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35
The EulerPoincaré equations and semidirect products with applications to continuum theories
 ADV. MATH
, 1998
"... We study Euler–Poincaré systems (i.e., the Lagrangian analogue of LiePoisson Hamiltonian systems) defined on semidirect product Lie algebras. We first give a derivation of the Euler–Poincaré equations for a parameter dependent Lagrangian by using a variational principle of Lagrange d’Alembert type. ..."
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Cited by 129 (61 self)
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We study Euler–Poincaré systems (i.e., the Lagrangian analogue of LiePoisson Hamiltonian systems) defined on semidirect product Lie algebras. We first give a derivation of the Euler–Poincaré equations for a parameter dependent Lagrangian by using a variational principle of Lagrange d’Alembert type. Then we derive an abstract KelvinNoether theorem for these equations. We also explore their relation with the theory of LiePoisson Hamiltonian systems defined on the dual of a semidirect product Lie algebra. The Legendre transformation in such cases is often not invertible; thus, it does not produce a corresponding Euler–Poincaré system on that Lie algebra. We avoid this potential difficulty by developing the theory of Euler–Poincaré systems entirely within the Lagrangian framework. We apply the general theory to a number of known examples, including the heavy top, ideal compressible fluids and MHD. We also use this framework to derive higher dimensional CamassaHolm equations, which have many potentially interesting analytical properties. These
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
The reduced EulerLagrange equations
 Fields Institute Comm
, 1993
"... Marsden and Scheurle [1993] studied Lagrangian reduction in the context of momentum map constraints—here meaning the reduction of the standard EulerLagrange system restricted to a level set of a momentum map. This provides a Lagrangian parallel to the reduction of symplectic manifolds. The present ..."
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Cited by 33 (16 self)
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Marsden and Scheurle [1993] studied Lagrangian reduction in the context of momentum map constraints—here meaning the reduction of the standard EulerLagrange system restricted to a level set of a momentum map. This provides a Lagrangian parallel to the reduction of symplectic manifolds. The present paper studies the Lagrangian parallel of Poisson reduction for Hamiltonian systems. For the reduction of a Lagrangian system on a level set of a conserved quantity, a key object is the Routhian, which is the Lagrangian minus the mechanical connection paired with the fixed value of the momentum map. For unconstrained systems, we use a velocity shifted Lagrangian, which plays the role of the Routhian in the constrained theory. Hamilton’s variational principle for the EulerLagrange equations breaks up into two sets of equations that represent a set of EulerLagrange equations with gyroscopic forcing that can be written in terms of the curvature of the connection for horizontal variations, and into the EulerPoincaré equations for the vertical variations. This new set of equations is what we call the reduced EulerLagrange equations, and it includes the EulerPoincaré and the Hamel equations as special cases. We illustrate this methodology for a rigid body with internal rotors and for a particle moving in a magnetic field. 1
Hamiltonian systems with symmetry, coadjoint orbits and plasma physics
 IN PROC. IUTAMIS1MM SYMPOSIUM ON MODERN DEVELOPMENTS IN ANALYTICAL MECHANICS
, 1982
"... The symplectic and Poisson structures on reduced phase spaces are reviewed, including the symplectic structure on coadjoint orbits of a Lie group and the LiePoisson structure on the dual of a Lie algebra. These results are applied to plasma physics. We show in three steps how the MaxwellVlasov equ ..."
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Cited by 31 (20 self)
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The symplectic and Poisson structures on reduced phase spaces are reviewed, including the symplectic structure on coadjoint orbits of a Lie group and the LiePoisson structure on the dual of a Lie algebra. These results are applied to plasma physics. We show in three steps how the MaxwellVlasov equations for a collisionless plasma can be written in Hamiltonian form relative to a certain Poisson bracket. First, the PoissonVlasov equations are shown to be in Hamiltonian form relative to the LiePoisson bracket on the dual of the (finite dimensional) Lie algebra of infinitesimal canonical transformations. Then we write Maxwell’s equations in Hamiltonian form using the canonical symplectic structure on the phase space of the electromagnetic fields, regarded as a gauge theory. In the last step we couple these two systems via the reduction procedure for interacting systems. We also show that two other standard models in plasma physics, ideal MHD and twofluid electrodynamics, can be
Geometric mechanics, Lagrangian reduction and nonholonomic systems
 in Mathematics Unlimited2001 and Beyond
, 2001
"... This paper surveys selected recent progress in geometric mechanics, focussing on Lagrangian reduction and gives some new applications to nonholonomic systems, that is, mechanical systems with constraints typified by rolling without slipping. Reduction theory for mechanical systems with symmetry has ..."
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Cited by 23 (5 self)
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This paper surveys selected recent progress in geometric mechanics, focussing on Lagrangian reduction and gives some new applications to nonholonomic systems, that is, mechanical systems with constraints typified by rolling without slipping. 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 mechanics ranges 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 utilizing their associated conservation laws. Reduction theory has been extremely useful in a wide variety of areas, from a deeper understanding of many
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.
EulerPoincaré dynamics of perfect complex fluids
, 2000
"... Lagrangian reduction by stages is used to derive the EulerPoincaré equations for the nondissipative coupled motion and micromotion of complex fluids. We mainly treat perfect complex fluids (PCFs) whose order parameters are continuous material variables. These order parameters may be regarded geomet ..."
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Cited by 19 (9 self)
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Lagrangian reduction by stages is used to derive the EulerPoincaré equations for the nondissipative coupled motion and micromotion of complex fluids. We mainly treat perfect complex fluids (PCFs) whose order parameters are continuous material variables. These order parameters may be regarded geometrically either as objects in a vector space, or as coset spaces of Lie symmetry groups with respect to subgroups that leave these objects invariant. Examples include liquid crystals, superfluids, YangMills magnetofluids and spinglasses. A LiePoisson Hamiltonian formulation of the dynamics for perfect complex fluids is obtained by Legendre transforming the EulerPoincaré formulation. These dynamics are also derived by using the Clebsch approach. In the Hamiltonian and Lagrangian formulations of perfect complex fluid dynamics Lie algebras containing twococycles arise as a characteristic feature. After discussing these geometrical formulations of the dynamics of perfect complex fluids, we give an example of how to introduce defects into the order parameter as imperfections (e.g., vortices) that carry their own momentum. The defects may move relative to the Lagrangian fluid material and thereby produce additional reactive forces and stresses.