Related papers: On globally invariant Euler--Lagrange equations fo…
We study the Euler-Lagrange equations for a parameter dependent $G$-invariant Lagrangian on a homogeneous $G$-space. We consider the pullback of the parameter dependent Lagrangian to the Lie group $G$, emphasizing the special invariance…
In this paper, we describe a geometric setting for higher-order lagrangian problems on Lie groups. Using left-trivialization of the higher-order tangent bundle of a Lie group and an adaptation of the classical Skinner-Rusk formalism, we…
The article concerns the problem if a~given system of differential equations is identical with the Euler--Lagrange system of an~appropriate variational integral. Elementary approach is applied. The main results involve the determination of…
We show that the exterior algebra bundle over a curved spacetime can be used as framework in which both the Dirac and the Einstein equations can be obtained. These equations and their coupling follow from the variational principle applied…
In a previous paper, the general approach for treatment of algebraic equations of different order in gravity theory was exposed, based on the important distinction between covariant and contravariant metric tensor components. In the present…
We deal with Lagrangian systems that are invariant under the action of a symmetry group. The mechanical connection is a principal connection that is associated to Lagrangians which have a kinetic energy function that is defined by a…
A general, consistent and complete framework for geometrical formulation of mechanical systems is proposed, based on certain structures on affine bundles (affgebroids) that generalize Lie algebras and Lie algebroids. This scheme covers and…
In this paper we will discuss some new developments in the design of numerical methods for optimal control problems of Lagrangian systems on Lie groups. We will construct these geometric integrators using discrete variational calculus on…
The application of a gauge covariant derivative to the Euler-Lagrange equation yields a shortcut to the equations of motion for a field subject to an external force. The gauge covariant derivative includes an external force as an intrinsic…
We study integrable Euler equations on the Lie algebra $\mathfrak{gl}(3,\mathbb{R})$ by interpreting them as evolutions on the space of hexagons inscribed in a real cubic curve.
In this paper, we consider a class of plane curves called log-aesthetic curves and their generalization which are used in computer aided geometric design. We consider these curves in the framework of the similarity geometry and characterize…
We generalize the fractional variational problem by allowing the possibility that the lower bound in the fractional derivative does not coincide with the lower bound of the integral that is minimized. Also, for the standard case when these…
The present article deals with general mechanics in an unconventional manner. At first, Newtonian mechanics for a point particle has been described in vectorial picture, considering Cartesian, polar and tangent-normal formulations in a…
We investigate higher-order geometric $k$-splines for template matching on Lie groups. This is motivated by the need to apply diffeomorphic template matching to a series of images, e.g., in longitudinal studies of Computational Anatomy. Our…
We consider tilings of Euclidean spaces by polygons or polyhedra, in particular, tilings made by a substitution process, such as the Penrose tilings of the plane. We define an isomorphism invariant related to a subgroup of rotations and…
A constructive procedure is proposed for formulation of linear differential equations invariant under global symmetry transformations forming a semi-simple Lie algebra f. Under certain conditions f-invariant systems of differential…
We propose a novel algorithmic method for constructing invariant variational schemes of systems of ordinary differential equations that are the Euler-Lagrange equations of a variational principle. The method is based on the invariantization…
Euler's equation relates the change in angular momentum of a rigid body to the applied torque. This paper fills a gap in the literature by using Lagrangian dynamics to derive Euler's equation in terms of generalized coordinates. This is…
It is known that some equations of differential geometry are derived from variational principle in form of Euler-Lagrange equations. The equations of geodesic flow in Riemannian geometry is an example. Conversely, having Lagrangian…
Whereas in a coordinate-dependent setting the Euler-Lagrange equations establish necessary conditions for solving variational problems in which both the integrands of functionals and the resulting paths are assumed to be sufficiently…