Results 1  10
of
94
Lax pairs for the AblowitzLadik system via orthogonal polynomials on the unit
"... Abstract. In [14] Nenciu and Simon found that the analogue of the Toda system in the context of orthogonal polynomials on the unit circle is the defocusing AblowitzLadik system. In this paper we use the CMV and extended CMV matrices defined in [5] and [13, 14], respectively, to construct Lax pair r ..."
Abstract

Cited by 23 (4 self)
 Add to MetaCart
Abstract. In [14] Nenciu and Simon found that the analogue of the Toda system in the context of orthogonal polynomials on the unit circle is the defocusing AblowitzLadik system. In this paper we use the CMV and extended CMV matrices defined in [5] and [13, 14], respectively, to construct Lax pair representations for this system. 1.
Various Approaches to Conservative and Nonconservative Nonholonomic Systems
 Reports on Mathematical Physics 42
, 1998
"... We propose a geometric setting for the Hamiltonian description of mechanical systems with a nonholonomic constraint, which may be used for constraints of general type (nonlinear in the velocities, and such that the constraint forces may not obey Chetaev's rule). Such constraints may be realized b ..."
Abstract

Cited by 22 (0 self)
 Add to MetaCart
We propose a geometric setting for the Hamiltonian description of mechanical systems with a nonholonomic constraint, which may be used for constraints of general type (nonlinear in the velocities, and such that the constraint forces may not obey Chetaev's rule). Such constraints may be realized by servomechanisms; therefore, the corresponding mechanical system may be nonconservative. In that setting, the kinematic properties of the constraint are described by a submanifold of the tangent bundle, mapped, by Legendre's transformation, onto a submanifold (called the Hamiltonian constraint submanifold) of the phase space (i.e., of the cotangent bundle to the configuration manifold). The dynamical properties of the constraint are described by a vector subbundle of the tangent bundle to the phase space along the Hamiltonian constraint submanifold. In order to be able to deal with systems obtained by reduction by a symmetry group, we generalize that setting by using a Poisson struc...
CMV: The unitary analogue of Jacobi matrices
 Comm. Pure Appl. Math
"... Abstract. We discuss a number of properties of CMV matrices, by which we mean the class of unitary matrices recently introduced by Cantero, Moral, and Velazquez. We argue that they play an equivalent role among unitary matrices to that of Jacobi matrices among all Hermitian matrices. In particular, ..."
Abstract

Cited by 18 (2 self)
 Add to MetaCart
Abstract. We discuss a number of properties of CMV matrices, by which we mean the class of unitary matrices recently introduced by Cantero, Moral, and Velazquez. We argue that they play an equivalent role among unitary matrices to that of Jacobi matrices among all Hermitian matrices. In particular, we describe the analogues of wellknown properties of Jacobi matrices: foliation by coadjoint orbits, a natural symplectic structure, algorithmic reduction to this shape, Lax representation for an integrable lattice system (AblowitzLadik), and the relation to orthogonal polynomials. As offshoots of our analysis, we will construct action/angle variables for the finite AblowitzLadik hierarchy and describe the longtime behaviour of this system. 1.
Unbounded growth of energy in nonautonomous Hamiltonian systems
 Nonlinearity
, 1998
"... The result of J. Mather on the existence of trajectories with unbounded energy for time periodic Hamiltonian systems on a torus is generalized to a class of multidimensional Hamiltonian systems with Hamiltonian polynomial in momenta. It is assumed that the leading homogeneous term of the Hamiltonian ..."
Abstract

Cited by 16 (3 self)
 Add to MetaCart
The result of J. Mather on the existence of trajectories with unbounded energy for time periodic Hamiltonian systems on a torus is generalized to a class of multidimensional Hamiltonian systems with Hamiltonian polynomial in momenta. It is assumed that the leading homogeneous term of the Hamiltonian is autonomous and the corresponding Hamiltonian system has a hyperbolic invariant torus possessing a transversal homoclinic trajectory. Under certain Melnikov type condition, the existence of trajectories with unbounded energy is proved. Instead of the variational methods of Mather, a geometrical approach based on KAM theory and the PoincaréMelnikov method is used. This makes it possible to study a more general class of Hamiltonian systems, but requires additional smoothness assumptions on the Hamiltonian.
Commuting Dual Billiards
 Geom. Dedicata
"... Mathematical billiards is a rich and beautiful subject. It is very extensive as well. The choice of material for this survey reflects the taste of the author, who has attempted to make the exposition as geometrical as possible. The survey consists of five chapters: the first provides some general ba ..."
Abstract

Cited by 14 (8 self)
 Add to MetaCart
Mathematical billiards is a rich and beautiful subject. It is very extensive as well. The choice of material for this survey reflects the taste of the author, who has attempted to make the exposition as geometrical as possible. The survey consists of five chapters: the first provides some general background; the second
NormalInternal Resonances in QuasiPeriodically Forced Oscillators: A Conservative Approach
, 2002
"... We perform a bifurcation analysis of normalinternal resonances in parametrised families of quasiperiodically forced Hamiltonian oscillators, for small forcing. The unforced system is a one degree of freedom oscillator, called the `backbone' system; forced, the system is a skewproduct flow with a ..."
Abstract

Cited by 14 (13 self)
 Add to MetaCart
We perform a bifurcation analysis of normalinternal resonances in parametrised families of quasiperiodically forced Hamiltonian oscillators, for small forcing. The unforced system is a one degree of freedom oscillator, called the `backbone' system; forced, the system is a skewproduct flow with a quasiperiodic driving with n basic frequencies. The dynamics of the forced system are simplified by averaging over the orbits of a linearisation of the unforced system. The averaged system turns out to have the same structure as in the wellknown case of periodic forcing (n = 1); for a real analytic system, the nonintegrable part can even be made exponentially small in the forcing strength. We investigate the persistence and the bifurcations of quasiperiodic ndimensional tori in the averaged system, filling normalinternal resonance `gaps' that had been excluded in previous analyses. However, these gaps cannot completely be filled up: secondary resonance gaps appear, to which the averaging analysis can be applied again. This phenomenon of `gaps within gaps' makes the quasiperiodic case more complicated than the periodic case.
VARIATIONAL PRINCIPLES FOR LIE–POISSON AND HAMILTON–POINCARÉ EQUATIONS
, 2002
"... Dedicated to Vladimir Arnold on his 65th birthday Abstract. As is wellknown, there is a variational principle for the Euler–Poincaré equations on a Lie algebra g of a Lie group G obtained by reducing Hamilton’s principle on G by the action of G by, say, left multiplication. The purpose of this pape ..."
Abstract

Cited by 12 (2 self)
 Add to MetaCart
Dedicated to Vladimir Arnold on his 65th birthday Abstract. As is wellknown, there is a variational principle for the Euler–Poincaré equations on a Lie algebra g of a Lie group G obtained by reducing Hamilton’s principle on G by the action of G by, say, left multiplication. The purpose of this paper is to give a variational principle for the Lie–Poisson equations on g ∗ , the dual of g, and also to generalize this construction. The more general situation is that in which the original configuration space is not a Lie group, but rather a configuration manifold Q on which a Lie group G acts freely and properly, so that Q → Q/G becomes a principal bundle. Starting with a Lagrangian system on T Q invariant under the tangent lifted action of G, the reduced equations on (T Q)/G, appropriately identified, are the Lagrange–Poincaré equations. Similarly, if we start with a Hamiltonian system on T ∗ Q, invariant under the cotangent lifted action of G, the resulting reduced equations on (T ∗ Q)/G are called the Hamilton–Poincaré equations. Amongst our new results, we derive a variational structure for the Hamilton–Poincaré equations, give a formula for the Poisson structure on these reduced spaces that simplifies previous formulas of Montgomery, and give a new representation for the symplectic structure on the associated symplectic leaves. We illustrate the formalism with a simple, but interesting example, that of a rigid body with internal rotors. 1.
Noncommutative integrability, moment map and geodesic flows
 Ann. Global Anal. Geom
"... The purpose of this paper is to discuss the relationship between commutative and noncommutative integrability of Hamiltonian systems and to construct new examples of integrable geodesic flows on Riemannian manifolds. In particular, we prove that the geodesic flow of the biinvariant metric on any bi ..."
Abstract

Cited by 12 (3 self)
 Add to MetaCart
The purpose of this paper is to discuss the relationship between commutative and noncommutative integrability of Hamiltonian systems and to construct new examples of integrable geodesic flows on Riemannian manifolds. In particular, we prove that the geodesic flow of the biinvariant metric on any biquotient of a compact Lie group is integrable in the noncommutative sense by means of polynomial integrals, and therefore, in the classical commutative sense by means of C ∞ –smooth integrals.