Null cubics and Lie quadratics

Noakes, Lyle

doi:10.1063/1.1537461

Cited by 49 publications

(63 citation statements)

References 16 publications

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“…Null Lie quadratics in E 3 (solutions of V = V x V) have been studied by the second author in [7]. Null elastic Lie quadratics in E 3 admit a much simpler, closed form description.…”

Section: Lemma 41 For Any a E 50(3) And Any T 0 E R (I) T \-Y A(v(mentioning

confidence: 99%

“…where L g : G -*• G is left multiplication by g e G, namely L g (h) = gh, and (dL s ) h : T h G -> T gh G is the derivative of L g at /i 6 G. As noted in [7], allowing for the opposite sign convention to [6] …”

Section: Elastica In Lie Groupsmentioning

confidence: 99%

“…https://doi.org/10.1017/S1446788700036417 [7] Elastica in 50 (3) 111 By analogy with the Lie quadratics of [7][8][9] and [11] (unconstrained solutions of V = [V, V] + C), a curve V : I -+9 satisfying (2.3) and (2.4) for some C e 9 and all t e I will be called an elastic Lie quadratic with constant C. If x : / -> G is an elastic curve, the curve V defined by (2.2) is an elastic Lie quadratic; it will be called the elastic Lie quadratic associated with x. For later reference, we re-state the observation (2.9) from the preceding proof.…”

Section: Elastica In Lie Groupsmentioning

confidence: 99%

“…The critical points of <t>, Riemannian cubics, are studied in [7][8][9]11] and references therein. When vo and v\ are unit vectors, the elastic problem is to minimise 4> over [3] Elastica in 50 (3) 107…”

Section: Introductionmentioning

confidence: 99%

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Elastica in SO(3)

Popiel¹,

Noakes²

2007

J. Aust. Math. Soc.

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In a Riemannian manifold M, elastica are solutions of the Euler-Lagrange equation of the following second order constrained variational problem: find a unit-speed curve in M, interpolating two given points with given initial and final (unit) velocities, of minimal average squared geodesic curvature. We study elastica in Lie groups G equipped with bi-invariant Riemannian metrics, focusing, with a view to applications in engineering and computer graphics, on the group 50(3) of rotations of Euclidean 3-space. For compact G, we show that elastica extend to the whole real line. For G = 50(3), we solve the Euler-Lagrange equation by quadratures.2000 Mathematics subject classification: primary 49K99,49Q99.

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“…Null Lie quadratics in E 3 (solutions of V = V x V) have been studied by the second author in [7]. Null elastic Lie quadratics in E 3 admit a much simpler, closed form description.…”

Section: Lemma 41 For Any a E 50(3) And Any T 0 E R (I) T \-Y A(v(mentioning

confidence: 99%

Section: Elastica In Lie Groupsmentioning

confidence: 99%

Section: Elastica In Lie Groupsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Elastica in SO(3)

Popiel¹,

Noakes²

2007

J. Aust. Math. Soc.

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“…The equation (1) [32]) and explored from a dynamical interpolation perspective in 1995 (see [17]). Interesting points related to this subject have been developed in the last few years, namely a geometric theory surprisingly close to the Riemannian theory of geodesics (see [2,3,4,12,14,15,16,19,30,31,34,35]). We recall, in particular, a result which says that if V denotes the velocity vector field of a cubic polynomial x, then…”

Section: Introductionmentioning

confidence: 99%

Cubic polynomials on Lie groups: reduction of the Hamiltonian system

Abrunheiro

Camarinha

Clemente-Gallardo

2011

J. Phys. A: Math. Theor.

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Abstract. This paper analyzes the optimal control problem of cubic polynomials on compact Lie groups from a Hamiltonian point of view and its symmetries. The dynamics of the problem is described by a presymplectic formalism associated with the canonical symplectic form on the cotangent bundle of the semidirect product of the Lie group and its Lie algebra. Using these control geometric tools, the relation between the Hamiltonian approach developed here and the known variational one is analyzed. After making explicit the left trivialized system, we use the technique of Marsden-Weinstein reduction to remove the symmetries of the Hamiltonian system. In view of the reduced dynamics, we are able to guarantee, by means of the Lie-Cartan theorem, the existence of a considerable number of independent integrals of motion in involution.

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Magnetic trajectories in three‐dimensional Lie groups

Turhan

2019

Math Methods in App Sciences

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We study magnetic trajectories in Lie groups equipped with bi‐invariant Riemannian metric. We define the Lorentz force of a magnetic field in a Lie group G, and then, we give the Lorentz force equation for the associated magnetic trajectories that are curves in G. When the manifold is a Lie group G equipped with bi‐invariant Riemannian metric, we derive differential equation system that characterizes magnetic flow associated with the Killing magnetic curves with regard to the Lie reduction of the curve γ in G.

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Null cubics and Lie quadratics

Cited by 49 publications

References 16 publications

Elastica in SO(3)

Elastica in SO(3)

Cubic polynomials on Lie groups: reduction of the Hamiltonian system

Magnetic trajectories in three‐dimensional Lie groups

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