Results 1 
9 of
9
Geometric phases, reduction and LiePoisson structure for the resonant threewave interaction
 Physica D
, 1998
"... Hamiltonian LiePoisson structures of the threewave equations associated with the Lie algebras su(3) and su(2, 1) are derived and shown to be compatible. Poisson reduction is performed using the method of invariants and geometric phases associated with the reconstruction are calculated. These resul ..."
Abstract

Cited by 16 (5 self)
 Add to MetaCart
Hamiltonian LiePoisson structures of the threewave equations associated with the Lie algebras su(3) and su(2, 1) are derived and shown to be compatible. Poisson reduction is performed using the method of invariants and geometric phases associated with the reconstruction are calculated. These results can be applied to applications of nonlinearwaves in, for instance, nonlinear optics. Some of the general structures presented in the latter part of this paper are implicit in the literature; our purpose is to put the threewave interaction in the modern setting of geometric mechanics and to explore some new things, such as explicit geometric phase formulas, as well as some old things, such as integrability, in this context.
Stepwise Precession of the Resonant Swinging Spring
, 2001
"... The swinging spring, or elastic pendulum, has a 2:1:1 resonance arising at cubic order in its approximate Lagrangian. The corresponding modulation equations are the wellknown threewave equations that also apply, for example, in lasermatter interaction in a cavity. We use Hamiltonian reduction a ..."
Abstract

Cited by 13 (5 self)
 Add to MetaCart
The swinging spring, or elastic pendulum, has a 2:1:1 resonance arising at cubic order in its approximate Lagrangian. The corresponding modulation equations are the wellknown threewave equations that also apply, for example, in lasermatter interaction in a cavity. We use Hamiltonian reduction and pattern evocation techniques to derive a formula that describes the characteristic feature of this system's dynamics, namely, the stepwise precession of its azimuthal angle. PACS numbers: 02.40.k, 05.45.a, 45.10.Db, 45.20.Jj Keywords: Classical mechanics, Variational principles, Averaged Lagrangian, Elastic Pendulum, Nonlinear Resonance. email: dholm@lanl.gov y email: Peter.Lynch@met.ie 1 D. D. Holm & P. Lynch Precession of the Swinging Spring 2 Contents 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 ..."
Abstract

Cited by 10 (10 self)
 Add to MetaCart
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...
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 ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
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]. ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
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].
Abstract
, 2001
"... The swinging spring, or elastic pendulum, has a 2:1:1 resonance arising at cubic order in its approximate Lagrangian. The corresponding modulation equations are the wellknown threewave equations that also apply, for example, in lasermatter interaction in a cavity. We use Hamiltonian reduction and ..."
Abstract
 Add to MetaCart
The swinging spring, or elastic pendulum, has a 2:1:1 resonance arising at cubic order in its approximate Lagrangian. The corresponding modulation equations are the wellknown threewave equations that also apply, for example, in lasermatter interaction in a cavity. We use Hamiltonian reduction and pattern evocation techniques to derive a formula that describes the characteristic feature of this systemâ€™s dynamics, namely, the stepwise precession of its azimuthal angle.
Mathematical Sciences HP Laboratories Bristol
, 1998
"... threewave interaction; geometric phases; reduction; LiePoisson structure Hamiltonian LiePoisson structures of the threewave equations associated with the Lie algebras su(3) ana su(2,1) are delivered and shown to be compatible. POIsson reduction is performed using the method of invariants and geom ..."
Abstract
 Add to MetaCart
threewave interaction; geometric phases; reduction; LiePoisson structure Hamiltonian LiePoisson structures of the threewave equations associated with the Lie algebras su(3) ana su(2,1) are delivered and shown to be compatible. POIsson reduction is performed using the method of invariants and geometric phases associated with the reconstruction are calculated. These results can be applied to applications of nonlinearwaves in, for Instance, nonlinear optics. Some of the general structures presented in the latter part of ~IS paper are implicit in the litet:atur~; our purpose IS to put the threewave Interaction In the modem setting of geometric mechanics and to explore some new things, such as integrability, in thIS context.