The Extended Rigid Body and the Pendulum Revisited
2020, Vol. 16, no. 1, pp. 133-159
Author(s): de la Cruz M., Gaspar N., Linares R.
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.
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