4 research outputs found
Lie systems: theory, generalisations, and applications
Lie systems form a class of systems of first-order ordinary differential
equations whose general solutions can be described in terms of certain finite
families of particular solutions and a set of constants, by means of a
particular type of mapping: the so-called superposition rule. Apart from this
fundamental property, Lie systems enjoy many other geometrical features and
they appear in multiple branches of Mathematics and Physics, which strongly
motivates their study. These facts, together with the authors' recent findings
in the theory of Lie systems, led to the redaction of this essay, which aims to
describe such new achievements within a self-contained guide to the whole
theory of Lie systems, their generalisations, and applications.Comment: 161 pages, 2 figure
Lagrangian-Hamiltonian unified formalism for field theory
The Rusk-Skinner formalism was developed in order to give a geometrical
unified formalism for describing mechanical systems. It incorporates all the
characteristics of Lagrangian and Hamiltonian descriptions of these systems
(including dynamical equations and solutions, constraints, Legendre map,
evolution operators, equivalence, etc.).
In this work we extend this unified framework to first-order classical field
theories, and show how this description comprises the main features of the
Lagrangian and Hamiltonian formalisms, both for the regular and singular cases.
This formulation is a first step toward further applications in optimal control
theory for PDE's.Comment: LaTeX file, 23 pages. Minor changes have been made. References are
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