Academic
Publications
The Stackel systems and algebraic curves

The Stackel systems and algebraic curves,A. V. Tsiganov

The Stackel systems and algebraic curves   (Citations: 13)
BibTex | RIS | RefWorks Download
We show how the Abel-Jacobi map provides all the principal properties of an ample family of integrable mechanical systems associated to hyperelliptic curves. We prove that derivative of the Abel-Jacobi map is just the St\"{a}ckel matrix, which determines $n$-orthogonal curvilinear coordinate systems in a flat space. The Lax pairs, $r$-matrix algebras and explicit form of the flat coordinates are constructed. An application of the Weierstrass reduction theory allows to construct several flat coordinate systems on a common hyperelliptic curve and to connect among themselves different integrable systems on a single phase space.
Published in 1997.
Cumulative Annual
View Publication
The following links allow you to view full publications. These links are maintained by other sources not affiliated with Microsoft Academic Search.
    • ...The system associated with the name of St¨ackel [13, 15, 16] is a holonomic system on the phase space R 2n , with the canonical variables q = (q1,...,qn) and p = (p1,...,pn):...
    • ...It allows reducing the solution of the equations of motion to a problem in algebraic geometry [2, 4, 15]...
    • ...which consist of integrals of first kind Abelian differentials on the hyperelliptic curves Cj (2.4) [2, 4, 15, 16]...

    A. V. Tsiganov. On maximally superintegrable systems

    • ...The system associated with the name of St¨ackel [19, 20] is a holonomic system on the phase space M = R 2n , with the canonical variables q = (q1, . . . , qn) and p = (p1, . . . , pn):...
    • ...It allows reducing solution of the equations of motion to a problem in algebraic geometry [20]...
    • ...which are usually the sums of integrals ϑij of the first kind Abelian differentials on the hyperelliptic curves Cj (3.5) [20, 24]...
    • ...If these curves are equal Cj = C then F may be identified with the Jacobian J(C) of C [20]...
    • ...As above the separated relations (5.9) coincide with the Jacobi relations for the uniform St¨ackel systems [19, 20, 21]...
    • ...such that the St¨ackel matrix is a lowest block of the corresponding Brill-Noether matrix [20, 21]...

    A V Tsiganov. Leonard Euler: addition theorems and superintegrable systems

    • ...The system associated with the name of St¨ackel [9, 10] is a holonomic system on the phase space M = R2n, with the canonical variables q = (q1, . . . , qn) and p =...
    • ...It allows reducing solution of the equations of motion to a problem in algebraic geometry [10]...
    • ...which are the sums of integrals ϑij of the first kind Abelian differentials on the hyperelliptic curves Cj (2.5) [10, 14], i.e...

    A. V. Tsiganov. Addition theorems and the Drach superintegrable systems

    • ...Recall the St¨ackel matrix is a n×n block of the transpose Brill-Noether matrix, which is a differential of the Abel-Jacobi map associated with a product F(�) of the algebraic curves Ci (2.11), see [21] and references within...
    • ...Let us consider uniform St¨ackel systems [21] for which the Lagrangian submanifold...
    • ...Remark 2 The St¨ackel matrix S (2.18) is one of the most studied matrices, which appears very often in various applications [2, 6, 7, 12, 21]...

    A. V. Tsiganov. Towards a classification of natural bi-hamiltonian systems

    • ...Therefore, we can suppose that the Steklov-Lyapunov system belong to the family of the St¨ackel systems [16] and there are the Lax matrices with rational dependence on the spectral parameter...
    • ...In our case the St¨ackel matrix S (2.12) is the lowest block of the transpose Brill-Noether matrix U C (2.13)) and, therefore, there are canonical coordinates in which equations of motion (2.8) are the Newton equations [16, 17]...
    • ...According to [10, 16, 17] the generic 2 × 2 Lax matrices for the uniform St¨ackel system are constructed using Hamiltonian H1 and the generating function e(λ) of the separated variables only...
    • ...parametric function and [ξ]MN is the linear combinations of the following Laurent projections [16, 17]...
    • ...The corresponding r-matrix is rational dynamical matrix [16, 17]...

    A. V. Tsiganov. On the Steklov-Lyapunov case of the rigid body motion

Sort by: