The Extended Rigid Body and the Pendulum Revisited

In this paper we revisit the construction by which the $SL(2,\mathbb{R})$ symmetry of the Euler equations allows a simple pendulum to be obtained from a rigid body. We begin by reviewing the original relation found by Holm and Marsden in which, starting from the two integrals of motion of the extended rigid body with Lie algebra $\mathfrak{iso}(2)$ and introducing a proper momentum map, it is possible to obtain both the Hamiltonian and the equations of motion of the pendulum. Important in this construction is the fact that both integrals of motion 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 integrals of motion and the corresponding momentum maps that produce the simple pendulum, showing that this system can also appear when the geometry of one of the integrals of motion is given by a hyperbolic cylinder and the other by an elliptic cylinder. As a result, we show that, from the extended rigid body with Lie algebra~$\mathfrak{iso}(1,1)$, it is possible to obtain the pendulum, but only in circulating movement. Finally, as a byproduct 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.