54 citations · 777 across the 87 of their papers we have counts for
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On the Darboux-Halphen system: Jacobi vs Lie
A. V. Tsiganov
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…
On integrable by Euler planar differential systems
A. V. Tsiganov
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 Calcul…
On the Lie noncommutative integrability
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-…
Comment on: "Stäckel and Eisenhart lifts, Haantjes geometry and Gravitation"
A. V. Tsiganov
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 Haantje…
Homogeneous potentials, Lagrange's identity and Poisson geometry
A. V. Tsiganov
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 fo…
Novikov equations for commuting differential operators of orders 3,4,5
G. B. Shabat, V. V. Sokolov, A. V. Tsiganov
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…
On novel Hamiltonian description of the nonholonomic Suslov problem
A. V. Tsiganov
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 fu…
On tensor invariants of the Clebsch system
A. V. Tsiganov
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 f…
On computation of Darboux polynomials for full Toda lattice
A. V. Tsiganov
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…
On tensor invariants for integrable cases of Euler, Lagrange and Kovalevskaya rigid body motion
A. V. Tsiganov
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…
Rotations and integrability
A. V. Tsiganov
We discuss some families of integrable and superintegrable systems in -dimensional Euclidean space which are invariant to rotations. The integrable invariant Hamilto…
Second order Killing tensors related to symmetric spaces
E. O. Porubov, A. V. Tsiganov
We discuss the pairs of quadratic integrals of motion belonging to the -dimensional space of independent integrals of motion in involution, that provide integrability of the cor…
Integrable systems on the sphere, ellipsoid and hyperboloid
A. V. Tsiganov
Affine transformations in Euclidean space generates a correspondence between integrable systems on cotangent bundles to the sphere, ellipsoid and hyperboloid embedded in . Usi…
Equivalent Integrable Metrics on the Sphere with Quartic Invariants
Andrey V. Tsiganov
We discuss canonical transformations relating well-known geodesic flows on the cotangent bundle of the sphere with a set of geodesic flows with quartic invariants. By adding variou…
On two-dimensional Hamiltonian systems with sixth-order integrals of motion
E. O. Porubov, A. V. Tsiganov
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…
Reducible Abelian varieties and Lax matrices for Euler's problem of two fixed centres
A. V. Tsiganov
Abel's quadratures for integrable Hamiltonian systems are defined up to a group law of the corresponding Abelian variety . If is isogenous to a direct product of Abelian var…
On integrable systems outside Nijenhuis and Haantjes geometry
A. V. Tsiganov
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…
Reduction of divisors and Clebsch system
A. V. Tsiganov
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…
Reduction of divisors and Kowalevski top
A. V. Tsiganov
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…
Reduction of divisors for classical superintegrable magnetic chain
A. V. Tsiganov
Variables of separation for classical magnetic chain obtained by Sklyanin form a generic positive divisor of degree on a genus non-hyperelliptic algebraic curve…
On rheonomic nonholonomic deformations of the Euler equations proposed by Bilimovich
A. V. Borisov, A. V. Tsiganov
In 1913 A.D. Bilimovich observed that rheonomic linear and homogeneous in generalized velocities constraints are ideal. As a typical example, he considered rheonomic nonholonomic d…
On the Chaplygin sphere in a magnetic field
A. V. Borisov A. V. Tsiganov
We consider the possibility of using Dirac's ideas of the deformation of Poisson brackets in nonholonomic mechanics. As an example, we analyze the composition of external forces th…
Superintegrable systems and Riemann-Roch theorem
A. V. Tsiganov
In algebraic geometry, there is a reduction algorithm that transforms the unreduced divisor into a unique reduced divisor, which existence is guaranteed by the Riemann-Roch theorem…
The Kepler problem: polynomial algebra of non-polynomial first integrals
A. V. Tsiganov
The sum of elliptic integrals simultaneously determines orbits in thr Kepler problem and the addition of divisors on elliptic curves. Periodic motion of a body in physical space is…