The rigid body and the pendulum revisited
Manuel de la Cruz Néstor Gaspar Román Linares





fisikito@gmail.com


nex3t.gr@gmail.com




 
lirr@xanum.uam.mx

Departamento de Física, Universidad Autónoma Metropolitana Iztapalapa,

San Rafael Atlixco 186, C.P. 09340, Ciudad de México, México,




            


In this paper we revisit the construction by which the SL(2,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,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.


The extended rigid body and the pendulum revisited
    Román Linares
    December 30, 2023
==================================================





§ INTRODUCTION



The simple pendulum and the torque free rigid body are two well understood physical systems in both classical and 
quantum mechanics. The first systematic study of the pendulum is attributed to Galileo Galilei around 1602 and its 
dynamical description culminated  with the development of the elliptic functions by Abel <cit.> and Jacobi 
<cit.>, which turn out to be the analytical solutions to the equation of motion of the pendulum 
(for a review of elliptic functions see for instance <cit.>  and 
<cit.> for the solutions of the pendulum). The quantization of the pendulum is based on the 
equivalence  between the Schrödinger equation and the Mathieu differential equation, result developed originally by 
Condon in 1928 <cit.> and source of subsequent  analysis of different aspects of the quantum system 
<cit.>. On the other hand in 1758 Euler showed that the equations of 
motion that describe the rotation of a rigid body form a vectorial quasilinear first-order ordinary differential equations set 
<cit.>. A geometric construction of the solution was given later on by Poinsot <cit.> and analytically these 
solutions are given, as for the simple pendulum, by elliptic functions (see for instance 
<cit.> and references therein). The quantization of the problem was attacked first by 
Kramers and Ittmann <cit.> and since then many authors have contributed to understand deeper many aspects 
of the problem <cit.>.

Despite the old age of these problems, from time to time there are some new physical aspects uncovered about these 
systems that contribute to our knowledge and understanding of physics in general. The list is long and here we point 
out just four examples: i) In 1973 Y. Nambu, taking the Liouville theorem as a guiding  principle, proposed a 
generalization of the classical Hamiltonian dynamics by supersede the usual two dimensional phase space 
with a n-dimensional one  <cit.>. The dynamics in the new phase space was formulated via an 
n-linear fully antisymmetric Nambu bracket with two or more “Hamiltonians"; as an example, Nambu applied his 
formalism to the free rigid body. ii) Hinted by an article by Deprit <cit.> and using the 
SL(2,ℝ) symmetry of the Euler equations, in 1991 Holm and Marsden built a new Hamiltonian in such a 
way that the new dynamical system, so called in the literature extended rigid body,  can be written as a Lie-Poisson 
system whose different Lie algebras structures can be SO(3), ISO(2) or Heis_3 <cit.>; particularly  
interesting is the ISO(2) case, where the phase space of the Eulerian top is filled with invariant elliptic cylinders, on 
each of which, the dynamics, in elliptic  coordinates, is the dynamics of a standard simple pendulum. iii) In 1995 R. 
Montgomery  computed the change in the geometric phase for the attitude of the rigid body when the angular 
momentum vector in the body frame performs one period of its motion <cit.>. iv) Finally in 2017 
<cit.>  the free rotation of a classical rigid body was used in the control of two-level quantum systems by 
means of external electromagnetic pulses. In particular, authors showed that the dynamics of a rigid body can be used 
to implement one-qubit quantum gates. 

In this paper we are interested in explore deeper the relation between the extended rigid body and the simple 
pendulum.  As we have mentioned above, in the original paper <cit.> Holm and Marsden showed that the 
different Lie algebras structures of the extended rigid body are SO(3), ISO(2) or Heis_3, however in 
<cit.> authors showed that the complete list of possible Lie algebras are all the ones related to SO(3) 
via analytical continuation and group contractions, which means that the algebra ISO(1,1) must be also included. 
Even more, according to <cit.> the pendulum can be obtained from the extended rigid body if, from the 
geometrical point of view, the surfaces of the two new Casimir functions have the shape of elliptic cylinders, which 
leads to the fact that the corresponding Lie algebra is ISO(2). Very recently using a representation of the rigid body 
in terms of two free parameters e_0 and κ <cit.> instead of the usual five parameters: 
energy E, magnitude of the angular momentum L, and the three principal moments of inertia I_1, I_2 and 
I_3, the classification of the inequivalent SL(2,ℝ) combinations of the Casimir functions  was discussed 
both algebraically and geometrically. It turns out that whereas the geometry of the Casimir function that 
represents the square of the angular momentum continues being an S^2 sphere, the geometry of the Casimir 
function associated to the kinetic energy which is an ellipsoid in the five parameters representation of the rigid body, is 
replaced by an elliptic hyperboloid that can be either of one or two sheets depending on the numerical values of both 
e_0 and κ, making the classification of the SL(2,ℝ) combinations of the Casimir functions richer. 
A result of this paper is to show that, considering all possible different geometries of the new Casimir functions, it is 
possible to obtain the pendulum also when the geometries associated to them are an elliptic cylinder and a hyperbolic 
cylinder. Specifically, when the cotangent space of the simple pendulum is given by the elliptic cylinder, the Lie algebra 
of the Hamiltonian vector fields associated to the coordinates in the rigid body-fixed frame is ISO(2), whereas if the 
cotangent space is given by the hyperbolic cylinder, the Lie algebra is ISO(1,1). Even more, we show 
explicitly that for the ISO(2) case we always obtain the whole set of motions of the pendulum whereas for the  
ISO(1,1) case we get the circulating motions but not the oscillatory ones. To the best of our knowledge this 
result has not been discussed previously in the literature. 
 
Our exposition is as self-contained as possible. Section <ref> is dedicated to summarize the main 
characteristics of the rigid body and its equations of motion, specially the SL(2,ℝ) symmetry 
of the later. In section <ref> we discuss the Hamiltonian structure of the simple pendulum as function of 
both a real time and an imaginary time.  The study of the whole SL(2,ℝ) transformations can be 
divided in three general sets where each set contains different fixed forms of  the SL(2,ℝ) matrices.  
One of the three sets does not give origin to the pendulum and therefore we focus on the other two. Section 
<ref> is devoted to the set of matrices where the relation between the rigid body and the simple pendulum can 
be stablished. This set can be subdivided in two general cases determined by the geometry of the Casimir functions, 
subsection  <ref> is dedicated to the case where both Casimir functions are given geometrically by elliptic 
cylinders whereas subsections <ref> and <ref> are dedicated to the study of the cases where 
one Casimir function has the geometry of an elliptic cylinder and the another Casimir function has the geometry of a 
hyperbolic cylinder. In section <ref> we discuss the third and last set; as we will argue, this set is a limiting 
case of the one in section <ref> and therefore there does also exist a relation between the simple pendulum and 
the rigid body. Our conclusions are given in section <ref>.







§ THE EULER EQUATIONS AND ITS SL(2,ℝ) SYMMETRY
 





The torque free rigid body motion is one of the best understood systems in physics and the amount of papers and 
books discussing their dynamical properties is overwhelming. However there are still some properties related to this  
system that deserve to be explored further. In this section we give a short summary of the characteristics of the system 
that are relevant for our analysis of the relation between the so called extended rigid body and  the 
simple pendulum. We base our discussion in <cit.>.




 §.§ The Euler equations and the Casimir functions



It is well known that in a body-fixed reference frame the motion of the rigid body is governed by the Euler equations 
(see for instance <cit.>)

    d L⃗/dt = L⃗× I^-1L⃗,

where L⃗ is the vector of angular momentum and I is the moment of inertia tensor. If  the body-fixed frame is 
oriented to coincide with the principal axes of inertia, the tensor I is diagonal. Without losing generality in the 
following discussion we consider that the principal moments of inertia satisfy the inequality

    I_1 < I_2 < I_3.

When written in a basis, the L_i  components of the angular momentum are the generators of a  SO(3) Lie 
algebra

    [L_i,L_j]=ε_ijk L_k.

The system has two Casimir functions, the rotational kinetic energy C_1 and the square of the angular momentum 
C_2, which are given in terms of the moments of inertia and the components of L⃗ as

    C_1(L_1, L_2, L_3) ≡L_1^2/2I_1+  L_2^2/2I_2+ L_3^2/2I_3,

and

    C_2(L_1,L_2,L_3) ≡  L_1^2+L_2^2+L_3^2,

respectively. The dynamical  problem is usually solved using the Poinsot construction, in an Euclidean space 
ℝ^3 whose coordinates are the components of the angular momentum. When the vector L⃗ moves 
relative to the axes of inertia of the top, it  lies along the curve of intersection of the surfaces C_1=E = constant (an 
ellipsoid with semiaxes √(2EI_1), √(2EI_2) and √(2EI_3)) and C_2=L^2 = constant (a sphere of 
radius L). Because 2EI_1< L^2 < 2EI_3, the radius of the sphere has a value between the minimum and 
maximum values of the semiaxes of the ellipsoid. At first sight the solutions of the Euler equations (<ref>) depend 
on five different parameters, the three moments of inertia, the energy E and the square of the angular momentum 
L^2. However it has been shown that the problem can be rewritten in such a way that the solutions depend   
only on two parameters <cit.>. The first parameter e_0 is related to the 
quotient E/L^2 whereas the second parameter κ codifies the values of the three moments of inertia, 
specifically, if I is the diagonal matrix I= diag(1/I_1, 1/I_2, 1/I_3), then the relation between the five 
parameters: { I_i}, E and L^2 with the four parameters: { e_i} and e_0 is given by

    1/I_i-1/3  I≡√(g_2/3) e_i 2E/L^2 - 1/3  I≡√(g_2/3) e_0,
  
with i=1,2,3 and

    g_2 ≡2/3[  ( 1/I_1- 1/I_2)^2 + ( 1/I_2- 1/I_3)^2   +
    ( 1/I_3- 1/I_1)^2  ] >0 .

Notice that the new four parameters are dimensionless, and abusing of the language we can call to the set {e_ i} 
dimensionless inertia parameters, analogously although e_0 is not the energy of the system we can 
call it the dimensionless energy parameter.  The dimensionless inertia parameters can be written in terms of only 
one angular parameter κ in the form

    e_1= cos(κ)  ,      e_2= cos(κ-2π/3) ,        e_3= cos(κ+2π/3),

and they are restricted to satisfy the conditions

    e_1+e_2+e_3=0,  e_1^2+e_2^2+e_3^2=3/2  
    e_1e_2e_3=(cos 3 κ) /4 .

Geometrically, in the three dimensional space { e_1,e_2,e_3}, the first condition in (<ref>) represents a 
plane crossing the origin and the second condition represents a sphere of radius √(3/2). The intersection of these 
two surfaces is a circle which is parameterized by the angular parameter κ. The role of g_2 (equation 
(<ref>)) is to change the size of the sphere and therefore the size of the intersecting circle. It is clear that given a 
specific rigid body or equivalently a set of values for the three principal moments of inertia I_i, the value of the 
angular parameter κ is completely determined.

In general κ∈ [0,2π], but notice that the condition (<ref>) in terms of the dimensionless inertia 
parameters is obtained if the free parameter κ takes values in the subinterval κ∈ (0,π/3), i.e. 

    e_3 < e_2 < e_1.

The other five subintervals of length π/3 produce the other five possible orders for the e_i's, for instance, if  
κ∈ (π/3,2π/3) the parameters ordering is e_3<e_1<e_2, etc. For any value of κ, at least one of 
the inertia parameters is positive, another is negative and the third one may be either, positive, null or negative. 
Throughout  all the paper we work in the interval κ∈ (0,π/3), for which e_1>0, e_3<0 and e_2 
can be either positive, negative or null.
 
Finally introducing the dimensionless coordinates u_i ≡ L_i/L, the Euler equations (<ref>) are rewritten as

    d u_i/dt  = 1/2√(g_2/3)  L  ε_ijk (e_k-e_j) u_j u_k .

It is possible to absorb the factor √(g_2/3)  L, by defining a dimensionless time parameter: 
x ≡ t  √(g_2/3)  L, obtaining 

    u̇_i =1/2ε_ijk (e_k-e_j) u_j u_k  ,

where (·) denotes derivative with respect to the dimensionless time x. In a similar fashion Casimir functions 
(<ref>) and (<ref>) read now as

    C_1(u_1,u_2,u_3)    ≡     e_1 u_1^2 + e_2 u_2^2 + e_3u_3^2,                                                     
    
    C_2(u_1,u_2,u_3)    ≡    u_1^2+u_2^2+u_3^2.

In the dimensionless angular momentum space ℝ^3: (u_1,u_2,u_3), C_2=1 and it 
represents a unitary sphere, whereas for the other Casimir: C_1=e_0 and the ellipsoid (<ref>), is replaced by an elliptic hyperboloid that can be either of one or two sheets depending on the numerical values of both 2E/L^2 and κ, or equivalently on the number of positive and negative coefficients (dimensionless inertia parameters) in the equation. In this latter case the Casimir surface can also have the geometry of an elliptic cone or a hyperbolic cylinder in the proper limit situations. A complete classification of the geometrical shapes of the Casimir surface C_1 can be 
found in <cit.>. 

Regarding the solutions of the Euler equations (<ref>), given the ordering (<ref>) of the 
dimensionless inertia parameters, the solutions depend of the relative value between e_0 and e_2. Here 
we are not going into details, but for purposes of completeness in our exposition, we present the explicit solutions, 
details can be found for instance in <cit.>.


  I. Case e_3<e_2<e_0<e_1   (i.e. 1/I_2<2E/L^2).

When the energy parameter e_0 is between the inertia parameters e_2 and e_1, the solutions are given by 

    u_1(τ)=  (τ ' ,m_c)  (τ,m), 
    u_2(τ)= (τ ' ,m_c)   (τ,m), 
    u_3 (τ)= (τ ' ,m_c)  (τ,m),

where τ is a dimensionless time parameter defined as

    τ= x √((e_1-e_2)(e_0-e_3)).

Here the amplitudes of the solutions are written as Jacobi elliptic functions at parameter τ ' = constant and are 
related to the dimensionless parameters e_i and e_0 in the form

    ^2(τ ' ,m_c) = e_0-e_3/e_1-e_3 , ^2(τ ' ,m_c) = e_1-e_0/e_1-e_3 , ^2(τ ' ,m_c)  = e_1-e_0/e_1-e_2.

The square modulus m and the complementary modulus m_c in the equation (<ref>) are given by

    m^2≡(e_2-e_3)/(e_0-e_3)(e_1-e_0)/(e_1-e_2)  = (e_1-e_0)/(e_0-e_3) k_1^2/k_2^2,  
    m_c^2 ≡(e_0-e_2)/(e_0-e_3)(e_1-e_3)/(e_1-e_2)= (e_0-e_2)/(e_0-e_3) 1/k_2^2,

which take values in the interval 0<m^2<1 and 0<m_c^2<1 and satisfy m^2+m_c^2=1. Here the quotients

    k_1^2=e_2-e_3/e_1-e_3 ,
     
    k_2^2=e_1-e_2/e_1-e_3,

satisfy in a similar way k_1^2+k_2^2=1.


  II. Case e_3<e_0<e_2<e_1   (i.e.  2E/L^2<1/I_2).

In this case the solutions are given by 

    u_1=( τ ' , im_c/m)  ( mτ,1/m),
    u_2=( τ ' , im_c/m)   ( mτ,1/m), 
    u_3 =( τ ' , im_c/m)  ( mτ,1/m),

where the square modulus of the elliptic function take values in the interval
0<1/m^2 < 1, with m^2 as defined in (<ref>) but due to the fact that e_3<e_0<e_2<e_1, 
m^2>1. The time parameter  τ is the one defined in (<ref>), whereas the amplitudes are written as Jacobi 
elliptic functions at parameter τ ' = constant

    ^2( τ ' , im_c/m) = e_1-e_0/e_1-e_3 , ^2 ( τ ' , im_c/m) = e_0-e_3/e_1-e_3 , ^2 ( τ ' , im_c/m)  = e_0-e_3/e_2-e_3.

Physical interpretation of the solutions is straightforward, the curves (<ref>) and (<ref>) are the 
parameterization of the intersection of the unitary sphere of angular momentum and the corresponding surface 
of the Casimir function (<ref>) <cit.> (see figure <ref>). 

At this point it is convenient to make clear the possible values that can take the square modulus m^2, the 
complementary modulus m_c^2, their inverse values and the quotients of these quantities. The reason for it, is 
that these quantities will appear when the connection between the rigid body and the simple pendulum be established.


Notice that, in an analogous way to the fact that m^2 and m_c^2 are complementary to each other in the sense that m^2+m_c^2=1, the couple 1/m^2 and -m_c^2/m^2 are complementary to each other, as well as the couple 
1/m_c^2 and -m^2/m_c^2, i.e.

    m^2+m_c^2=1, 1/m^2-m_c^2/m^2=1, 1/m_c^2-m^2/m_c^2=1.





 §.§ SL(2 , IR) symmetries of the Euler equations



In order to make manifest the gauge symmetries of the Euler equations, we notice that its dimensionless  form

    u̇⃗̇= u⃗×ϵu⃗

with ϵ a diagonal matrix of the form ϵ =diag(e_1,e_2,e_3), can be rewritten as the gradient of 
two scalar functions

    u̇⃗̇=∇ l  ×∇ h,

where

    h≡1/2 C_1 (u_1, u_2, u_3)- 1/2 e_0 =0    ⇒   ϵu⃗= ∇ h = 
     (e_1u_1,e_2u_2,e_3u_3),       
    
    l ≡1/2 C_2 (u_1, u_2, u_3) - 1/2 =0    ⇒   u⃗=∇ l = (u_1,u_2,u_3) .

This form of writing the Euler equations makes explicit its invariance under any SL(2,  IR)  transformation 
<cit.>.
It is straightforward to check that the transformation

    ( 
    [ H; N ]) = ( 
    [ a b; c d ])
     ( 
    [ h; l ]),

with ad-bc=1 leads to

    u̇⃗̇=∇ N×∇ H.

At this point we can consider either of the new Casimir surfaces  N  or H as the Hamiltonian 
surface. The systems that have this property are called bi-hamiltonian (see for instance 
<cit.>). Usually the one that is chosen as the hamiltonian surface is H, 
thus, given a dynamical system with Hamiltonian H, any dynamical quantity Q evolves with time according to

    Q̇=X_ HQ,

where the generator X_G is given by

    X_G=( ∇ N×∇ G) ·∇.

Notice that we are denoting with a different letter to the Hamiltonian or Casimir function: 
H=H(u_1,u_2,u_3) and to the Casimir surface: H (u_1,u_2,u_3) ≡ H (u_1,u_2,u_3)- E=0.

According to the transformation (<ref>), the generator X_G in components form is expressed as

    X_G    =   (ce_1+d)u_1( ∂ G/∂ u_2∂/∂ u_3 -  
    ∂ G/∂ u_3∂/∂ u_2) + 
    (ce_2+d)u_2( ∂ G/∂ u_3∂/∂ u_1 -  
    ∂ G/∂ u_1∂/∂ u_3) 
           + (ce_3+d)u_3( ∂ G/∂ u_1∂/∂ u_2 -  
    ∂ G/∂ u_2∂/∂ u_1).

In order to determine the structure of the Lie algebra, we calculate the Lie-Poisson brackets for the Hamiltonian vector fields associated with the coordinate functions u_i

    X_u_1   =    (ce_3+d) u_3 ∂_2- (ce_2+d) u_2 ∂_3, 
    
    X_u_2   =    (ce_1+d) u_1 ∂_3- (ce_3+d) u_3 ∂_1, 
    
    X_u_3   =    (ce_2+d) u_2 ∂_1- (ce_1+d) u_1 ∂_2.

Explicitly, the Lie-Poisson brackets  are given by

    [X_u_1,X_u_2]=(ce_3+d)X_u_3,  
    [X_u_2,X_u_3]=(ce_1+d)X_u_1, 
    [X_u_3,X_u_1]=(ce_2+d)X_u_2.

Regarding the classification of the different Lie algebras that arise from the SL(2,  IR) invariant orbits, it turns out that, besides the SO(3) Lie algebra (<ref>) which corresponds to the rigid body, the algebras 
SO(2,1), ISO(2), ISO(1,1) and Heis_3, also emerge. The systems corresponding to these algebras are termed under the name extended rigid bodies. The whole classification can be found, for instance, in 
<cit.>.

Because is relevant for our discussion, it is important to emphasize that if instead we take the Casimir surface 
N as the Hamiltonian surface, then the generator X_G is given by

    X_G= - ( ∇ H×∇ G) ·∇⇒Q̇=X_ NQ.

Clearly X_ N=-X_ H. If we denote as X̃_u_i to the Hamiltonian vector fields associated to the 
coordinates, we obtain in this case

    X̃_u_1   =    -(ae_3+b) u_3 ∂_2+ (ae_2+b) u_2 ∂_3, 
    X̃_u_2   =    -(ae_1+b) u_1 ∂_3+ (ae_3+b) u_3 ∂_1, 
    X̃_u_3   =    -(ae_2+b) u_2 ∂_1+ (ae_1+b) u_1 ∂_2.

With these vector fields the Lie-Poisson brackets are given explicitly by

    [X̃_u_1,X̃_u_2]=-(ae_3+b)X̃_u_3,  
    [X̃_u_2,X̃_u_3]=-(ae_1+b)X̃_u_1, 
    [X̃_u_3,X̃_u_1]=-(ae_2+b)X̃_u_2.

The classification of the different Lie-Algebras coincide with the five mentioned above. Details are completely 
analogous to the previous ones and we do not discuss them further.






§ THE PENDULUM





As in the case of the extended rigid body, solutions of the simple pendulum system are given in terms of Jacobi elliptic 
functions which are defined in the whole complex plane ℂ and are doubly periodic. Due to the relevance 
of the expressions of the pendulum energy in both real and imaginary time, for the purposes of this paper, 
in this section we review briefly the main characteristics of the mathematical formulation of the simple pendulum. This 
is a well understood system and there are many interesting papers and books on the subject 
<cit.>. In our discussion we follow 
<cit.> mainly.




 §.§ Real time pendulum and solutions



Let us start considering a pendulum of point mass m and length r, in a constant downwards gravitational field, of 
magnitude -g (g>0). If θ is the polar angle measured counterclockwise respect to the vertical line and 
θ̇ stands for the time derivative of this angular position, the lagrangian of the system is given by

    L(θ,θ̇)= 1/2mr^2 θ̇^2-mgr(1-cosθ).

Here the zero of the potential energy is set at the lowest vertical position of the pendulum (θ =2 n π, 
with n ∈ℤ). The equation of motion for this system is 

    θ̈+g/rsinθ = 0,

which after integration gives origin to the conservation of total mechanical energy

    E=1/2 m r^2 θ̇^2+2mgrsin^2 ( θ/2)=constant.

Physical solutions of equation  (<ref>) exist only if E ≥ 0. We can rewrite the equation in dimensionless 
form, in terms of the dimensionless energy parameter: k_E^2 ≡E/2mgr, and the dimensionless real time 
variable: 
x ≡√(g/r) t ∈ℝ, obtaining 

    ( p/2)^2 + sin^2 ( θ/2) = k_E^2,

where p is the dimensionless angular velocity: p(x) ≡ dθ/ d x. By inspection of the potential we conclude 
that the pendulum has four different types of solutions depending of the value of the constant k_E^2:


∙ Static equilibrium (θ̇=0): The trivial behavior occurs when either k_E^2=0 or 
k_E^2=1. In the first case, necessarily θ̇=0. For the case k_E^2=1 we consider also the 
situation where θ̇=0. In both cases, the pendulum does not move, it is in static equilibrium. When 
θ =2 n π the equilibrium is stable and when  θ = (2n+1) π the equilibrium is unstable. 


∙ Oscillatory motions (0< k_E^2<1): In these cases the pendulum swings to and fro, 
respect to a point of stable equilibrium. The analytical solutions are given by

    θ (x)   =   2  arcsin [k_E  (x-x_0,k_E)], 
    
    p(x)    =    2   k_E  (x-x_0,k_E),
 
where the square modulus k^2 of the elliptic functions is given directly by the energy parameter: k^2 ≡ k^2_E. 
Here x_0 is a second constant of integration and appears when equation (<ref>) is integrated out. It 
means physically that we can choose the zero of time arbitrarily. The period of the movement is 4K, or restoring the 
dimension of time, 4K√(g/r), with K the quarter period of the elliptic function sn(x-x_0,k_E). 


∙ Asymptotical motion (k_E^2=1 and θ̇≠0): In this case the angle θ 
takes values in the open interval  (-π, π) and therefore, sin(θ/2)  ∈ (-1,1). The particle just 
reach the highest point of the circle. The analytical solutions are given by

    θ (x)   =   ±  2  arcsin [tanh(x-x_0)], 
    
    p(x)    =   ± 2  (x-x_0).
 
The sign ± corresponds to the movement from (∓π→±π).  Notice that tanh(x-x_0), 
takes values in the open interval (-1,1) if: 
x-x_0 ∈ (-∞, ∞). For instance if θ→π, x-x_0 →∞ 
and tanh(x-x_0) goes asymptotically to 1. It is clear that this movement is not periodic. In the 
literature it is common to take x_0=0.


∙ Circulating motions (k_E^2>1): In these cases the momentum of the particle is large 
enough to carry it over the highest point of the circle, so that it moves round and round the circle, always in 
the same direction. The solutions that describe these motions are of the form

    θ (x)   =   ± 2 [( k_E (x-x_0) , 1/k_E) ] 
     arcsin[ ( k_E (x-x_0) , 1/k_E) ], 
    
    
     p(x)    =   ± 2   k_E  ( k_E (x-x_0) , 1/k_E) ,
 
where the global sign (+) is for the counterclockwise motion and the (-) sign for the motion in the clockwise 
direction. The symbol sgn(x) stands for the piecewise sign function which we define in the form

    [( k_E (x-x_0) , 1/k_E) ] = {[                               +1 (4n-1)K ≤ k_E (x-x_0) < (4n+1)K,;                               -1 (4n+1)K ≤ k_E (x-x_0) < (4n+3)K, ].

and its role is to shorten the period of the function sn( k_E (x-x_0) , 1/k_E ) by half. This fact 
is in agreement with the expression for  p(x) because the period of the elliptic function  
dn( k_E (x-x_0) , 1/k_E ) is 2K/k_E instead of 4K/k_E, which is the period of the elliptic function  
sn( k_E (x-x_0) , 1/k_E ). The square modulus k^2 of the elliptic functions is equal to the inverse of the energy 
parameter 0<k^2=1/k_E^2<1.

These are all the possible motions of the simple pendulum. It is straightforward to check that the solutions satisfy the equation of  conservation of energy (<ref>) by using the following relations between the Jacobi functions (in these relations the modulus satisfies 0<k^2 <1) and its analogous relation for hyperbolic functions (which are obtained in the limit case k=1)


    ^2(x,k) + ^2(x,k)    =    1, 
    tanh^2(x)+^2(x)    =   1, 
    
     k^2 ^2(x,k)  + ^2(x,k)     =    1.





 §.§ Imaginary time pendulum



In the analysis above, time was considered a real variable, and therefore in the solutions of the simple pendulum  only the real quarter period K appeared. But Jacobi elliptic functions are defined in ℂ and, for instance, the function (z,k) of square modulus 0 < k^2 < 1, besides the real primitive period 4K, owns a pure imaginary primitive period 2iK_c,  where K and K_c are  the so called quarter periods (see for instance <cit.>). In 1878 Paul Appell clarified the physical meaning of the imaginary time and the imaginary period in the 
oscillatory solutions of the pendulum <cit.>, by introducing an ingenious trick, he reversed the direction 
of the gravitational field: g → -g,  i.e. now the gravitational field is upwards. In order the Newton equations of 
motion remain invariant under this change in the force, we must replace the real time variable t by a purely 
imaginary one: τ≡± it. Implementing these changes in the equation of motion (<ref>) leads to 
the equation

    d^2 θ/dτ^2 - g/rsinθ =0.

Writing this equation in dimensionless form requires the introduction of the pure imaginary time variable
y ≡±τ√(g/r)=± ix. Integrating once the resulting dimensionless equation of motion gives origin to 
the following equation

    1/4( d θ/dy)^2 - sin^2 ( θ/2) = -k_E^2,

which looks like equation (<ref>) but with an inverted potential. In order to solve  this equation, we 
should flip the sign in the potential and rewrite the equation in such a way that it looks similar to equation 
(<ref>). To achieve this aim we start by shifting the value of the potential energy one unit such that its 
minimum value be zero. Adding a unit of energy to both sides of the equation leads to

    (ℙ/2)^2 + cos^2 ( θ/2) = 1-k_E^2.

Here ℙ is the momentum as function of imaginary time. The second step is to rewrite the potential energy 
in such a form it coincides with the potential energy of (<ref>) and, in this way, allowing us to compare solutions. We can accomplish this by a simple translation of the graph, for instance, by translating it an angle of π/2 
to the right. Defining θ'= θ - π, we obtain

    (ℙ/2)^2 + sin^2 ( θ'/2) = 1-k_E^2.

It is clear that 0< 1-k_E^2 <1 for oscillatory motions and 1-k_E^2 <0 for the circulating ones. It is not the purpose 
of this paper to review the whole set of solutions with imaginary time. The reader interested in the detailed construction 
can see <cit.> for instance.







§ THE PENDULUM FROM THE EXTENDED RIGID BODY (C≠ 0 AND D ≠ 0)





In this section we review the relation between the rigid body and the simple pendulum as originally discussed by Holm and Marsden <cit.>, and we extend it to pendulums of imaginary time. In every case we discuss the geometrical and the algebraic characteristics of the relations. 

A lesson learned from <cit.> is that, in order to have the pendulum from the rigid body it is necessary that the 
SL(2,ℝ) transformation of the Casimir functions lead to new ones that geometrically represent 
two perpendicular elliptic cylinders.  In our parameterization, the mathematical conditions for this to happen is that

    ae_i+b=0,  ce_j+d=0, 
     i ≠ j.
 
However, due to the fact that the inertia parameters are not necessarily positive, these conditions are not attached to 
elliptical cylinders only, but also to hyperbolic cylinders  <cit.>. In order to have control over all 
different possibilities to get pendulums from the rigid body it is necessary to list all different SL(2,ℝ) 
transformations that fulfill conditions  (<ref>). The transformations are divided into three general sets, 
where each set contains different fixed  forms of the SL(2,ℝ) matrices. The three sets are determined by 
the conditions: i) c≠ 0 and d ≠ 0. ii)  c≠ 0 and d = 0 and iii) c=0 and d ≠ 0 <cit.>. 
In the latter case a generic SL((2,ℝ) group element has the form

    g= ( 
    [   a   b;   0 1/a ]),  b ∈ℝ.

We conclude that the surface N is a unitary sphere and therefore in this case it is not possible to degenerate 
the surface to a cylinder. As a consequence the pendulum can not arise from group elements of this kind and we 
must analyze only the cases i) and ii). Because the case discussed by Holm and Marsden belongs to the first set of 
conditions, we analyze it in this section and leave the set: c≠ 0 and d = 0 for section <ref>.

The first step in our analysis is to fully classify the different cases belonging to the set c ≠ 0 and d ≠ 0 
in which the Casimir surfaces H and N as defined in equation (<ref>) fulfill conditions 
(<ref>). It is clear that there are 6 different forms to satisfy these conditions. However 
these are not all independent, in fact, the difference between conditions: ae_i+b=0 and ce_j+d=0, with respect to 
conditions: ae_j+b=0 and ce_i+d=0, is that they interchange the role of the Casmir functions H and N. Thus 
we can restrict ourselves to the analysis of the three cases (<ref>) for i<j. Because solutions 
(<ref>) and (<ref>) depend on the relative value of the parameter e_0 respect to e_2, the 
classification of geometries for both H and N in general also depends on this relative value  
<cit.>. This fact increases the number of cases to five. We show in table <ref> the 
different possible geometries for the Casimir H and in table <ref> the corresponding ones for the Casimir 
N. It is important to stress that in this classification we are using the fact that the space 
{u_1,u_2,u_3} is ℝ^3.




It is clear the five different possibilities we are referring to correspond to the intersections: 1-6, 1-7, 1-8, 2-8 y 3-8. The one discussed by Holm and Marsden corresponds to the case of the intersection of two elliptical cylinders 1-8, however we have four more possibilities which  correspond to the intersection of an elliptical cylinder and a hyperbolic cylinder.  For the best of our knowledge these four cases have not been discussed in the literature; one of the aims of this 
section is to give them an interpretation as pendulums. At this point it is convenient to stress the two main differences 
between our analysis and the original discussion in <cit.>. 1) The discussion of Holm and Marsden was 
given using  the moments of inertia (<ref>) as parameters whereas here we are using a different 
parameterization, the one in terms of the dimensionless inertia parameters {e_i}, which produce slight differences 
in the discussion. 2) Additionally to the original momentum map { u_i}→{ u_i (θ,p)} whose main 
characteristic  is to have a real pendulum momentum p or equivalently a real angular coordinate θ and a real 
time t, in our discussion we will consider also momentum maps where the real pendulum momentum p is rewritten 
as p=i ℙ with the new momentum: ℙ=-ip purely imaginary. As discussed in subsection 
<ref> this momentum is a consequence of introducing a purely imaginary time. When time is real the final 
dimensionless angular momentum space  remains ℝ^3  whereas the associated pendulum phase space is 
ℝ× S^1. When time is purely imaginary the final angular  momentum space is 
ℝ^2 × i ℝ whereas the pendulum phase space is built with a real coordinate θ and
a purely imaginary momentum ℙ. We start discussing the original case of Holm and Marsden (1-8).





 §.§ Intersection of two elliptic cylinders



Choosing the a, b, c and d nonvanishing values of the group element g ∈ SL(2,ℝ) to satisfy 

    a e_1+b=ce_3+d=0,

fix two of the three free parameters of g.  Substituting b and d in the expressions (<ref>) for the 
Casimir surfaces we obtain

    H:        -a/2 (e_1-e_2)    u_2^2 -a/2 (e_1-e_3)   u_3^2 = -a/2( e_1-e_0), 
     N:            c/2( e_1-e_3)   u_1^2 + c/2 (e_2-e_3)   u_2^2 =     c/2 (e_0-e_3).

Notice that the surfaces do not depend of the specific values of both a and c. However we can fix one of these parameters in terms of the other using the condition ad-bc=1, which in this case produce the condition ac=1/(e_1-e_3). In other words, Holm and Marsden considered an element g ∈ SL(2,ℝ) of the type

    g= ( 
    [    1/c(e_1-e_3) -e_1/c(e_1-e_3);               c           -ce_3 ]),

which depends only on one free parameter c ≠ 0. Originally this parameter was fixed to the value c=1 although it 
is not necessary to fix it because the SL(2,ℝ) combination of h and l leads to expressions that do not 
depend on c. It is clear that given the ordering (<ref>), the coefficients in (<ref>) have the 
property a>0, b<0 and d>0.

The equations that determine the Casimir surfaces can be written finally as

    H:        k_2^2   u_2^2 +   u_3^2 = e_1-e_0/e_1-e_3, 
     N:         u_1^2 + k_1^2   u_2^2 = e_0-e_3/ e_1-e_3 ,

which generically represent two elliptic cylinders, H with axis on cordinate u_1 and N with 
axis along u_3. Here k_1^2 and k_2^2 are the ones defined in (<ref>). At this point the transverse sections 
of the cylinders are ellipses whose semi-major and semi-minor axes depend on the value of the parameter e_0, 
i.e. they have variable size and the intersections are exactly the same as the ones in fig. <ref> because 
the SL(2,ℝ) transformations change the geometry of the Casimirs but leave invariant the intersections 
<cit.>. 

The strategy is now to implement a variable change in such a way that one of the cylinders becomes circular using 
only trigonometric functions. Once this aim is achieved, the second cylinder can not be mapped to a circular cylinder 
simultaneously without introducing elliptic functions. This strategy can be applied in two  different ways, in one case 
H becomes the circular cylinder, whereas in the second case the Casimir whose geometry becomes the 
circular cylinder is  N. Interestingly these two cases produce the same physical system as expected for a 
bi-hamiltonian system, although the physical origin differs in both cases as we will explain. The variable changes or 
technically the momentum maps we implement <cit.>, are analogous to the one performed by Holm and 
Marsden. As a result of the mapping one of the resulting Casimir functions obtain the form of the Hamiltonian of the 
simple pendulum and we can identify directly if the energy produces an oscillating or a circulating movement. 
Additionally we also introduce momentum maps for imaginary time.


∙ Simple pendulum with real time and Hamiltonian H: Consider the momentum map

    u_1≡√(e_0-e_3/e_1-e_3)cos( θ/2) , 
    u_2≡1/k_1√(e_0-e_3/e_1-e_3)sin( θ/2),  
    u_3≡k_2/k_1√(e_0-e_3/e_1-e_3) p/2.

The expression for the Casimir surface N is explicitly satisfied whereas the expression for the Casimir 
H becomes

    H: sin^2  ( θ/2) + ( p/2)^2= 
    e_1-e_0/e_0-e_3·e_2-e_3/e_1-e_2≡ m^2.

N is interpreted as the cotangent space and has the geometry of a circular cylinder of unitary radius. On the other side, according to equation (<ref>) H is the Hamiltonian of a pendulum. This is the 
analogous of the relation found by Holm and Marsden <cit.>. Because situation 1 in table <ref> 
is valid for any value of e_0 in the interval e_3 < e_0 < e_1 we have that 0<m^2<1 for e_0 > e_2, m^2=1 
for e_0=e_2 and m^2>1 for e_0<e_2 (see table <ref>). Therefore for m^2<1 we have an oscillatory 
movement of the pendulum, for m^2=1 we have a critical movement and for m^2>1 we have a circulating one. 
Fig. <ref> shows the complete phase space of the pendulum.


Regarding the generators (<ref>) of the system, it is straightforward to check they can be redefined as

    Y_u_1   ≡   1/c α X_u_1= -(e_1-e_3)/α  k^2_1  u_2  ∂_3, 
    
    Y_u_2   ≡   k_1/c α X_u_2= (e_1-e_3)/α  k_1   u_1 ∂_3, 
    
    
    Y_u_3   ≡   1/c k_1 (e_1-e_3) X_u_3= k_1   u_2  ∂_1 - 1/k_1  u_1  ∂_2,
 
which satisfy the Lie-Poisson algebra ISO(2)

    [Y_u_1,Y_u_2]=0,  [Y_u_2,Y_u_3]=Y_u_1,     [Y_u_3,Y_u_1]=Y_u_2.

A Lie-Poisson structure for the pendulum can be introduced on the cylindrical surface N=1 
<cit.>, in terms of the variables θ, p by defining the Lie-Poisson bracket as 

    { f,g }_ N = - ∇ N· (∇ f ×∇ g).
 
An straightforward calculation gives 

    { f,g }_ N=-8(e_1-e_3)(e_2-e_3)/√((e_1-e_2)(e_0-e_3))(∂ f/∂θ∂ g/∂ p-∂ f/∂ p∂ g/∂θ),

which shows that the variables θ and p are canonically conjugate up to a scale factor:  
{ p,θ}_ N=8(e_1-e_3)(e_2-e_3)/√((e_1-e_2)(e_0-e_3)). In terms of this 
bracket the canonical equations of motion are given by

    d θ/d τ = { H,θ}_ Nd p/d τ = { H,p }_ N.

Combining these we obtain the Newton equation of motion for the simple pendulum

    d^2θ/dτ^2=-16(e_1-e_3)^2(e_2-e_3)^2/(e_1-e_2)(e_0-e_3)sinθ.




∙ Simple pendulum with imaginary time and Hamiltonian H: Consider the momentum map

    u_1≡√(e_0-e_3/e_1-e_3)sin( θ'/2) , 
    u_2≡1/k_1√(e_0-e_3/e_1-e_3)cos( θ'/2),  
    u_3≡  i k_2/k_1√(e_0-e_3/e_1-e_3) ℙ/2.

The expression for the Casimir surface N is again explicitly satisfied whereas the form of the Casimir 
H becomes

    H: sin^2  ( θ'/2) + ( ℙ/2)^2= m_c^2 ,

where m_c^2 is the complementary modulus as defined in (<ref>). The Casimir H 
represents the Hamiltonian of the simple pendulum with imaginary time (<ref>). Again because 
(<ref>) is valid for any value of e_0, we have all different movements of the pendulum. In particular 
for the oscillatory motions: e_2<e_0 and 0<m_c^2<1, for the asymptotical motion: e_0=e_2 and m_c^2=0, 
whereas for the circulating movements: e_0<e_2 and m^2_c<0 (see table <ref>). Notice that the explicit 
presence of the imaginary number i in coordinate u_3 geometrically plays the role of changing the elliptic cylinder 
into a hyperbolic cylinder although the former is defined in a real space ℝ^3 whereas the later is defined in 
a complex space ℝ^2× i ℝ. Regarding the Lie-Poisson algebra it is clear that here we have 
the same ISO(2) algebra (<ref>). The two dimensional Lie-Poisson bracket in this case has a purely  
imaginary scale factor

    { f,g }_ N=-8i (e_1-e_3)(e_2-e_3)/√((e_1-e_2)(e_0-e_3))(∂ f/∂θ∂ g/∂ℙ-
    ∂ f/∂ℙ∂ g/∂θ),

which is in agreement with the fact that the Newton equation for the simple pendulum with imaginary time has an extra 
minus sign in the force term

    d^2θ/dτ^2=16(e_1-e_3)^2(e_2-e_3)^2/(e_1-e_2)(e_0-e_3)sinθ.

Because the rigid body is a bi-Hamiltonian system it is interesting to obtain the simple pendulum but now having the 
Casimir function N as the hamiltonian and the Casimir function H as the cotangent space.


∙ Simple pendulum with real time and Hamiltonian N: Consider the momentum map

    u_1 ≡k_1/k_2√(e_1-e_0/e_1-e_3) p/2,   
    u_2 ≡1/k_2√(e_1-e_0/e_1-e_3)sin( θ/2), 
    u_3 ≡√(e_1-e_0/e_1-e_3)cos( θ/2).

Now the expression for the Casimir function H is explicitly satisfied whereas the expression for the Casimir 
N becomes

    N:  ( p/2) ^2 + sin^2( θ/2) = 1/m^2,

which according to equation (<ref>) corresponds to the energy of a simple pendulum. Because 
situation 8 in table <ref> is valid for any value of e_0 in the interval e_3<e_0<e_1, in equation 
(<ref>) we have all different movements of the pendulum. It is worth to stress that whereas for values e_0<e_2 
for which 0<m^2<1 the movements in (<ref>) are oscillatory (see fig. <ref>), in (<ref>) 
for the same values of e_0: 1<1/m^2< ∞ and therefore we have circulating pendulums (see 
fig. <ref>). Something analogous occur for the circulating movements in (<ref>) and the 
oscillatory ones in (<ref>). The asymptotical cases are given for e_0=e_2 for which m^2=1 in both 
(<ref>) and (<ref>).

Let us emphasize that the circular cylinder H has unitary radius. In this case the Hamiltonian vector field 
associated with the coordinates u_i's are given by equations (<ref>). Redefining these vector fields as

    Ỹ_u_1   ≡   1/a (e_1-e_3) k_2X̃_u_1= 1/k_2  u_3  ∂_2 - k_2   
    u_2 ∂_3, 
    Ỹ_u_2   ≡   1/a αX̃_u_2 = -(e_1-e_3)/α  u_3  ∂_1, 
    
    Ỹ_u_3   ≡   1/a α k_2X̃_u_3= e_1-e_3/α  k_2   u_2  ∂_1,
 
they satisfy the ISO(2) Lie-Poisson algebra

    [Ỹ_u_1,Ỹ_u_2]=Ỹ_u_3,  [Ỹ_u_2,Ỹ_u_3]= 0,     
    [Ỹ_u_3,Ỹ_u_1]=Ỹ_u_2.

Regarding the Lie-Poisson bracket in terms of the variables (θ, p) on the cylindrical surface H=1,  
and defined as

    { f,g }_ H = - ∇ H· (∇ f ×∇ g),
 
we have 

    { f,g }_ H=8(e_1-e_3)(e_1-e_2)/√((e_1-e_0)(e_2-e_3))(∂ f/∂θ∂ g/∂ p-∂ f/∂ p∂ g/∂θ).

The Newton equation associated to this Lie-Poisson bracket is

    d^2θ/dτ^2=-16(e_1-e_3)^2(e_1-e_2)^2/(e_1-e_0)(e_2-e_3)sinθ.




∙ Simple pendulum with imaginary time and Hamiltonian N: Consider the momentum map

    u_1 ≡ i  k_1/k_2√(e_1-e_0/e_1-e_3) ℙ/2,   
    u_2 ≡1/k_2√(e_1-e_0/e_1-e_3)cos( θ'/2), 
    u_3 ≡√(e_1-e_0/e_1-e_3)sin( θ'/2).

The expression for the Casimir function H is explicitly satisfied whereas the expression for the Casimir 
N becomes

    N:  ( ℙ/2) ^2 + sin^2( θ'/2) =- m_c^2/m^2.

This is the Hamiltonian of a simple pendulum with imaginary time. As in the previous cases, because (<ref>) is valid for any value of e_0, it includes all different movements of the pendulum. Because the cotangent space is H, the ISO(2) Lie-Poisson algebra is given by (<ref>), and the Lie-Poisson bracket in terms of the 
variables (θ, ℙ) is similar to (<ref>) and differs only by a factor of i in the scale factor.


The geometrical interpretation of these results is straightforward.  If we start with a specific SO(3) rigid 
body, the three moments of inertia {I_i} are given, which fix the parameter κ and therefore the three 
dimensionless parameters of inertia {e_i}.  The different solutions are obtained for different values of e_0. After 
the SL(2,ℝ) gauge transformation of the Casimir surfaces, for the momentum maps 
(<ref>) and (<ref>) we obtain a unitary circular cylinder N with the axis along the u_3 
direction representing the cotangent space. On the other side, geometrically H is 
an element of a set of elliptic-(hyperbolic) cylinders with axis along the u_1 direction which physically are energy 
surfaces, because H is the Hamiltonian of the simple pendulum (see equations (<ref>) and  
(<ref>)). Intersections of the single circular cylinder N with the set of elliptic-(hyperbolic) cylinders 
H represent all the movements of the pendulum. On the other side, for the momentum maps (<ref>) and 
(<ref>) the surface H becomes an unitary circular cylinder with axis along the u_1 direction 
representing the cotangent space and N is an element of a set of elliptic-(hyperbolic) cylinders that represent 
the Hamiltonian of the simple pendulum. 




 §.§ Intersection of a hyperbolic cylinder and an elliptic cylinder I



Let us consider the cases 2-8 and 3-8, which correspond to the intersection of a hyperbolic cylinder and an elliptical 
cylinder. In these cases the SL(2,ℝ) group element has the form

    g= ( 
    [    1/c(e_2-e_3) -e_2/c(e_2-e_3);               c           -ce_2 ]).

As a consequence the Casimir surfaces  are given by the expressions

    H:        k_2^2   u_1^2 -  k_1^2  u_3^2 = e_0-e_2/e_1-e_3 , 
     N:        u_1^2 + k_1^2   u_2^2 = e_0-e_3/e_1-e_3.

Notice that the Casimir N is exactly the same as (<ref>), as it should be since in both cases we are dealing 
with the situation 8 (see table <ref>), which represents an elliptic cylinder of unitary radius with axis along 
u_3. The cases 2-8 and 3-8 differ in the sign of the right hand side of the Casimir H (<ref>), 
for the case 2-8 we have e_0 - e_2>0, whereas for the case 3-8 we have e_0 - e_2<0. Geometrically the different 
signs change the orientation of the hyperbolic cylinders (see fig <ref>).  Again we expect to have two 
different momentum maps, one in which N is the cotangent space and H is the Hamiltonian and a 
second one where  H is the cotangent space and N is the Hamiltonian. 




∙ Simple pendulum with real time and Hamiltonian H: As we have discussed previously, 
under the momentum map (<ref>)

    u_1≡√(e_0-e_3/e_1-e_3)cos( θ/2), 
    u_2≡1/k_1√(e_0-e_3/e_1-e_3)sin( θ/2),  
    u_3≡k_2/k_1√(e_0-e_3/e_1-e_3) ( p/2),

the expression for the Casimir N is automatically satisfied.  On the other side the Casimir surface 
(<ref>) coincides with the Hamiltonian (<ref>) and (<ref>) 

    H: sin^2  ( θ/2) + ( p/2)^2 = m^2 ,

where 0<m^2<1 for the case 2-8 (e_0 - e_2>0) and therefore represent the oscillatory movements, whereas 
1<m^2< ∞ for the case 3-8  (e_0 - e_2<0) which represent the circulating movements. As usual the case 
e_0=e_2 produce the separatrix. Regarding the Hamiltonian vector fields associated to the coordinates, they are 
the same as (<ref>) and therefore they satisfy the Lie-Poisson algebra ISO(2) (<ref>). A 
straightforward calculation gives the two dimensional Lie-Poisson bracket (<ref>) and the Newton equation 
(<ref>). These results show that this case and IV-A represent the same dynamics, although they come from 
a different geometry for the Casimir H.


∙ Simple pendulum with imaginary time and Hamiltonian H: As in section <ref>, 
under the momentum map

    u_1≡√(e_0-e_3/e_1-e_3)sin( θ'/2), 
    u_2≡1/k_1√(e_0-e_3/e_1-e_3)cos( θ'/2),  
    u_3≡ i k_2/k_1√(e_0-e_3/e_1-e_3) ( ℙ/2),

the expression for the Casimir N is automatically satisfied whereas the Casimir H becomes

    H: sin^2  ( θ'/2) + ( ℙ/2)^2 = m_c^2.

where -∞ < m_c^2<1.  Notice that the geometrical role of writing the coordinate u_3 in terms of an imaginary momentum ℙ is to map the hyperbolic cylinder H to an elliptic cylinder with axis along 
the u_2 direction. According to equation (<ref>), H represents the Hamiltonian for a simple 
pendulum of imaginary time and energy m^2=1-m_c^2, as the one in (<ref>). As expected, the two 
dimensional Lie-Poisson structure and Newton equations coincide with (<ref>) and (<ref>).


Next natural mapping is the one that maps the rigid body system to a cotangent space H, with hamiltonian 
N. By inspection of the Casimir (<ref>), this can be achieved introducing a purely imaginary coordinate. 
Doing this allows to change the hyperbolic nature of the cylinders defined in ℝ^3 to elliptic cylinders 
defined in ℝ^2 × iℝ.


∙ Circulating simple pendulum with real time, imaginary coordinate and Hamiltonian N: 
Consider the momentum map

    u_1≡1/k_2√(e_0-e_2/e_1-e_3)sin( θ/2), 
    u_2≡1/k_1 k_2√(e_0-e_2/e_1-e_3)p/2,  
    u_3≡i/k_1√(e_0-e_2/e_1-e_3)cos( θ/2).

Here we are assuming: e_0-e_2>0 (situation 2 in table <ref>). For such a map the Casimir H is 
automatically satisfied whereas the Casimir N takes the form

    N: sin^2  ( θ/2) + ( p/2)^2 = 1/m_c^2,

where 1< 1/m_c^2 < ∞ (see table <ref>) and therefore the Hamiltonian equation represents only 
circulating movements of the simple pendulum. 

Because the Hamiltonian is given by N, the suitable Hamiltonian vector fields associated to the coordinates are 
given by equation (<ref>). Redefining them as

    Ỹ_u_1   ≡   k_2/a β k_1X̃_u_1= 1/β√((e_1-e_2)(e_2-e_3))  u_3 
    ∂_2,
    
    Ỹ_u_2   ≡   1/a √((e_1-e_3)(e_2-e_3))X̃_u_2= -k_2/k_1 u_1 ∂_2 - 
    k_1/k_2 u_3 ∂_1,  
    Ỹ_u_3   ≡   1/a βX̃_u_3 = 1/β (e_1-e_2)     u_1 ∂_2,
 
they satisfy the ISO(1,1) Lie-Poisson algebra 

    [Ỹ_u_1,Ỹ_u_2]=Ỹ_u_3,  [Ỹ_u_2,Ỹ_u_3]= -Ỹ_u_1,  
       [Ỹ_u_3,Ỹ_u_1] = 0.

Regarding the two-dimensional Lie-Poisson bracket, it has the form

    { f,g }_ N=-  8 i(e_1-e_2)(e_2-e_3)/√((e_0-e_2)(e_1-e_3))(∂ f/∂θ∂ g/∂ p-∂ f/∂ p∂ g/∂θ),

while the equation of motion goes as:

    d^2θ/dτ^2=16(e_2-e_3)^2(e_1-e_2)^2/(e_1-e_3)(e_0-e_2)sinθ,

At this point the sign in the Newton equation seems to be wrong, but this sign is reflecting the fact that we have 
applied a complex map in coordinate u_3. 

Although we have presented the analysis assuming e_2-e_0<0, it is worthing to point out that the same physical 
situation emerge also for the case e_2-e_0>0. The only difference is to introduce a mapping where the complex 
coordinate is u_1 instead of u_3. We get as a conclusion of this subsection that, for the situation where the 
intersection of Casimir functions is among a hyperbolic cylinder H and a elliptic cylinder N, if the 
Hamiltonian is given by H and the cotangent space is given by N then we get the whole motions of the 
simple pendulum and the Lie algebra of the extended rigid body is ISO(2), but in the case where H is the 
Hamiltonian and H is the cotangent space we only get the circulating motions and the Lie algebra is 
ISO(1,1).


∙ Circulating simple pendulum with imaginary time, imaginary coordinate and Hamiltonian N: 
Consider the mapping 

    u_1≡1/k_2√(e_0-e_2/e_1-e_3)cos( θ'/2), 
    u_2≡i/k_1 k_2√(e_0-e_2/e_1-e_3)ℙ/2,  
    u_3≡i/k_1√(e_0-e_2/e_1-e_3)sin( θ'/2).

For such a map, where we are assuming e_0-e_2>0, the Casimir H is automatically satisfied whereas the 
Casimir N takes the form

    N: ( ℙ/2)^2  +  sin^2  ( θ'/2) = - m^2/m_c^2.

It is straightforward to notice that -m^2/m_c^2<0, and therefore we have as expected only circulating movements. This case also comes from the extend rigid body with Lie algebra SO(1,1). Regarding the two dimensional 
Lie-Poisson bracket it has the following expression

    { f,g }_ N=  8 (e_1-e_2)(e_2-e_3)/√((e_0-e_2)(e_1-e_3))(∂ f/∂θ∂ g/∂ℙ-∂ f/∂ℙ∂ g/∂θ),

while the Newton equation of motion goes as (<ref>) but with an extra minus sign in the right hand side of the equation. Again the discordance of the Newton equation with respect to equation (<ref>) is understood by the fact that the momentum map (<ref>) considers a complex transformation in coordinate u_3.





 §.§ Intersection of a hyperbolic cylinder and an elliptic cylinder II



As final cases we study the intersections between the elliptic cylinder and hyperbolic cylinder 1-6 and 1-7, for which the 
SL(2,ℝ) group elements take the form

    g= ( 
    [    1/c(e_1-e_3) -e_1/c(e_1-e_3);               c           -ce_2 ]).

For these intersections the expressions for the Casimir functions are 

    H:        k_2^2   u_2^2 +   u_3^2 = e_1-e_0/e_1-e_3 , 
     N:        - k_2^2   u_1^2 + k_1^2   u_3^2 = e_2-e_0/e_1-e_3.

The surface H has the same expression as (<ref>) as it should be since we are working with the situation 
1 of table <ref>, which corresponds to an elliptic cylinder.


∙ Simple pendulum with real time and Hamiltonian N: Consider the momentum map 
(<ref>)

    u_1≡k_1/k_2√( e_1-e_0/e_1-e_3) p/2, 
    u_2≡1/k_2√(e_1-e_0/e_1-e_3)sin( θ/2),  
    u_3≡√(e_1-e_0/e_1-e_3)cos( θ/2) .

Clearly the Casimir H is automatically satisfied whereas the Casimir 

    N: ( p/2)^2 +  sin^2  ( θ/2)  = 1/m^2 .

For values of e_0 <e_2, the quotient 1/m^2<1 and therefore N represents the Hamiltonian for the simple 
pendulum in oscillatory motion whereas for e_0>e_2,  1/m^2>1 and N represents the Hamiltonian for the 
simple pendulum in circulating motion (<ref>). For the limiting case e_0=e_2, 1/m^2=1 and we 
obtain the asymptotical motion.

As expected, in this case the two dimensional Lie-Poisson bracket coincides with (<ref>) and therefore the 
Newton equation of motion is given by (<ref>). 


∙ Simple pendulum with imaginary time and Hamiltonian N: 
Consider the momentum map (<ref>)

    u_1≡ i k_1/k_2√( e_1-e_0/e_1-e_3) ℙ/2, 
    u_2≡1/k_2√(e_1-e_0/e_1-e_3)cos( θ'/2),  
    u_3≡√(e_1-e_0/e_1-e_3)sin( θ'/2) .

Under this map the Casimir H is automatically satisfied whereas the Casimir N becomes

    N: ( ℙ/2)^2 +  sin^2  ( θ'/2)  = -  m_c^2/m^2 ,

which is precisely the Hamiltonian of simple pendulum with imaginary time (<ref>). Because we are using 
the same cotangent space (<ref>) and (<ref>), the rest of physical quantities such as the two dimensional 
Lie-Poisson bracket and the Newton equation of motion coincide with the ones mentioned below equation  
(<ref>).

Following the previous cases, next natural situation is the one that maps the rigid body system to a cotangent space 
N, with Hamiltonian H. By inspection of Casimir (<ref>), this can be achieved introducing a 
momentum map with a purely imaginary coordinate.


∙ Circulating pendulum with real time, imaginary coordinate and Hamiltonian H: Consider the 
momentum map

    u_1≡i/k_2√( e_2-e_0/e_1-e_3)cos( θ/2) , 
    u_2≡1/k_1 k_2√(e_2-e_0/e_1-e_3) p/2,  
    u_3≡1/k_1√(e_2-e_0/e_1-e_3)sin( θ/2) .

Here we are assuming e_2-e_0>0 (situation 7 in table <ref>).

Under this map the Casimir N is satisfied, while the Hamiltonian H has the following expression

    H: ( p/2)^2 +  sin^2  ( θ/2)  = -  m^2/m_c^2 .

For values e_2-e_0>0 the quotient 1<- m^2/m_c^2<∞ (see table <ref>) and we have a simple 
pendulum in circulating motion. Regarding the two dimensional Lie-Poisson bracket it has the following expression

    { f,g }_ N=  8i (e_1-e_2)(e_2-e_3)/√((e_2-e_0)(e_1-e_3))(∂ f/∂θ∂ g/∂ p-∂ f/∂ p∂ g/∂θ),

whereas the Newton equation of motion reads

    d^2θ/dτ^2=16(e_1-e_2)^2(e_2-e_3)^2/(e_2-e_0)(e_1-e_3)sinθ.

As the analogous physical situation in section <ref>, the extra minus sign that appears in 
the Newton equation (<ref>) is due to the complex nature of the map (<ref>). 
As for the case e_0>e_2, we can go through it but the physical result will be the same, we only get the 
circulating movements of the pendulum.


∙ Circulating pendulum with imaginary time, imaginary coordinate and Hamiltonian H: 
Consider the momentum map in which we assume e_2>e_0

    u_1≡i/k_2√( e_2-e_0/e_1-e_3)sin( θ'/2) , 
    u_2≡i/k_1 k_2√(e_2-e_0/e_1-e_3) ℙ/2,  
    u_3≡1/k_1√(e_2-e_0/e_1-e_3)cos( θ'/2) .

Under this map the Casimir N is satisfied, while the Hamiltonian H has the following expression

    H: ( ℙ/2)^2 +  sin^2  ( θ'/2)  =   1/m_c^2 .

It is clear that -∞ < 1/m_c^2 <0 and therefore as expected we only have circulating motions.

For completeness, the two dimensional Lie-Poisson bracket is

    { f,g }_ N=  -8 (e_1-e_2)(e_2-e_3)/√((e_2-e_0)(e_1-e_3))(∂ f/∂θ∂ g/∂ℙ-∂ f/∂ℙ∂ g/∂θ),

whereas the equation of motion reads

    d^2θ/dτ^2=- 16(e_1-e_2)^2(e_2-e_3)^2/(e_2-e_0)(e_1-e_3)sinθ.

Notice again the extra minus sign in (<ref>) due to complex nature of the map (<ref>).





§ THE PENDULUM FROM THE EXTENDED RIGID BODY (C ≠ 0 AND D  = 0)




Finally let us analyze the second general set of SL(2,ℝ) transformations. Regarding the conditions 
(<ref>) the second one becomes modified to ce_i=0. Since in this case 
c ≠ 0 then necessarily e_i=0. Even more, because we are restricted to the interval κ∈ (0,π/3) 
the only possibility is to have e_2 = 0 and because the restrictions  (<ref>) we also have 
e_1=-e_3=√(3)/2. We conclude that the conditions (<ref>) can be satisfied only in two situations. 
In both of them e_2=0 and either ae_1+b=0 or ae_3+b=0. Due to the relation between e_1 and e_3 these 
two situations correspond to a transformation (<ref>) with different sign of the coefficient a. So without 
losing generality we can restrict ourselves to the case a>0 and analyze one of the situations. The second one can be 
obtained from the same formulas by changing the sign of a. In summary the conditions (<ref>) 
become

    ae_1+b=0,  e_2=0,

and a generic  SL(2,ℝ) element has the form

    g= ( 
    [ 1/ce_1   -1/c;      c      0 ]).

Notice that this transformation can be obtained from (<ref>) modulo a factor of 1/2 in the first arrow of the 
matrix, which does not change the geometry of the Casimir surface H and therefore  (<ref>) corresponds 
to the SL(2,ℝ) transformation that takes the original Casimir surfaces to an elliptic and a hyperbolic 
cylinders. Notice that for e_1=-e_3 and e_2=0, the quotients (<ref>) become equal k_1^2=k_2^2=1/2. 

It is clear that we do not have to developed this case further, since we can obtain it from a limiting case of the ones 
previously studied. Here we have two situations again, either e_0<0 (situation 1-7 of subsection 
<ref>) or e_0>0 (situation 1-6 of subsection <ref>). 





§ CONCLUSIONS




In this paper we have revisited the relation between the extended rigid body and the simple pendulum with the aim to give an exhaustive list of all different ways in which the relation takes place. We started in section <ref> reviewing the basics of the rigid body system in its two parameters formulation. The first parameter e_0 is related to the quotient E/L^2 where E is the energy of the motion and L^2 is the square of the magnitude of angular momentum. The second parameter κ codifies the values of the three moments of inertia. We work in this formulation of the rigid body because it allows us to have a good control on the different geometries of the two Casimir functions of the system <cit.>.

The original construction to establish the relation between the extended rigid body and the pendulum was discussed by Holm and Marsden <cit.> and uses the SL(2,ℝ) symmetry of the Euler equations to find linear combinations of the two Casimir functions and transform them to new ones, denoted in this paper as H and N. For one specific class of SL(2,ℝ) transformations, both Casimirs have the geometry of an elliptic cylinder, this case corresponds to the extended rigid body with ISO(2) Lie algebra. By a proper change of coordinates, or momentum map, and taking one of the cylinders as the cotangent space and the another one as the Hamiltonian, it is possible to obtain both the two dimensional Hamiltonian of the simple pendulum and its canonical equations of motion, even more, taking different values for the principal moments of inertia allows to get the different solutions of the simple pendulum: oscillatory, circulating and critical. Our present work revisits the construction proposed by Holm and Marsden in the two parameters formulation and, since we are working with a bi-hamiltonian system, we give the momentum maps for the two different physical situations, i.e. when N is the cotangent space and H is the Hamiltonian and the situation where we invert the role of the Casimirs, H as the cotangent space and N as the Hamiltonian. This exercise allows us to understand in a precise way what kind of movements in the rigid body are mapped to certain type of movements of the simple pendulum. Specifically, we show that movements in the rigid body, for which e_2<e_0, are mapped to oscillatory movements of the pendulum if  H is the Hamiltonian, but are mapped to circulating movements of the pendulum if N is the Hamiltonian instead. Something similar occurs for movements with e_2>e_0.

Going one step beyond and using the whole SL(2,ℝ) transformations of the Euler equations, in this paper we study all different cases in which there is a relation between the solutions of the extended rigid body and the ones of the simple pendulum or at least part of it.  As a result, we have found that if we keep the geometry of one of the Casimirs an elliptic cylinder and we consider the other Casimir a hyperbolic cylinder, it is also possible to get the simple pendulum. To be specific, if H represents the elliptic cylinder and N represents the hyperbolic cylinder then we have again two situations: i) If H is the cotangent space and N is the Hamiltonian we can provide a momentum map that gives origin to the whole movements of the pendulum; this case also corresponds to an extended rigid body with Lie algebra ISO(2). ii) If instead H is the Hamiltonian and N is the cotangent space, then we can provide a momentum map considering one of the coordinates as purely imaginary in such a way that we get the Hamiltonian of the pendulum, although the energies of the system correspond only to the circulating movements and not the oscillatory ones; this case corresponds to the extended rigid body with Lie algebra ISO(1,1). To the best of our knowledge this is a physical situation not previously discussed in the literature. In 
every case  we also give the momentum map that relates the extended rigid body to the simple pendulum with 
imaginary time.

As for future work, because the analysis performed in this paper has been established only classically it would be 
interesting to extend the construction to the quantum framework. We expect this analysis could produce interesting relations between Lame differential equation, which governs the quantum behaviour of the rigid body, and the Mathieu differential equation, which governs the quantum behaviour of the simple pendulum. A second possible direction of research is to investigate whether or not the dynamics of the pendulum could be used also to implement one-qubit quantum gates as the rigid body does <cit.>. 

The work of M. de la C. and N. G. is supported by the Ph.D. scholarship program of the Universidad 
Autónoma Metropolitana. The work of R. L. is  partially supported from CONACyT Grant No. 237351 
“Implicaciones físicas de la estructura del espacio-tiempo".


