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Reduction of Poisson manifolds
 Letters in Mathematical Physics
, 1986
"... Reduction in the category of Poisson manifolds is defined and some basic properties are derived. The context is chosen to include the usual theorems on reduction of symplectic manifolds, as well as results such as the Dirac bracket and the reduction to the LiePoisson bracket. 1 ..."
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Cited by 52 (3 self)
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Reduction in the category of Poisson manifolds is defined and some basic properties are derived. The context is chosen to include the usual theorems on reduction of symplectic manifolds, as well as results such as the Dirac bracket and the reduction to the LiePoisson bracket. 1
Lagrangian reduction and the double spherical pendulum
 ZAMP
, 1993
"... This paper studies the stability and bifurcations of the relative equilibria of the double spherical pendulum, which has the circle as its symmetry group. This example as well as others with nonabelian symmetry groups, such as the rigid body, illustrate some useful general theory about Lagrangian re ..."
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Cited by 40 (20 self)
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This paper studies the stability and bifurcations of the relative equilibria of the double spherical pendulum, which has the circle as its symmetry group. This example as well as others with nonabelian symmetry groups, such as the rigid body, illustrate some useful general theory about Lagrangian reduction. In particular, we establish a satisfactory global theory of Lagrangian reduction that is consistent with the classical local Routh theory for systems with an abelian symmetry group. 1
1981] A partial differential equation with infinitely many periodic orbits: chaotic oscillations of a forced
 Horseshoes in Perturbations on Hamiltonian Systems With Two Degrees of Freedom,” Commun
, 1981
"... This paper delineates a class of timeperiodically perturved evolution equations in a Banach space whose associated Poincaré map contains a Smale horseshoe. This implies that such systems possess periodic orbits with arbitrarily high period. The method uses techniques originally due to Melnikov and ..."
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Cited by 11 (3 self)
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This paper delineates a class of timeperiodically perturved evolution equations in a Banach space whose associated Poincaré map contains a Smale horseshoe. This implies that such systems possess periodic orbits with arbitrarily high period. The method uses techniques originally due to Melnikov and applies to systems of the form ˙x = f0(x) + εf1(x, t), where ˙x = f0(x) is Hamiltonian and has a homoclinic orbit. We give an example from structural mechanics: sinusoidally forced vibrations of a buckled beam. 1
Branches of Stable ThreeTori Using Hamiltonian Methods in Hopf Bifurcation on a Rhombic Lattice
 of Systems
, 1996
"... This paper uses Hamiltonian methods to find and determine the stability of some new solution branches for an equivariant Hopf bifurcation on C 4 . The normal form has a symmetry group given by the semidirect product of D2 with T 2 \Theta S 1 . The Hamiltonian part of the normal form is comple ..."
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Cited by 10 (10 self)
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This paper uses Hamiltonian methods to find and determine the stability of some new solution branches for an equivariant Hopf bifurcation on C 4 . The normal form has a symmetry group given by the semidirect product of D2 with T 2 \Theta S 1 . The Hamiltonian part of the normal form is completely integrable and may be analyzed using a system of invariants. The idea of the paper is to perturb relative equilibria in this singular Hamiltonian limit to obtain new three frequency solutions to the full normal form for parameter values near the Hamiltonian limit. The solutions obtained have fully broken symmetry, that is, they do not lie in fixed point subspaces. The methods developed in this paper allow one to determine the stability of this new branch of solutions. An example shows that the branch of threetori can be stable. 1 Introduction A standard approach in the bifurcation analysis of spatiallyextended systems (such as RayleighBenard convection in an infinite plane) is to res...
Normal forms for threedimensional parametric instabilities
 Physica D
, 1994
"... We derive and analyze several low dimensional Hamiltonian normal forms describing system symmetry breaking in ideal hydrodynamics. The equations depend on two parameters (ɛ, λ), where ɛ is the strength of a system symmetry breaking perturbation and λ is a detuning parameter. In many cases the result ..."
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Cited by 9 (8 self)
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We derive and analyze several low dimensional Hamiltonian normal forms describing system symmetry breaking in ideal hydrodynamics. The equations depend on two parameters (ɛ, λ), where ɛ is the strength of a system symmetry breaking perturbation and λ is a detuning parameter. In many cases the resulting equations are completely integrable and have an interesting Hamiltonian structure. Our work is motivated by threedimensional instabilities of rotating columnar fluid flows with circular streamlines (such as the Burger vortex) subjected to precession, elliptical distortion or offcenter displacement. 1
Geometry and control of threewave interactions
 in The Arnoldfest
, 1997
"... The integrable structure of the threewave equations is discussed in the setting of geometric mechanics. LiePoisson structures with quadratic Hamiltonian are associated with the threewave equations through the Lie algebras su(3) and su(2, 1). A second structure having cubic Hamiltonian is shown to ..."
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Cited by 4 (1 self)
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The integrable structure of the threewave equations is discussed in the setting of geometric mechanics. LiePoisson structures with quadratic Hamiltonian are associated with the threewave equations through the Lie algebras su(3) and su(2, 1). A second structure having cubic Hamiltonian is shown to be compatible. The analogy between this system and the rigidbody or Euler equations is discussed. Poisson reduction is performed using the method of invariants and geometric phases associated with the reconstruction are calculated. We show that using piecewise continuous controls, the transfer of energy among three 1 waves can be controlled. The so called quasiphasematching control strategy, which is used in a host of nonlinear optical devices to convert laser light from one frequency to another, is described in this context. Finally, we discuss the connection between piecewise constant controls and billiards.
Park City Lectures on Mechanics, Dynamics, and Symmetry
, 1998
"... This paper was also one of the first to notice deep links between reduction and integrable systems, a subject continued by, for example, Bobenko, Reyman and SemenovTianShansky [1989]. ..."
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Cited by 4 (1 self)
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This paper was also one of the first to notice deep links between reduction and integrable systems, a subject continued by, for example, Bobenko, Reyman and SemenovTianShansky [1989].