On natural Poisson bivectors on the sphere

Abstract

We discuss the concept of natural Poisson bivectors, which allows us to consider the overwhelming majority of known integrable systems on the sphere in framework of bi-Hamiltonian geometry. IntroductionThe Hamilton-Jacobi theory seems to be one of the most powerful methods of investigation the dynamics of mechanical (holonomic and nonholonomic) and control systems. Besides its fundamental aspects such as its relation to the action integral and generating functions of symplectic maps, the theory is known to be very useful in integrating the Hamilton equations using the variables separation technique. The milestones of this technique include the works of Stäckel, Levi-Civita, Eisenhart, Woodhouse, Kalnins, Miller, Benenti and others. The majority of results was obtained for a very special class of integrable systems, important from the physical point of view, namely for the systems with quadratic in momenta integrals of motion. The Kowalevski, Chaplygin and Goryachev results on separation of variables for the systems with higher order integrals of motion missed out of this scheme.Bi-Hamiltonian structures can be seen as a dual formulation of integrability and separability, in the sense that they substitute a hierarchy of compatible Poisson structures to the hierarchy of functions in involution, which may be treated either as integrals of motion or as variables of separation for some dynamical system. The Eisenhart-Benenti theory was embedded into the bi-Hamiltonian set-up using the lifting of the conformal Killing tensor that lies at the heart of Benenti's construction [8,15]. The concept of natural Poisson bivectors allows us to generalize this construction and to study systems with quadratic and higher order integrals of motion in framework of a single theory [31].The aim of this note is to bring together all the known examples of natural Poisson bivectors on the sphere, because a good example is the best sermon. Some of these Poisson bivectors have been obtained and presented earlier in different coordinate systems and notations. Here we propose the unified description of this known and few new bivectors using so-called geodesic Π and potential Λ matrices [31]. In some sense we propose new form for the old content and believe that this unification is a first step to the geometric analysis of various natural systems on the sphere, which reveals what they have in common and indicates the most suitable strategy to obtain and to analyze their solutions.The corresponding integrable natural systems on two-dimensional unit sphere S 2 are related to rigid body dynamics. In order to describe these systems we will use the angular momentum vector J = (J 1 , J 2 , J 3 ) and the Poisson vector x = (x 1 , x 2 , x 3 ) in a moving frame of coordinates attached to the principal axes of inertia [4]. The Poisson brackets between these variablesmay be associated to the Lie-Poisson algebra of the three-dimensional Euclidean algebra e(3) with two Casimir elementsx k J k .(1.2)Below we always put C 2 = 0.

On quadratures of integrable systems on a sphere with higher degree integrals of motion

“…Следуя [19][20][21]23], определим новые вещественные координаты q 1,2 на кокасательном расслоении T * S 2 , в виде корней следующего полинома B(λ) = (λ − q 1 )(λ − q 2 ) = λ 2 …”

“…Согласно [23] , в случае α = 1 координаты q 1,2 (2.1) являются переменными разделения для уравнения Гамильтона-Якоби при…”

“…Второй интеграл движения при этом равен Согласно [11,23], каноническое преобразование (1.8) приводит данную функцию Гамильто-на (2.9) к натуральному виду…”

“…Согласно [19,23], координаты q 1,2 (2.1) являются переменными разделения для данной интегрируемой системы при α = 2 и b = 0. В этом случае q 1,2 будут корнями полинома (2.1)…”

“…В ряде работ [19][20][21][22][23]25] были построены новые переменные разделения для несколь-ких интегрируемых систем натурального вида на двумерной единичной сфере, обладающих интегралами движения третьей и четвертой степени по импульсам. Основной целью этих работ было развитие аппарата бигамильтоновой геометрии, позволяющего вычислять пе-ременные разделения и разделенные уравнения, используя при этом только выражения для интегралов движения и определение кинематической скобки Пуассона, относительно которой они находятся в инволюции.…”