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On a generalization of Helmholtz conditions

Radka Malíková (2009)

Acta Mathematica Universitatis Ostraviensis

Helmholtz conditions in the calculus of variations are necessary and sufficient conditions for a system of differential equations to be variational ‘as it stands’. It is known that this property geometrically means that the dynamical form representing the equations can be completed to a closed form. We study an analogous property for differential forms of degree 3, so-called Helmholtz-type forms in mechanics ( n = 1 ), and obtain a generalization of Helmholtz conditions to this case.

On asymptotic motions of robot-manipulator in homogeneous space

Anton Dekrét, Ján Bakša (2008)

Applications of Mathematics

In this paper the notion of robot-manipulators in the Euclidean space is generalized to the case in a general homogeneous space with the Lie group G of motions. Some kinematic subspaces of the Lie algebra 𝒢 (the subspaces of velocity operators, of Coriolis acceleration operators, asymptotic subspaces) are introduced and by them asymptotic and geodesic motions are described.

On Carnot's theorem in time dependent impulsive mechanics.

Stefano Pasquero (2005)

Extracta Mathematicae

We show that the validity of the Carnot's theorem about the kinetic energy balance for a mechanical system subject to an inert impulsive kinetic constraint, once correctly framed in the time dependent geometric environment for Impulsive Mechanics given by the left and right jet bundles of the space-time bundle N, is strictly related to the frame of reference used to describe the system and then it is not an intrinsic property of the mechanical system itself. We analyze in details the class of frames...

On differential inclusions of velocity hodograph type with Carathéodory conditions on Riemannian manifolds

Yuri E. Gliklikh, Andrei V. Obukhovski (2004)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

We investigate velocity hodograph inclusions for the case of right-hand sides satisfying upper Carathéodory conditions. As an application we obtain an existence theorem for a boundary value problem for second-order differential inclusions on complete Riemannian manifolds with Carathéodory right-hand sides.

On Galilean connections and the first jet bundle

James Grant, Bradley Lackey (2012)

Open Mathematics

We see how the first jet bundle of curves into affine space can be realized as a homogeneous space of the Galilean group. Cartan connections with this model are precisely the geometric structure of second-order ordinary differential equations under time-preserving transformations - sometimes called KCC-theory. With certain regularity conditions, we show that any such Cartan connection induces “laboratory” coordinate systems, and the geodesic equations in this coordinates form a system of second-order...

On some cohomological properties of the Lie algebra of Euclidean motions

Marta Bakšová, Anton Dekrét (2009)

Mathematica Bohemica

The external derivative d on differential manifolds inspires graded operators on complexes of spaces Λ r g * , Λ r g * ⊗ g , Λ r g * ⊗ g * stated by g * dual to a Lie algebra g . Cohomological properties of these operators are studied in the case of the Lie algebra g = s e ( 3 ) of the Lie group of Euclidean motions.

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