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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)
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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.
Geometry and integrability of Euler–Poincaré–Suslov equations
, 2001
"... We consider nonholonomic geodesic flows of leftinvariant metrics and leftinvariant nonintegrable distributions on compact connected Lie groups. The equations of geodesic flows are reduced to the Euler–Poincaré–Suslov equations on the corresponding Lie algebras. The Poisson and symplectic structure ..."
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We consider nonholonomic geodesic flows of leftinvariant metrics and leftinvariant nonintegrable distributions on compact connected Lie groups. The equations of geodesic flows are reduced to the Euler–Poincaré–Suslov equations on the corresponding Lie algebras. The Poisson and symplectic structures give raise to various algebraic constructions of the integrable Hamiltonian systems. On the other hand, nonholonomic systems are not Hamiltonian and the integration methods for nonholonomic systems are much less developed. In this paper, using chains of subalgebras, we give constructions that lead to a large set of first integrals and to integrable cases of the Euler–Poincaré–Suslov equations. Further, we give examples of nonholonomic geodesic flows that can be seen as a restrictions of integrable subRiemannian geodesic flows. ∗ permanent address 1 0