A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: THE GENERAL THEORY OF ORBITS

http://www.gbv.de/dms/hebis-darmstadt/toc/113948433.pdf

The Problem of Integrable Discretization: Hamiltonian Approach

The special quasiperiodic solution of the (2+1)-dimensional Kadometsev–Petviashvili equation is separated into three systems of ordinary differential equations, which are the second, third, and fourth members in the well-known confocal involutive hierarchy associated with the nonlinearized Zakharov–Shabat eigenvalue problem. The explicit theta function solution of the Kadometsev–Petviashvili equation is obtained with the help of this separation technique. A generating function approach is introduced to prove the involutivity and the functional independence of the conserved integrals which are essential for the Liouville integrability.

Relation between the Kadometsev–Petviashvili equation and the confocal involutive system

Riemann surface, 29 addition theorem for sn(u), 28 additive number theory, 15 algebraic function, 29 balanced hypergeometric series, 18 Bell number, 17, 107 Bernoulli counting scheme, 111 Bernoulli number, 17 Bernoulli polynomial, 17 beta integral, 19 binomial series, 18 branched covering map, 28 branchpoint, 29 Catalan number, 16 Chu-Vandermonde summation formula, 19 commutative ordinary differential operators, 117 complete symmetric polynomial, 12 completely multiplicative, 15 complex structure, 29 conformally equivalent, 29 conjugate partition, 5 contiguous relation, 20 Dedekind sum, 107 digamma function, 21 Dirichlet region, 26 double gamma function, 27 Eisenstein series, 26 elementary symmetric polynomial, 12 elliptic function, 26 elliptic integral, 25 Euler number, 22 Euler product, 15 Euler’s dilogarithm, 19 Euler’s transformation formula, 19 Euler-Maclaurin summation formula, 15, 21 Eulerian number, 17, 107 Fermat measure, 53 Fermat’s last theorem, 27 fractional differentiation, 18 function element, 28 fundamental region, 26 Galois field, 65, 121 Gauss second summation theorem, 21 Gauss summation formula, 20 Gegenbauer polynomial, 51 generalised Stirling number, 16 generalised Stirling polynomial, 16 generalized hypergeometric series, 18 generalized Laguerre polynomial, 22 generalized Vandermonde determinant, 125 Genocchi number, 17

The history of q-calculus and a new method

An explicit symmetry constraint for the Lax pairs and the adjoint Lax pairs of AKNS systems

Constrained flows of integrable PDEs and bi-Hamiltonian structure of the Garnier system

A systematic method of constructing finite‐dimensional integrable systems starting from a bi‐Hamiltonian hierarchy of soliton equations is introduced. The existence of two Hamiltonian structures of the hierarchy leads to a bi‐Hamiltonian formulation of the resulting finite‐dimensional systems. The case of coupled KdV hierarchies is studied in detail. A surprising connection with separable Jacobi potentials is uncovered and described.

How to construct finite-dimensional bi-Hamiltonian systems from soliton equations: Jacobi integrable potentials

Restricted flows of soliton hierarchies associated with the energy-dependent Schrodinger spectral problem are determined explicitly. A remarkable connection with separable potentials is used for proving complete integrability of the restricted flows. A previously unknown Lagrangian and Hamiltonian formulation of the Neumann system is found. Whole families of generalizations of the Neumann and Garnier systems are given.

Restricted flows of soliton hierarchies: coupled KdV and Harry Dym case

Bi-Hamiltonian formulation of the Hénon-Heiles system and its multidimensional extensions

A detailed description is given of integrable cases of the generalized Henon–Heiles systems which differs from the standard H–H ones by the term α/q22. Their connection with fifth‐order one‐component soliton equations is discussed. Lax representations are constructed, and the bi‐Hamiltonian formulation of dynamics is given. It is also shown that the gH–H system can be mapped onto another system of Newton equations with a nonstandard Hamiltonian structure.

Stefan Rauch-Wojciechowski

Papers

Constrained flows of integrable PDEs and bi-Hamiltonian structure of the Garnier system

How to construct finite-dimensional bi-Hamiltonian systems from soliton equations: Jacobi integrable potentials

Restricted flows of soliton hierarchies: coupled KdV and Harry Dym case

Bi-Hamiltonian formulation of the Hénon-Heiles system and its multidimensional extensions

A generalized Henon–Heiles system and related integrable Newton equations