arXiv subjects
Publications and source records attributed to A. V. Tsiganov.
The Lie theory of non-commutative integrability is used to reconstruct some integrable systems of ordinary differential equations in three dimensional Eucledian space. The Darboux-Brioschi-Halphen system is an example of the Lie integrable system associated with the simple Lie algebra sl(2,R). Other examples are related with solvable three dimensional real Lie algebras of Bianchi B class.
Two constructions of the Darboux-Halphen system are discussed. In the Jacobi construction we start with transcendental functions which are fixed as the first integrals. In the Lie construction we use a single-valued representation of the simple Lie algebra $sl(2,\mathbb R)$ which is non-integrable in Lie's terminology.
The subject of our discussion is the theory of differential equations as set out in two classical Euler's textbooks "Institutiones Calculi Differentialis" and "Institutiones Calculi Integralis".
We present some new Poisson bivectors that are invariants by the flow of the nonholonomic Suslov problem. Two rank four invariant Poisson bivectors have globally defined Casimir functions and, therefore, define cubic Poisson brackets on the five dimensional state space with standard symplectic leaves. For the Suslov gyrostat in the potential field we found rank two Poisson bivectors having only two globally defined Casimir functions and, therefore, we say about formal Hamiltonian description in these cases.
The Lagrange identity expresses the second derivative of the moment of inertia of a system of material points through kinetic energy and homogeneous potential energy, from which follows the Jacobi well-known result on the instability of a system of gravitating bodies. In this work, it is proven that if a Hamiltonian system satisfies the Lagrange identity, then it possesses additional tensor invariants that are not expressed through the basic invariants existing for all Hamiltonian systems. A new class of Hamiltonian systems with inhomogeneous potentials is considered, which also possess similar additional tensor invariants.
One of the oldest methods for constructing integrable Hamiltonian systems, proposed by Jacobi, recently is being presented as a novel Stäckel lift construction related with Haantjes geometry. It may cause some confusion.
We consider Novikov equations for commutative ring generated by differential operators of orders 3,4,5. We present an explicit Hamiltonian form of these equations. Using the method of compatible Poisson brackets, we find a separation of variables on a hyperelliptic curve of genus 2 for the Novikov equations.
We present some new Poisson bivectors that are invariants by the Clebsch system flow. Symplectic integrators on their symplectic leaves exactly preserve the corresponding Casimir functions, which have different physical meanings. The Kahan discretization of the Clebsch system is discussed briefly.
Multi-symplectic integrators are typically regarded as a discretization of the Hamiltonian partial differential equations. This is due to the fact that, for generic finite-dimensional Hamiltonian systems, there exists only one independent symplectic structure. In this note, the second invariant symplectic form is presented for the nonintegrable Henon-Heiles system, Kepler problem, integrable and non-integrable Toda type systems. This approach facilitates the construction of a multi-symplectic integrator, which effectively preserves both symplectic forms for these benchmark problems.
One of the oldest methods for computing invariants of ordinary differential equations is tested using the full Toda lattice model. We show that the standard method of undetermined coefficients and modern symbolic algebra tools together with sufficient computing power allow to compute Darboux invariants without any additional information.
We discuss some families of integrable and superintegrable systems in $n$-dimensional Euclidean space which are invariant to $m\geq n-2$ rotations. The integrable invariant Hamiltonian $H=\sum p_i^2+V(q)$ commutes with $n-2$ integrals of motion $M_α$ and an additional integral of motion $G$, which are polynomials of first and fourth-order in momenta, respectively.
We discuss global tensor invariants of a rigid body motion in the cases of Euler, Lagrange and Kovalevskaya. These invariants are obtained by substituting tensor fields with cubic on variable components into the invariance equation and solving the resulting algebraic equations using computer algebra systems. According to the Poincaré-Cartan theory of invariants, the existence of invariant geometric structures raises the question of using them to study the dynamics.
We discuss the pairs of quadratic integrals of motion belonging to the $n$-dimensional space of independent integrals of motion in involution, that provide integrability of the corresponding Hamiltonian equations of motion by quadratures. In contrast to the Eisenhart theory, additional integrals of motion are polynomials of the fourth, sixth and other orders in momenta. The main focus is on the second-order Killing tensors corresponding to quadratic integrals of motion and relating to the special combinations of rotations and translations in Euclidean space.
Affine transformations in Euclidean space generates a correspondence between integrable systems on cotangent bundles to the sphere, ellipsoid and hyperboloid embedded in $R^n$. Using this correspondence and the suitable coupling constant transformations we can get real integrals of motion in the hyperboloid case starting with real integrals of motion in the sphere case. We discuss a few such integrable systems with invariants which are cubic, quartic and sextic polynomials in momenta.
There are a few Lax matrices of the Clebsch system. Poles of the Baker-Akhiezer function determine the class of equivalent divisors on the corresponding spectral curves. According to the Riemann-Roch theorem, each class has a unique reduced representative. We discuss properties of such reduced divisor on the spectral curve of $3\times 3$ Lax matrix having a natural generalization to $gl^*(n)$ case.
We study non-invariant Killing tensors with non-zero Nijenhuis torsion in the three-dimensional Euclidean space. Generalizing the corresponding integrable systems we construct two new families of superintegrable systems in $n$-dimensional Euclidean space.
We obtain 21 two-dimensional natural Hamiltonian systems with sextic invariants, which are polynomial of the sixth order in momenta. Following to Bertrand, Darboux, and Drach these results of the standard brute force experiments can be applied to construct a new mathematical theory.
In the modern theory of the Kowalevski top there are two elliptic curves introduced by Kowalevski and by Reyman and Semenov-Tian-Shansky. The Kowalevski variables of separation and poles of the Baker-Akhiezer function define two classes of linearly equivalent divisors on these elliptic curves. According to the Riemann-Roch theorem each class has a unique reduced representative and we construct such reduced divisors for the Kowalevski top.