The Stäckel systems and algebraic curves

Abstract

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 Stäckel 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.arch-ive/9712003

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“…Remark 2 The Stäckel matrix S (2.18) is one of the most studied matrices, which appears very often in various applications [2,6,7,12,21]. But we have to keep in mind that there are many other Stäckel matrices associated with hyperelliptic or non-hyperelliptic curves.…”

“…Let us consider uniform Stäckel systems [21] for which the Lagrangian submanifold F (α) is the n-th symmetric product of a hyperelliptic curve C. For the standard basis of the holomorphic differentials on C associated with the Veronese map the corresponding Stäckel matrix is equal to…”

“…Recall the Stäckel 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 (α) of the algebraic curves C i (2.11), see [21] and references within. There is one to one correspondence between j-th rows of the Stäckel matrix S and a basis of the differentials on the j-th curve C j .…”

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“…In our case the Stäckel 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].…”

“…Therefore, we can suppose that the Steklov-Lyapunov system belong to the family of the Stäckel systems [16] and there are the Lax matrices with rational dependence on the spectral parameter.…”