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Algebraic Curve
Coordinate System
hyperelliptic curve
Integrable System
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Related Publications
(3)
On maximally superintegrable systems
Duality between integrable Stackel systems
Bihamiltonian geometry and separation of variables for Toda lattices
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The Stackel systems and algebraic curves
The Stackel systems and algebraic curves,A. V. Tsiganov
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The Stackel systems and algebraic curves
(
Citations: 13
)
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A. V. Tsiganov
We show how the AbelJacobi map provides all the principal properties of an ample family of integrable
mechanical systems
associated to hyperelliptic curves. We prove that derivative of the AbelJacobi 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.
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Citation Context
(7)
...The system associated with the name of St¨ackel
[1
3, 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
[1
9, 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
[1
9, 20, 21]...
...such that the St¨ackel matrix is a lowest block of the corresponding BrillNoether mat
ri
x [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 BrillNoether matrix, which is a differential of the AbelJacobi map associated with a product F(�) of the algebraic curves Ci (2.
11
), see [21] and references within...
...Let us consider uniform St¨ackel syst
em
s [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,
1
2, 21]...
A. V. Tsiganov
.
Towards a classification of natural bihamiltonian systems
...Therefore, we can suppose that the SteklovLyapunov system belong to the family of the St¨ackel syst
em
s [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 BrillNoether matrix U C (2.13)) and, therefore, there are canonical coordinates in which equations of motion (2.8) are the Newton equati
on
s [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 rmatrix is rational dynamical matrix [
16
, 17]...
A. V. Tsiganov
.
On the SteklovLyapunov case of the rigid body motion
References
(26)
Mathematical Methods of Classical Mechanics
(
Citations: 1311
)
V. I. Arnold
Published in 1975.
Symplectic forms in the theory of solitons
(
Citations: 38
)
I. M. Krichever
,
D. H. Phong
Published in 1997.
Arnold''s formula for algebraically completely integrable systems
(
Citations: 6
)
J. P. Francoise
Published in 1987.
Classical adiabatic angles and quantal adiabatic phase
(
Citations: 30
)
M. V. Berry
Journal:
Journal of Physics Amathematical and General  J PHYSAMATH GEN
, vol. 18, no. 1, pp. 1527, 1985
Separable systems of St篓ackel
(
Citations: 93
)
L. P. Eisenhart
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Citations
(13)
On natural Poisson bivectors on the sphere
(
Citations: 2
)
A V Tsiganov
Published in 2010.
On the superintegrable Richelot systems
A. V. Tsiganov
Journal:
Journal of Physics Amathematical and Theoretical  J PHYS AMATH THEOR
, vol. 43, no. 5, 2010
Integrable Euler top and nonholonomic Chaplygin ball
A V Tsiganov
Published in 2010.
On maximally superintegrable systems
(
Citations: 8
)
A. V. Tsiganov
Journal:
Regular & Chaotic Dynamics  REGUL CHAOTIC DYN
, vol. 13, no. 3, pp. 178190, 2008
Leonard Euler: addition theorems and superintegrable systems
(
Citations: 3
)
A V Tsiganov
Published in 2008.