# The rigid body and the pendulum revisited

## TL;DR

This paper explores the $SL(2, ext{R})$ symmetry in the Euler equations to derive the simple pendulum from the rigid body, revealing new geometric configurations and the possibility of pendulum systems with imaginary time.

## Contribution

It extends previous work by classifying all Casimir-based constructions of the pendulum, including hyperbolic geometries and systems with imaginary time, using algebraic and geometric methods.

## Key findings

- Multiple Casimir configurations produce the pendulum.
- The pendulum can be derived from $ISO(2)$ and $ISO(1,1)$ Lie algebras.
- New momentum maps for pendulum with imaginary time.

## Abstract

In this paper we revisit the construction by which the $SL(2,\mathbb{R})$ symmetry of the Euler equations allows to obtain the simple pendulum from the rigid body. We begin reviewing the original relation found by Holm and Marsden in which, starting from the two Casimir functions of the extended rigid body with Lie algebra $ISO(2)$ and introducing a proper momentum map, it is possible to obtain both the Hamiltonian and equations of motion of the pendulum. Important in this construction is the fact that both Casimirs have the geometry of an elliptic cylinder. By considering the whole $SL(2,\mathbb{R})$ symmetry group, in this contribution we give all possible combinations of the Casimir functions and the corresponding momentum maps that produce the simple pendulum, showing that this system can also appear when the geometry of one of the Casimirs is given by a hyperbolic cylinder and the another one by an elliptic cylinder. As a result we show that from the extended rigid body with Lie algebra $ISO(1,1)$, it is possible to obtain the pendulum but only in circulating movement. Finally, as a by product of our analysis we provide the momentum maps that give origin to the pendulum with an imaginary time. Our discussion covers both the algebraic and the geometric point of view.