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The Hamiltonian approach to isomonodromic deformation systems is extended to include generic rational covariant derivative operators on the Riemann sphere with irregular singularities of arbitrary Poincaré rank. The space of rational connections with given pole degrees carries a natural Poisson structure corresponding to the standard classical rational R-matrix structure on the dual space L * g l (r) of the loop algebra L g l (r). Nonautonomous isomonodromic counterparts of isospectral systems generated by spectral invariants are obtained by identifying deformation parameters as Casimir elements on the phase space. These are shown to coincide with higher Birkhoff invariants determining local asymptotics near to irregular singular points, together with the pole loci. Pairs consisting of Birkhoff invariants, together with the corresponding dual spectral invariant Hamiltonians, appear as "mirror images" matching, at each pole, the negative power coefficients in the principal part of the Laurent expansion of the fundamental meromorphic differential on the associated spectral curve with the corresponding positive power terms in the analytic part. Infinitesimal isomonodromic deformations are shown to be generated by the sum of the Hamiltonian vector field and an explicit derivative vector field that is transversal to the symplectic foliation. The Casimir elements serve as coordinates complementing those along the symplectic leaves, defining a local symplectomorphism between them. The explicit derivative vector fields preserve the Poisson structure and define a flat transversal connection, spanning an integrable distribution whose leaves may be identified as the orbits of a free Abelian local group action. The projection of infinitesimal isomonodromic deformation vector fields to the quotient manifold under this action gives commuting Hamiltonian vector fields corresponding to the spectral invariants dual to the Birkhoff invariants and the pole loci. [ABSTRACT FROM AUTHOR]

The addition formulae for KP τ-functions, when evaluated at lattice points in the KP flow group orbits in the infinite dimensional Sato-Segal-Wilson Grassmannian, give infinite parametric families of solutions to discretizations of the KP hierarchy. The CKP hierarchy may similarly be viewed as commuting flows on the Lagrangian sub-Grassmannian of maximal isotropic subspaces with respect to a suitably defined symplectic form. Evaluating the τ-functions at a sublattice of points within the KP orbit, the resulting discretization gives solutions both to the hyperdeterminantal relations (or Kashaev recurrence) and the hexahedron (or Kenyon–Pemantle) recurrence. [ABSTRACT FROM AUTHOR]

This work is motivated by the relation between the KP and BKP integrable hierarchies, whose τ-functions may be viewed as flows of sections of dual determinantal and Pfaffian line bundles over infinite dimensional Grassmannians. In finite dimensions, we show how to relate the Cartan map, which, for a vector space V of dimension N, embeds the Grassmannian G r V 0 (V + V * ) of maximal isotropic subspaces of V + V*, with respect to the natural scalar product, into the projectivization of the exterior space Λ(V), and the Plücker map, which embeds the Grassmannian GrV(V + V*) of all N-planes in V + V* into the projectivization of ΛN(V + V*). The Plücker coordinates on G r V 0 (V + V * ) are expressed bilinearly in terms of the Cartan coordinates, which are holomorphic sections of the dual Pfaffian line bundle P f * → G r V 0 (V + V * , Q). In terms of affine coordinates on the big cell, this is equivalent to an identity of Cauchy–Binet type, expressing the determinants of square submatrices of a skew symmetric N × N matrix as bilinear sums over the Pfaffians of their principal minors. [ABSTRACT FROM AUTHOR]

Using a combination of Hamiltonian reduction under continuous and discrete symmetry groups, the Lax equations for the motion of generalized tops are derived as part of a general approach to moment map embeddings of integrable Hamiltonian systems in loop algebras. [ABSTRACT FROM AUTHOR]

A classical finite-dimensional integrable Hamiltonian system, corresponding to the motion of a particle constrained to an n-dimensional sphere Σ[sup n, sub μ = 0], χ²[sub μ] = 1, with the Hamiltonian H = Σ[sub μ (½ y²[sub μ] + u²[sub μ]/x²[sub μ] + ∈α[sub μ]x²[sub μ]) (where u[sub μ], α[sub μ], and ∈ are constant[sub μ] and y, are the momenta conjugate to x[sub μ]), is integrated using several different methods. These are the following: (1) The projection of geodesic (free) flow on a larger space, namely the sphere S[sup 2n + 1] (for ∈ = 0). The flow is obtained in terms of elementary functions. (2) Separation of variables in the Hamilton-Jacobi equation in elliptic coordinates or, alternatively, the use of a complete set of integrals of motion in involution to reduce Hamilton's equations to quadratures. The flow is obtained in terms of Abelian integrals which are then inverted in terms of generalized θ functions. The relation between the different methods and results is clarified using methods of algebraic geometry, in particular the geometry of quadrics. [ABSTRACT FROM AUTHOR]