness.

## Abstract

This paper provides a theoretical background for Lagrangian Descriptors (LDs). The goal of achieving rigourous proofs that justify the ability of LDs to detect invariant manifolds is simplified by introducing an alternative definition for LDs. The definition is stated for $n$-dimensional systems with general time dependence, however we rigorously prove that this method reveals the stable and unstable manifolds of hyperbolic points in four particular 2D cases: a hyperbolic saddle point for linear autonomous systems, a hyperbolic saddle point for nonlinear autonomous systems, a hyperbolic saddle point for linear nonautonomous systems and a hyperbolic saddle point for nonlinear nonautonomous systems. We also discuss further rigorous results which show the ability of LDs to highlight additional invariants sets, such as $n$-tori. These results are just a simple extension of the ergodic partition theory which we illustrate by applying this methodology to well-known examples, such as the planar field of the harmonic oscillator and the 3D ABC flow. Finally, we provide a thorough discussion on the requirement of the objectivity (frame-invariance) property for tools designed to reveal phase space structures and their implications for Lagrangian descriptors.