Integrable discretizations of some cases of the rigid body dynamics

A heavy top with a fixed point and a rigid body in an ideal fluid are important examples of Hamiltonian systems on a dual to the semidirect product Lie algebra $e(n)=so(n)\ltimes\mathbb R^n$. We give a Lagrangian derivation of the corresponding equations of motion, and introduce discrete time analogs of two integrable cases of these systems: the Lagrange top and the Clebsch case, respectively. The construction of discretizations is based on the discrete time Lagrangian mechanics on Lie groups, accompanied by the discrete time Lagrangian reduction. The resulting explicit maps on $e^*(n)$ are Poisson with respect to the Lie--Poisson bracket, and are also completely integrable. Lax representations of these maps are also found.Comment: arXiv version is already officia

Argument shift method and sectional operators: applications to differential geometry

This paper does not contain any new results, it is just an attempt to present, in a systematic way, one construction which establishes an interesting relationship between some ideas and notions well-known in the theory of integrable systems on Lie algebras and a rather different area of mathematics studying projectively equivalent Riemannian and pseudo-Riemannian metrics

Geodesic flows and Neumann systems on Stiefel varieties: geometry and integrability

We study integrable geodesic flows on Stiefel varieties Vn,r = SO(n)/SO(n−r ) given by the Euclidean, normal (standard), Manakov-type, and Einstein metrics.We also consider natural generalizations of the Neumann systems on Vn,r with the above metrics and proves their integrability in the non-commutative sense by presenting compatible Poisson brackets on (T ∗Vn,r )/SO(r ). Various reductions of the latter systems are described, in particular, the generalized Neumann system on an oriented Grassmannian Gn,r and on a sphere Sn−1 in presence of Yang–Mills fields or a magnetic monopole field. Apart from the known Lax pair for generalized Neumann systems, an alternative (dual) Lax pair is presented, which enables one to formulate a generalization of the Chasles theorem relating the trajectories of the systems and common linear spaces tangent to confocal quadrics. Additionally, several extensions are considered: the generalized Neumann system on the complex Stiefel variety Wn,r = U(n)/U(n − r ), the matrix analogs of the double and coupled Neumann systems.Peer ReviewedPostprint (published version

Argument shift method and sectional operators: applications to differential geometry

This text does not contain any new results, it is just an attempt to present, in a systematic way, one construction which makes it possible to use some ideas and notions well-known in the theory of integrable systems on Lie algebras to a rather different area of mathematics related to the study of projectively equivalent Riemannian and pseudo-Riemannian metrics. The main observation can be formulated, yet without going into details, as follows: The curvature tensors of projectively equivalent metrics coincide with the Hamiltonians of multi-dimensional rigid bodies. Such a relationship seems to be quite interesting and may apparently have further applications in differential geometry. The wish to talk about this relation itself (rather than some new results) was one of motivations for this paper