The method of separation of variables facilitates the integration of the Hamilton-Jacobi equation by reducing its solution to a series of quadratures in the separable coordinates. The case in which the metric tensor is diagonal in the separable coordinates, that is, orthogonal separability, is fundamental. Recent theory by Benenti has established a concise geometric (coordinate-independent) characterisation of orthogonal separability of the Hamilton-Jacobi equation on a pseudoRiemannian manifold. It generalises an approach initiated by Eisenhart and developed by Kalnins and Miller. Benenti has shown that the orthogonal separability of a system via a point transformation is equivalent to the existence of a Killing tensor with real simple eigen values and orthogonally integrable eigenvectors. Applying a moving frame formalism, we develop a method that produces the orthogonal separable coordinates for low dimensional Hamiltonian systems. The method is applied to a two dimensional Riemannian manifold of arbitrary curvature. As an illustration, we investigate Euclidean 2-space, and the two dimensional surfaces of constant curvature, recovering known results. Using our formalism, we also derive the known superseparable potentials for Euclidean 2-space. Some of the original results presented in this thesis were announced in [8, 9, 10].

The method of separation of variables facilitates the integration of the Hamilton-Jacobi equation by reducing its solution to a series of quadratures in the separable coordinates. The case in which the metric tensor is diagonal in the separable coordinates, that is, orthogonal separability, is fundamental. Recent theory by Benenti has established a concise geometric (coordinate-independent) characterisation of orthogonal separability of the Hamilton-Jacobi equation on a pseudoRiemannian manifold. It generalises an approach initiated by Eisenhart and developed by Kalnins and Miller. Benenti has shown that the orthogonal separability of a system via a point transformation is equivalent to the existence of a Killing tensor with real simple eigen values and orthogonally integrable eigenvectors. Applying a moving frame formalism, we develop a method that produces the orthogonal separable coordinates for low dimensional Hamiltonian systems. The method is applied to a two dimensional Riemannian manifold of arbitrary curvature. As an illustration, we investigate Euclidean 2-space, and the two dimensional surfaces of constant curvature, recovering known results. Using our formalism, we also derive the known superseparable potentials for Euclidean 2-space. Some of the original results presented in this thesis were announced in [8, 9, 10].

<p>In case of the PDE's the concept of solving by separation of variables</p><p>has a well defined meaning. One seeks a solution in a form of a</p><p>product or sum and tries to build the general solution out of these</p><p>particular solutions. There are also known systems of second order</p><p>ODE's describing potential motions and certain rigid bodies that are</p><p>considered to be separable. However, in those cases, the concept of</p><p>separation of variables is more elusive; no general definition is</p><p>given.</p><p>In this thesis we study how these systems of equations separate and find that their separation usually can be reduced to sequential separation of single first order ODE´s. However, it appears that other mechanisms of separability are possible.</p>

In case of the PDE's the concept of solving by separation of variables has a well defined meaning. One seeks a solution in a form of a product or sum and tries to build the general solution out of these particular solutions. There are also known systems of second order ODE's describing potential motions and certain rigid bodies that are considered to be separable. However, in those cases, the concept of separation of variables is more elusive; no general definition is given. In this thesis we study how these systems of equations separate and find that their separation usually can be reduced to sequential separation of single first order ODE´s. However, it appears that other mechanisms of separability are possible.

Dans cette thèse nous donnons une solution des chaînes quantiques de spin-1/2 XXZ et XYZ ouvertes avec les termes de bord intégrables les plus généraux. En utilisant la méthode de la Séparation des Variable (SoV), à la Sklyanin, on est capable, dans le cas inhomogène, de construire l’ensemble complet des états propres et des valeurs propres associés. La caractérisation de ces quantités est faite par un système maximal de N équations quadratiques, où N est la taille du système. Des méthodes différentes, comme l’ansatz de Bethe algébrique (ABA) ou autres généralisations de l’ansatz de Bethe, ont été utilisés dans le passé pour résoudre ces problèmes. Aucune méthode a pu effectivement reproduire l’ensemble complet des états propres et valeur propres dans le cas de conditions au bord les plus génériques. Une expression, sous forme d’un déterminant à la Vandermonde, pour les produits scalaires entre les états en représentation de SoV est aussi obtenue. La formule pour les produits scalaires représente la première étape pour approcher le problème relié au calcul des facteurs de forme et fonctions de corrélations. / In this thesis we give accounts on the solution of the open XXZ and XYZ quantum spin-1/2 chains with the most generic integrable boundary terms. By using the the Separation of Variables method (SoV), due to Sklyanin, we are able, in the inhomogeneous case, to build the complete set of eigenstates and the associated eigenvalues. The characterization of these quantities is made through a maximal system of N quadratic equations, where N is the size of the chain. Different methods, like the Algebraic Bethe ansatz (ABA) or other generalized Bethe ansatz techniques, have been used, in the past, in order to tackle these problems. None of them resulted effective in the reproduction of the full set of eigenstates and eigenvalues in the case of most general boundary conditions. A Vandermonde determinant formula for the scalar products of SoV states is obtained as well. The scalar product formula represents a first step towards the calculation of form factors and correlation functions.

Numerical solutions of mathematical modelizations of heat and mass transfer in cubical and cylindrical reactors of solar adsorption refrigeration systems are studied. For the resolution of the equations describing the coupling between heat and mass transfer, Bubnov-Galerkin method is used. An exact solution for time dependent heat transfer in cylindrical multilayered annulus is presented. Separation of variables method has been used to investigate the temperature behavior. An analytical double series relation is proposed as a solution for the temperature distribution, and Fourier coefficients in each layer are obtained by solving some set of equations related to thermal boundary conditions at inside and outside of the cylinder.

This thesis is devoted to creating a systematic way of determining all inequivalent orthogonal coordinate systems which separate the Hamilton-Jacobi equation for a given natural Hamiltonian defined on three-dimensional spaces of constant, non-zero curvature. To achieve this, we represent the problem with Killing tensors and employ the recently developed invariant theory of Killing tensors. Killing tensors on the model spaces of spherical and hyperbolic space enjoy a remarkably simple form; even more striking is the fact that their parameter tensors admit the same symmetries as the Riemann curvature tensor, and thus can be considered algebraic curvature tensors. Using this property to obtain invariants and covariants of Killing tensors, together with the web symmetries of the associated orthogonal coordinate webs, we establish an equivalence criterion for each space. In the case of three-dimensional spherical space, we demonstrate the surprising result that these webs can be distinguished purely by the symmetries of the web. In the case of three-dimensional hyperbolic space, we use a combination of web symmetries, invariants and covariants to achieve an equivalence criterion. To completely solve the equivalence problem in each case, we develop a method for determining the moving frame map for an arbitrary Killing tensor of the space. This is achieved by defining an algebraic Ricci tensor. Solutions to equivalence problems of Killing tensors are particularly useful in the areas of multiseparability and superintegrability. This is evidenced by our analysis of symmetric potentials defined on three-dimensional spherical and hyperbolic space. Using the most general Killing tensor of a symmetry subspace, we derive the most general potential “compatible” with this Killing tensor. As a further example, we introduce the notion of a joint invariant in the vector space of Killing tensors and use them to characterize a well-known superintegrable potential in the plane. xiii

Dans cette thèse, nous proposons de nouveaux résultats de systèmes superintégrables séparables en coordonnées polaires. Dans un premier temps, nous présentons une classification complète de tous les systèmes superintégrables séparables en coordonnées polaires qui admettent une intégrale du mouvement d'ordre trois. Des potentiels s'exprimant en terme de la sixième transcendante de Painlevé et de la fonction elliptique de Weierstrass sont présentés. Ensuite, nous introduisons une famille infinie de systèmes classiques et quantiques intégrables et exactement résolubles en coordonnées polaires. Cette famille s'exprime en terme d'un paramètre k. Le spectre d'énergie et les fonctions d'onde des systèmes quantiques sont présentés. Une conjecture postulant la superintégrabilité de ces systèmes est formulée et est vérifiée pour k=1,2,3,4. L'ordre des intégrales du mouvement proposées est 2k où k ∈ ℕ. La structure algébrique de la famille de systèmes quantiques est formulée en terme d'une algèbre cachée où le nombre de générateurs dépend du paramètre k. Une généralisation quasi-exactement résoluble et intégrable de la famille de potentiels est proposée. Finalement, les trajectoires classiques de la famille de systèmes sont calculées pour tous les cas rationnels k ∈ ℚ. Celles-ci s'expriment en terme des polynômes de Chebyshev. Les courbes associées aux trajectoires sont présentées pour les premiers cas k=1, 2, 3, 4, 1/2, 1/3 et 3/2 et les trajectoires bornées sont fermées et périodiques dans l'espace des phases. Ainsi, les résultats obtenus viennent renforcer la possible véracité de la conjecture. / In this thesis, we propose new superintegrable systems separable in polar coordinates. After the introduction, in chapter 2, we present a complete classification of all separable systems in polar coordinates which admit a third order integral in addtion to the second order one responsible for the separation of variables. New potentials expressed in terms of the sixth Painlevé transcendent and of the Weierstrass elliptic function are obtained. In chapter 3 we introduce an infinite family of integrable and exactly sovable classical and quantum systems separable in polar coordinates. This family is described in term of a parameter k. The energy spectrum and the wave functions of the quantum systems are obtained. A conjecture postulating the superintegrability of these systems is formulated and is verified for the first cases k = 1,2,3,4. The order of the integrals is 2k where k ∈ ℕ. The algebraic structure of the family of quantum systems is formulated in term of a hidden algebra where the number of generators depends on the parameter k. A quasi-exactly solvable and integrable generalization of the family of potentials is proposed. Finally in chapter 4, the classical trajectories of the family of systems are calculated for all the rational cases k ∈ ℚ. Those are expressed in term of Chebyshev polynomials. We plot the curves associated with the trajectories for k=1,2,3,4,1/2, 1/3 and 3/2. The bounded curves are closed and periodic in the two dimensional phase space. Those results obtained reinforce the possible veracity of the conjecture.

Ce mémoire est une poursuite de l’étude de la superintégrabilité classique et quantique dans un espace euclidien de dimension deux avec une intégrale du mouvement d’ordre trois. Il est constitué d’un article. Puisque les classifications de tous les Hamiltoniens séparables en coordonnées cartésiennes et polaires sont déjà complétées, nous apportons à ce tableau l’étude de ces systèmes séparables en coordonnées paraboliques. Premièrement, nous dérivons les équations déterminantes d’un système en coordonnées paraboliques et ensuite nous résolvons les équations obtenues afin de trouver les intégrales d’ordre trois pour un potentiel qui permet la séparation en coordonnées paraboliques. Finalement, nous démontrons que toutes les intégrales d’ordre trois pour les potentiels séparables en coordonnées paraboliques dans l’espace euclidien de dimension deux sont réductibles. Dans la conclusion de l’article nous analysons les différences entre les potentiels séparables en coordonnées cartésiennes et polaires d’un côté et en coordonnées paraboliques d’une autre côté. Mots clés: intégrabilité, superintégrabilité, mécanique classique, mécanique quantique, Hamiltonien, séparation de variable, commutation. / This thesis is a contribution to the study of classical and quantum superintegrability in a two-dimensional Euclidean space involving a third order integral of motion. It consists of an article. Because the classifications of all separable hamiltonians into Cartesian and polar coordinates are already complete, we bring to this picture the study of those systems in parabolic coordinates. First, we derive the determinating equations of a system into parabolic coordinates, after which we solve the obtained equations in order to find integrals of order three for potentials, which allow the separations of variables into the parabolic coordinates. Finally, we prove that all the third order integrals for separable potentials in parabolic coordinates in the Euclidean space of dimension two are reducible. In the conclusion of this article, we analyze the differences between the separable potentials in Cartesian and polar coordinates and the separable potentials in parabolic coordinates. Keywords: integrability, superintegrability, classical mechanics, quantum mechanics, Hamiltonian, separation of variables, commutation.

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