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3-parametric robot manipulator with intersecting axes

Jerzy Gądek (1995)

Applications of Mathematics

A p -parametric robot manipulator is a mapping g of ℝ p into the homogeneous space P = ( C 6 × C 6 ) / Diag ( C 6 × C 6 ) represented by the formula g ( u 1 , u 2 , ⋯ , u p ) = exp ( u 1 X 1 ) · ⋯ · exp ( u p X p ) , where C 6 is the Lie group of all congruences of E 3 and X 1 , X 2 , ⋯ , X p are fixed vectors from the Lie algebra of C 6 . In this paper the 3 -parametric robot manipulator will be expressed as a function of rotations around its axes and an invariant of the motion of this robot manipulator will be given. Most of the results presented here have been obtained during the author’s stay at Charles University in Prague....

A canonical connection on sub-Riemannian contact manifolds

Michael Eastwood, Katharina Neusser (2016)

Archivum Mathematicum

We construct a canonically defined affine connection in sub-Riemannian contact geometry. Our method mimics that of the Levi-Civita connection in Riemannian geometry. We compare it with the Tanaka-Webster connection in the three-dimensional case.

A geometric analysis of dynamical systems with singular Lagrangians

Monika Havelková (2011)

Communications in Mathematics

We study dynamics of singular Lagrangian systems described by implicit differential equations from a geometric point of view using the exterior differential systems approach. We analyze a concrete Lagrangian previously studied by other authors by methods of Dirac’s constraint theory, and find its complete dynamics.

A new geometric setting for classical field theories

M. de León, J. C. Marrero, D. Martín de Diego (2003)

Banach Center Publications

A new geometrical setting for classical field theories is introduced. This description is strongly inspired by the one due to Skinner and Rusk for singular lagrangian systems. For a singular field theory a constraint algorithm is developed that gives a final constraint submanifold where a well-defined dynamics exists. The main advantage of this algorithm is that the second order condition is automatically included.

Affine connections on almost para-cosymplectic manifolds

Adara M. Blaga (2011)

Czechoslovak Mathematical Journal

Identities for the curvature tensor of the Levi-Cività connection on an almost para-cosymplectic manifold are proved. Elements of harmonic theory for almost product structures are given and a Bochner-type formula for the leaves of the canonical foliation is established.

Affinor structures in the oscillation theory

Boris N. Shapukov (2002)

Banach Center Publications

In this paper we consider the system of Hamiltonian differential equations, which determines small oscillations of a dynamical system with n parameters. We demonstrate that this system determines an affinor structure J on the phase space TRⁿ. If J² = ωI, where ω = ±1,0, the phase space can be considered as the biplanar space of elliptic, hyperbolic or parabolic type. In the Euclidean case (Rⁿ = Eⁿ) we obtain the Hopf bundle and its analogs. The bases of these bundles are, respectively, the projective...

An affine framework for analytical mechanics

Paweł Urbański (2003)

Banach Center Publications

An affine Cartan calculus is developed. The concepts of special affine bundles and special affine duality are introduced. The canonical isomorphisms, fundamental for Lagrangian and Hamiltonian formulations of the dynamics in the affine setting are proved.

Dirac structures and dynamical r -matrices

Zhang-Ju Liu, Ping Xu (2001)

Annales de l’institut Fourier

The purpose of this paper is to establish a connection between various objects such as dynamical r -matrices, Lie bialgebroids, and Lagrangian subalgebras. Our method relies on the theory of Dirac structures and Courant algebroids. In particular, we give a new method of classifying dynamical r -matrices of simple Lie algebras 𝔤 , and prove that dynamical r -matrices are in one-one correspondence with certain Lagrangian subalgebras of 𝔤 ⊕ 𝔤 .

Divergence operators and odd Poisson brackets

Yvette Kosmann-Schwarzbach, Juan Monterde (2002)

Annales de l’institut Fourier

We define the divergence operators on a graded algebra, and we show that, given an odd Poisson bracket on the algebra, the operator that maps an element to the divergence of the hamiltonian derivation that it defines is a generator of the bracket. This is the “odd laplacian”, Δ , of Batalin-Vilkovisky quantization. We then study the generators of odd Poisson brackets on supermanifolds, where divergences of graded vector fields can be defined either in terms of berezinian volumes or of graded connections. Examples...

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