Lagrangian Reduction on Homogeneous Spaces with Advected Parameters

Vizman, Cornelia

doi:10.3842/sigma.2015.009

Cited by 3 publications

(5 citation statements)

References 14 publications

(24 reference statements)

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“…For an arbitrary action of G on a parameter manifold M, the EP equations for symmetry breaking are obtained [9]. These results were adapted to the case of homogeneous spaces in [10].…”

Section: Euler-poincaré Equations For Symmetry Breakingmentioning

confidence: 96%

“…In this section we recall the Euler-Poincaré equations on homogeneous spaces [8] and the parameter dependent version which leads to Euler-Poincaré equations for symmetry breaking [10].…”

Section: Euler-poincaré Equations On Homogeneous Spacesmentioning

confidence: 99%

“…Reduction can be performed for parameter dependent Lagrangians on homogeneous spaces too [10]. We use the same approach to adapt to homogeneous spaces a result from [3], where space-time strands on Lie groups are treated dynamically.…”

Section: Introductionmentioning

confidence: 98%

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Invariant variational problems on homogeneous spaces

Vizman

2015

Journal of Geometry and Physics

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“…For an arbitrary action of G on a parameter manifold M, the EP equations for symmetry breaking are obtained [9]. These results were adapted to the case of homogeneous spaces in [10].…”

Section: Euler-poincaré Equations For Symmetry Breakingmentioning

confidence: 96%

“…In this section we recall the Euler-Poincaré equations on homogeneous spaces [8] and the parameter dependent version which leads to Euler-Poincaré equations for symmetry breaking [10].…”

Section: Euler-poincaré Equations On Homogeneous Spacesmentioning

confidence: 99%

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Invariant variational problems on homogeneous spaces

Vizman

2015

Journal of Geometry and Physics

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“…Integrable systems on symmetric spaces of finite dimensional Lie algebras have been well studied in the literature, see [12][13][14][15][16]. A more general but similar construction would be on homogeneous spaces, as studied recently in [17,18] also in the context of reduction by symmetry. This construction is different from the semidirect product G-strands explored in [3] for the special Euclidean group SE(3) := SO(3) R 3 with Lie algebra commutation relations [so(3), so(3)] ⊂ so(3) and [so(3), R 3 ] ⊂ R 3 , because for symmetric spaces, an additional commutation relation occurs involving the complementary space p…”

Section: Introductionmentioning

confidence: 99%

G -Strands on symmetric spaces

Arnaudon

Holm

Ivanov³

2017

Proc. R. Soc. A.

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We study the G-strand equations that are extensions of the classical chiral model of particle physics in the particular setting of broken symmetries described by symmetric spaces. These equations are simple field theory models whose configuration space is a Lie group, or in this case a symmetric space. In this class of systems, we derive several models that are completely integrable on finite dimensional Lie group G, and we treat in more detail examples with symmetric space SU(2)/S 1 and SO(4)/SO(3). The latter model simplifies to an apparently new integrable nine-dimensional system. We also study the G-strands on the infinite dimensional group of diffeomorphisms, which gives, together with the Sobolev norm, systems of 1 + 2 Camassa-Holm equations. The solutions of these equations on the complementary space related to the Witt algebra decomposition are the odd function solutions.

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“…The problem of reduction with symmetry breaking has been studied in [14,11] for variational principles of arbitrary order and where the parameter space is an arbitrary smooth manifold, for contact Lagrangian systems in [2], in the context of optimization problems on manifolds in [10], and for optimal control applications in [3], [13], [7], [21]. For homogeneous spaces, reduction with symmetry breaking has only been studied in [22].…”

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confidence: 99%

Collision Avoidance of Multiagent Systems on Riemannian Manifolds

Goodman¹,

Colombo²

2022

SIAM J. Control Optim.

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This paper studies the reduction by symmetry of variational problems on Lie groups and Riemannian homogeneous spaces. We derive the reduced equations of motion in the case of Lie groups endowed with a left-invariant metric, and on Lie groups that admits a bi-invariant metric. We repeated this analysis for Riemannian homogeneous spaces, where we derive the reduced equations by considering an alternative variational problem written in terms of a connection on the horizontal bundle of the underlying Lie group. We study also the case that the underlying Lie group admits a bi-invariant metric, and consider the special case that the homogeneous space is in fact a Riemannian symmetric space. These ideas are applied to geodesics for a rigid body on SO(3) to derive geodesic equations on the dual of its Lie algebra (a vector space), the heavy-top in SE(3) to derive reduced equations of motion on the unit sphere S 2 , geodesics on S 2 as a Riemannian symmetric space endowed with a bi-invariant metric and optimal control problems for applications to robotic manipulators.

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Lagrangian Reduction on Homogeneous Spaces with Advected Parameters

Cited by 3 publications

References 14 publications

Invariant variational problems on homogeneous spaces

Invariant variational problems on homogeneous spaces

G -Strands on symmetric spaces

Collision Avoidance of Multiagent Systems on Riemannian Manifolds

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