Bäcklund transformations for the nonholonomic Veselova system

Abstract

We present auto and hetero Bäcklund transformations of the nonholonomic Veselova system using standard divisor arithmetic on the hyperelliptic curve of genus two. As a by-product one gets two natural integrable systems on the cotangent bundle to the unit two-dimensional sphere whose additional integrals of motion are polynomials in the momenta of fourth order.

Duffing Oscillator and Elliptic Curve Cryptography

“…In [21][22][23][24] we apply the same algorithms of elliptic and hyperelliptic curve cryptography to study discrete versions of the Lagrange top, Hénon -Heiles system, nonholonomic Veselova and Chaplygin systems, etc. These standard algorithms of hyperelliptic curve cryptography can be also applied to the so-called cubic-quintic Duffing oscillator q + 4Aq + 8Bq 3 + 12Cq 5 = 0, which can be found in the modeling of free vibrations of a restrained uniform beam with intermediate lumped mass, the nonlinear dynamics of slender elastica, the generalized Pochhammer -Chree (PC) equation, the generalized compound KdV equation in nonlinear wave systems, etc.…”

Hamiltonization and Separation of Variables for a Chaplygin Ball on a Rotating Plane

“…The main advantage of the Jacobi method is that using variables of separation, we obtain not only quadratures, but also families of compatible Poisson brackets, recursion operators, algebras of Haantjes operators, master symmetries, Lax matrices, new integrable systems and exact discretization of the original equations of motion [41,42,43].…”

Bäcklund transformations and divisor doubling

“…We can identify this Hamiltonian with well-known second integrable Hénon-Heiles system with quartic additional integral H 2 [25,27]. According [26,28,29,30,31] we can use these cryptographic protocols in order to get new integrable systems on the plane, sphere and ellipsoid with polynomial integrals of motion of sixth, fourth and third order in momenta.…”