# Energy Distribution in Intrinsically Coupled Systems: The Spring Pendulum Paradigm

## TL;DR

This paper introduces a new analytical approach to study energy exchange in nonlinear coupled systems, exemplified by the spring pendulum, revealing how energy distribution varies with system parameters and coupling strength.

## Contribution

The paper presents a novel method to analyze energy components and their exchange in nonlinear coupled systems, demonstrated through the bi-dimensional spring pendulum model.

## Key findings

- Identified three energy components in the spring pendulum system.
- Analyzed energy exchange across different trajectories and phase space.
- Mapped regions of strong and weak coupling in parameter space.

## Abstract

Intrinsically nonlinear coupled systems present different oscillating components that exchange energy among themselves. We present a new approach to deal with such energy exchanges and to investigate how it depends on the system control parameters. The method consists in writing the total energy of the system, and properly identifying the energy terms for each component and, especially, their coupling. To illustrate the proposed approach, we work with the bi-dimensional spring pendulum, which is a paradigm to study nonlinear coupled systems, and is used as a model for several systems. For the spring pendulum, we identify three energy components, resembling the spring and pendulum like motions, and the coupling between them. With these analytical expressions, we analyze the energy exchange for individual trajectories, and we also obtain global characteristics of the spring pendulum energy distribution by calculating spatial and time average energy components for a great number of trajectories (periodic, quasi-periodic and chaotic) throughout the phase space. Considering an energy term due to the nonlinear coupling, we identify regions in the parameter space that correspond to strong and weak coupling. The presented procedure can be applied to nonlinear coupled systems to reveal how the coupling mediates internal energy exchanges, and how the energy distribution varies according to the system parameters.