Related papers: Second Order Lagrangian Dynamics On Double Cross P…
We apply methods of the so-called `inverse problem of the calculus of variations' to the stabilization of an equilibrium of a class of two-dimensional controlled mechanical systems. The class is general enough to include, among others, the…
A relative Rota-Baxter operator on Lie 2-groups is introduced as a pair of relative Rota-Baxter operators on the underlying Lie groups which is also a Lie groupoid morphism. Such an operator induces a factorization theorem for Lie 2-groups…
This article is the third in a series the aim of which is to use Lie group theory to obtain exact analytic solutions of Delay Ordinary Differential Systems (DODSs). Such a system consists of two equations involving one independent variable…
We describe the solutions to a family of rotationally symmetric second order partial differential equations in the complex plane that arises from a four-dimensional complex Lie algebra whose spanning set generates the algebra from which…
A new Lagrangian formulation with complex currents is developed and yields a direct and simple method for modeling three-phase permanent-magnet and induction machines. The Lagrangian is the sum a mechanical one and of a magnetic one. This…
Divergence-free symmetric tensors seem ubiquitous in Mathematical Physics. We show that this structure occurs in models that are described by the so-called "second" variational principle, where the argument of the Lagrangian is a closed…
We suggest an infinite-dimensional extension of the gauge transformations which includes non-Abelian tensor gauge fields. Extended gauge transformations of non-Abelian tensor gauge fields form a new large group which has natural geometrical…
Previous work in the literature has studied the Hamiltonian structure of an R-squared model of gravity with torsion in a closed Friedmann-Robertson-Walker universe. Within the framework of Dirac's theory, torsion is found to lead to a…
Two-dimensional gas dynamics equations in mass Lagrangian coordinates are studied in this paper. The equations describing these flows are reduced to two Euler-Lagrange equations. Using group classification and Noether's theorem,…
The purpose of this paper is to describe explicitly the solution for linear control systems on Lie groups. In case of linear control systems with inner derivations, the solution is given basically by the product of the exponential of the…
We study differential operators, whose coefficients define noncommutative algebras. As algebra of coefficients, we consider crossed products, corresponding to action of a discrete group on a smooth manifold. We give index formulas for…
Motivated by a search for Lie group structures on groups of Poisson diffeomorphisms [24], we investigate linearizability of Poisson structures of Poisson groupoids around the unit section. After extending the Lagrangian neighbourhood…
A geometric approach to Sundman transformation defined by basic functions for systems of second-order differential equations is developed and the necessity of a change of the tangent structure by means of the function defining the Sundman…
We prove a structure theorem for Lie n-algebras possessing an invariant inner product. We define the notion of a double extension of a metric Lie n-algebra by another Lie n-algebra and prove that all metric Lie n-algebras are obtained from…
In this paper, we study the Lagrangian functions for a class of second-order differential systems arising from physics. For such systems, we present necessary and sufficient conditions for the existence of Lagrangian functions. Based on the…
We will study the controllability problem of a bilinear control system on $\mathbb{R}^2:$ the main result is the characterization of the Lie algebra rank condition for the system. On the other hand, using elementary techniques, we recover…
We consider Lagrangians in Hamilton's principle defined on the tangent space $TG$ of a Lie group $G$. Invariance of such a Lagrangian under the action of $G$ leads to the symmetry-reduced Euler-Lagrange equations called the Euler-Poincar\'e…
We study the Lie point symmetries of a general class of partial differential equations (PDE) of second order. An equation from this class naturally defines a second-order symmetric tensor (metric). In the case the PDE is linear on the first…
We consider the description of second-class constraints in a Lagrangian path integral associated with a higher-order $\Delta$-operator. Based on two conjugate higher-order $\Delta$-operators, we also propose a Lagrangian path integral with…
The article introduces a method to learn dynamical systems that are governed by Euler--Lagrange equations from data. The method is based on Gaussian process regression and identifies continuous or discrete Lagrangians and is, therefore,…