Double Pendulum and θ -Divisor

SummaryThe equations of motion of integrable systems involving hyperelliptic Riemann surfaces of genus 2 and one relevant degree of freedom are integrated in the framework of the Jacobi inversion problem, using a reduction to the θ -divisor on the Jacobi variety, i.e., to the set of zeros of the θ -function. Explicit solutions are given in terms of Kleinian σ -functions and their derivatives. The procedure is applied to the planar double pendulum without gravity, but it is worked out for any Abelian integral of first or second kind.

[1]  B. Konopelchenko,et al.  Singular sector of the Burgers–Hopf hierarchy and deformations of hyperelliptic curves , 2002, nlin/0205012.

[2]  Holger R. Dullin,et al.  Melnikov's method applied to the double pendulum , 1994 .

[3]  David Grant,et al.  Formal groups in genus two. , 1990 .

[4]  On directional derivatives of the theta function along its divisor , 1992 .

[5]  V. Buchstaber,et al.  Graded lie algebras that define hyperelliptic sigma functions , 2002 .

[6]  A. Perelomov.  Integrable systems of classical mechanics and Lie algebras , 1989 .

[7]  P. H. Richter,et al.  Chaos in Classical Mechanics: The Double Pendulum , 1984 .

[8]  B. Konopelchenko,et al.  Singular sector of the Kadomtsev–Petviashvili hierarchy, ∂̄ operators of nonzero index, and associated integrable systems , 2000 .

[9]  P. Moerbeke,et al.  Birkhoff Strata, Backlund-transformations, and Regularization of Isospectral Operators , 1994 .

[10]  Boris Dubrovin,et al.  Theta functions and non-linear equations , 1981 .

[11]  P. H. Richter,et al.  Forces in the Double Pendulum , 2000 .

[12]  Victor Matveevich Buchstaber,et al.  Kleinian functions, hyperelliptic Jacobians and applications , 1997 .

[13]  John D. Fay.  Theta Functions on Riemann Surfaces , 1973 .

[14]  J. C. Eilbeck,et al.  Linear r-matrix algebra for systems separable in parabolic coordinates , 1993 .

[15]  P. Richter,et al.  Application of Greene's method and the MacKay residue criterion to the double pendulum , 1994 .