arXiv subjects

Michal Jozwikowski

Publications and source records attributed to Michal Jozwikowski.

5 recordsLinked to original sources

arXiv2016

A note on actions of some monoids

Smooth actions of the multiplicative monoid $(\mathbb{R},\cdot)$ of real numbers on manifolds lead to an alternative, and for some reasons simpler, definition of a vector bundle, a double vector bundle and related structures like a graded bundle [Grabowski and Rotkiewicz, J. Geom. Phys. 2011]. For these reasons it is natural to study smooth actions of certain monoids closely related with the monoid $(\mathbb{R},\cdot)$ . Namely, we discuss geometric structures naturally related with: smooth and holomorphic actions of the monoid of multiplicative complex numbers, smooth actions of the monoid of second jets of punctured maps $(\mathbb{R},0)\rightarrow (\mathbb{R},0)$, smooth action of the monoid of real 2 by 2 matrices and smooth actions of multiplicative reals on a supermanifold. In particular cases we recover the notions of a holomorphic vector bundle, a complex vector bundle and a non-negatively graded manifold.

math.DG↗

arXiv2012

Jacobi vector fields for Lagrangian systems on algebroids

We study the geometric nature of the Jacobi equation. In particular we prove that Jacobi vector fields (JVFs) along a solution of the Euler-Lagrange (EL) equations are themselves solutions of the EL equations but considered on a non-standard algebroid (different from the tangent bundle Lie algebroid). As a consequence we obtain a simple non-computational proof of the relation between the null subspace of the second variation of the action and the presence of JVFs (and conjugate points) along an extremal. We work in the framework of skew-symmetric algebroids.

math.DG↗

arXiv2011

Pontryagin Maximum Principle - a generalization

The fundamental theorem of the theory of optimal control, the Pontryagin maximum principle (PMP), is extended to the setting of almost Lie (AL) algebroids, geometrical objects generalizing Lie algebroids. This formulation of the PMP yields, in particular, a scheme comprising reductions of optimal control problems similar to the reduction for the rigid body in analytical mechanics. On the other hand, in the presented approach the reduced and unreduced PMPs are parts of the same universal formalism. The framework is based on a very general concept of homotopy of measurable paths and the geometry of AL algebroids.

math.OC↗

arXiv2011

Optimal Control Theory on almost-Lie Algebroids

We extend the Pontryagin Maximum Principle (PMP) to the geometric setting of almost-Lie (AL) algebroids -- objects which generalize Lie algebroids. The result may be understood as a very general reduction scheme for optimal control problems (OCPs). It covers the standard PMP, as well as gives necessary optimality conditions for symmetric OCPs on Lie groups, principal bundles, and Lie groupoids. We do not assume the symmetry of boundary conditions. The ideas are based on a very general concept of homotopy of admissible paths on AL algebroids. Our framework works for OCPs with fixed-end-points and general boundary conditions.

math.OC↗

arXiv2011

Generalized Livens theorem and constrained Hamiltonian dynamics on algebroids

We generalise Livens theorem, showing that Hamiltonian equation on the vector bundle $E^\ast\rightarrow M$, dual to a general algebroid $E\rightarrow M$, can be derived by means of a variational principle. The framework can be used to describe Hamiltonian dynamics associated with (both vaconomic and nonholonomic) constraints in the bundle $E$.

math-ph↗