Rotations of Asymmetric Molecules and the Hydrogen Atom in Free and Confined Configurations

Classification of the classicalSL(2,R)gauge transformations in the rigid body

“…Our analysis is classical and can be considered as a first step in the direction to generalize the quantum results obtained for the symmetrical rigid bodies [10], to the asymmetrical ones. In fact there are some recent results in the study of the quantum spectrum of the rotational motion of asymmetric molecules [14,[16][17][18][19], that could be generalized to the extended rigid body problem, and could bring some light in the quantization of the system using the Nambu brackets for the asymmetric cases. Although the rigid body problem has been extensively studied over the years, from the classical mechanics point of view there are still some interesting issues that could be addressed.…”

“…These cases can be obtained as limiting situations of the general asymmetric rigid body taking two inertia parameters equal. There are two of such limits in the parameters ordering (17) we are discussing here. The so called prolate limit e 3 = e 2 and the oblate limit e 2 = e 1 .…”

“…In the following without loosing generality we will assume that the inertia parameters satisfy inequalities (17). The condition for intersection between the sphere of angular momentum and the energy surface becomes in these parameters e 3 < e 0 < e 1 .…”

“…We include in our discussion the Bianchi nomenclature for the different Lie algebras [15]. Finally it would be very interesting to find the relation between the Nambu dynamics and some recent results in the area of atomic physics and asymmetric rotational molecular spectrums [16][17][18][19]. This issue will be addressed elsewhere.…”

The Extended Rigid Body and the Pendulum Revisited

“…A geometric construction of the solution was given later on by Poinsot [17] and analytically these solutions are given, as for the simple pendulum, by elliptic functions (see for instance [18][19][20][21] and references therein). The quantization of the problem was attacked first by Kramers and Ittmann [22] and since then many authors have contributed to understand deeper many aspects of the problem [23][24][25][26][27][28][29][30].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 [31].…”