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22
The Orbit Bundle Picture of Cotangent Bundle Reduction
, 2000
"... Cotangent bundle reduction theory is a basic and well developed subject in which one performs symplectic reduction on cotangent bundles. One starts with a (free and proper) action of a Lie group G on a configuration manifold Q, considers its natural cotangent lift to T ∗ Q and then one seeks realiza ..."
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Cited by 21 (15 self)
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Cotangent bundle reduction theory is a basic and well developed subject in which one performs symplectic reduction on cotangent bundles. One starts with a (free and proper) action of a Lie group G on a configuration manifold Q, considers its natural cotangent lift to T ∗ Q and then one seeks realizations of the corresponding symplectic or Poisson reduced space. We further develop this theory by explicitly identifying the symplectic leaves of the Poisson manifold T ∗ Q/G, decomposed as a Whitney sum bundle, T ∗ (Q/G) � �g ∗ over Q/G. The splitting arises naturally from a choice of connection on the Gprincipal bundle Q → Q/G. The symplectic leaves are computed and a formula for the reduced symplectic form is found.
Reduction theory and the LagrangeRouth Equations
 J. Math. Phys
, 2000
"... Reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of Euler, Jacobi, Lagrange, Hamilton, Routh, Poincaré and others. The modern vision of mechanics includes, besides the traditional mechanics of particles and rigid bodies, field theories such as e ..."
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Cited by 19 (7 self)
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Reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of Euler, Jacobi, Lagrange, Hamilton, Routh, Poincaré and others. The modern vision of mechanics includes, besides the traditional mechanics of particles and rigid bodies, field theories such as electromagnetism, fluid mechanics, plasma physics, solid mechanics as well as quantum mechanics, and relativistic theories, including gravity. Symmetries in these theories vary from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas, to subtle symmetries underlying integrable systems. Reduction theory concerns the removal of symmetries and their associated conservation laws. Variational principles along with symplectic and Poisson geometry, provide fundamental tools for this endeavor. Reduction theory has been extremely useful in a wide variety of areas, from a deeper understanding of many physical theories, including new variational and Poisson structures, stability theory, integrable systems, as well as geometric phases.
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 ..."
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Cited by 16 (5 self)
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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.
Symmetries in motion: Geometric foundations of motion control
, 1998
"... Some interesting aspects of motion and control for systems such as those found in biological and robotic locomotion, attitude control of spacecraft and underwater vehicles, and steering of cars and trailers, involve geometric concepts. When an animal or a robot moves its joints in a periodic fashion ..."
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Cited by 14 (9 self)
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Some interesting aspects of motion and control for systems such as those found in biological and robotic locomotion, attitude control of spacecraft and underwater vehicles, and steering of cars and trailers, involve geometric concepts. When an animal or a robot moves its joints in a periodic fashion, it can move forward or rotate in place. When the amplitude of the motion increases, the resulting net displacements normally increase as well. These observations lead to the general idea that when certain variables in a system move in a periodic fashion, motion of the whole object can result. This property can be used for control purposes; the position and attitude of a satellite, for example, are often controlled by periodic motions of parts of the satellite, such as spinning rotors. Geometric tools that have been useful to describe this phenomenon are \connections", mathematical objects that are extensively used in general relativity and other parts of theoretical physics. The theory of connections, which isnow part of the general subject of geometric mechanics, has also been helpful in the study of the stability or instability ofa system and in its bifurcations under parameter variations. This approach, currently in a period of rapid evolution, has been used, for example, to design stabilizing feedback control systems in the attitude dynamics of spacecraft and
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 ..."
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Cited by 13 (5 self)
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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
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].
2005] Cotangent bundle reduction
 Article in the Encyclopedia of Mathematical Physics. JeanPierre Françoise, Greg Naber, Tsou Sheung Tsun (editors
"... This encyclopedia article briefly reviews without proofs some of the main results in cotangent bundle reduction. The article recalls most the necessary prerequisites to understand the main results. The general symplectic reduction theory presented in [13] becomes much richer and has many application ..."
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Cited by 2 (2 self)
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This encyclopedia article briefly reviews without proofs some of the main results in cotangent bundle reduction. The article recalls most the necessary prerequisites to understand the main results. The general symplectic reduction theory presented in [13] becomes much richer and has many applications if the symplectic manifold is the cotangent bundle (T ∗Q, ΩQ = −dΘQ) of a manifold Q. The canonical oneform ΘQ on T ∗Q is given by ΘQ(αq) ( ) ( ()) Vαq = αq TαqπQ Vαq, for any q ∈ Q, αq ∈ T ∗ q Q, and tangent vector Vαq ∈ Tαq(T ∗Q), where πQ: T ∗Q → Q is the cotangent bundle projection and TαqπQ: Tαq(T ∗Q) → TqQ is its tangent map (or derivative) at q. In natural cotangent bundle coordinates (qi, pi), we have ΘQ = pidqi and ΩQ = dqi ∧ dpi. Let Φ: G × Q → Q be a left smooth action of the Lie group G on the manifold and Q. Denote by g · q = Φ(g, q) the action of g ∈ G on the point q ∈ Q and by Φg: Q → Q the diffeomorphism of Q induced by g. The lifted left action G × T ∗Q → T ∗Q, given by g · αq = T ∗ g·qΦg −1(αq) for g ∈ G and αq ∈ T ∗ q Q, preserves ΘQ, and admits the equivariant momentum map J: T ∗Q → g ∗ whose expression is 〈J(αq), ξ 〉 = αq((ξQ(q)), where ξ ∈ g, the Lie algebra of G, 〈 , 〉 : g ∗ × g → R is the duality pairing between the dual g ∗ and g, and ξQ(q) = dΦ(exp tξ, q)/dtt=0 is the value of the infinitesimal generator vector field ξQ of the Gaction at q ∈ Q. Throughout this article it is assumed that the Gaction on Q, and hence on T ∗Q, is free and proper. Recall also that ((T ∗Q)µ, (ΩQ)µ) denotes the reduced manifold at µ ∈ g ∗ [13], where (T ∗Q)µ: = J−1 (µ)/Gµ is the orbit space of the Gµaction on the momentum level manifold J−1 (µ) and Gµ: = {g ∈ G  Ad ∗ g µ = µ} is the isotropy subgroup of the coadjoint representation of G on g ∗. The left coadjoint representation of g ∈ G on µ ∈ g ∗ is denoted by Ad ∗ g−1 µ. Cotangent bundle reduction at zero is already quite interesting and has many applications. Let ρ: Q → Q/G be the Gprincipal bundle projection defined by the proper free action of G on Q, usually referred to as the shape space bundle. Zero is a regular value of J and the map ϕ0: ((T ∗ Q)0, (ΩQ)0) →
The 1:±2 Resonance
, 2007
"... On the linear level elliptic equilibria of Hamiltonian systems are mere superpositions of harmonic oscillators. Nonlinear terms can produce instability, if the ratio of frequencies is rational and the Hamiltonian is indefinite. In this paper we study the frequency ratio ±1/2 and its unfolding. In p ..."
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Cited by 1 (1 self)
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On the linear level elliptic equilibria of Hamiltonian systems are mere superpositions of harmonic oscillators. Nonlinear terms can produce instability, if the ratio of frequencies is rational and the Hamiltonian is indefinite. In this paper we study the frequency ratio ±1/2 and its unfolding. In particular we show that for the indefinite case (1:−2) the frequency ratio map in a neighbourhood of the origin has a critical point, i.e. the twist condition is violated for one torus on every energy surface near the energy of the equilibrium. In contrast, we show that the frequency map itself is nondegenerate (i.e. the Kolmogorov nondegeneracy condition holds) for every torus in a neighbourhood of the equilibrium point. As a by product of our analysis of the fequency map we obtain another proof of fractional monodromy in the 1:−2 resonance. 1
Geometric Foundations of Motion and Control
"... Some interesting aspects of motion and control such as those found in biological and robotic locomotion, and attitude control of spacecraft, involve geometric concepts. When an animal or a robot moves its joints in a periodic fashion, it can rotate or move forward. This observation leads to the gene ..."
Abstract
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Some interesting aspects of motion and control such as those found in biological and robotic locomotion, and attitude control of spacecraft, involve geometric concepts. When an animal or a robot moves its joints in a periodic fashion, it can rotate or move forward. This observation leads to the general idea that when one variable in a system moves in a periodic fashion, motion of the whole object can result. This property can be used for control purposes; the position and attitude of a satellite, for example, are often controlled by periodic motions of parts of the satellite, such as spinning rotors. One of the geometric tools that has been used to describe this phenomenon is that of connections, a notion that is extensively used in general relativity and other parts of theoretical physics. This tool, part of the general subject of geometric mechanics, has been helpful in the study of the stability or instability of a system and in its bifurcations, that is, changes in the nature of the systems dynamics, as some parameter changes. Geometric mechanics, currently in a period of rapid evolution, has been used, for example, to design stabilizing feedback control systems in attitude dynamics. The theory is also being developed for systems with rolling constraints such as those found in a simple rolling wheel. This article explains how some of these tools of geometric mechanics are used in the study of motion control and locomotion generation. 1