Related papers: Notes on exterior differential systems
A brief introduction to exterior differential systems for graduate students familiar with manifolds and differential forms. For complete files, see https://github.com/Ben-McKay/introduction-to-exterior-differential-systems
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…
These notes deal with a few aspects of Lie algebras and Lie groups, including some matters related to exponentiation.
We extend the theory of exterior differential systems from manifolds and their tangent bundles to Lie algebroids. In particular, we define the concept of an integral manifold of such an exterior differential system. We support our…
A powerful method for solving non-linear first-order ordinary differential equations, which is based on geometrical understanding of the corresponding dynamics of the so called Lie systems, is developed. This method allows us not only to…
Fundamentals on Lie group methods and applications to differential equations are surveyed. Many examples are included to elucidate their extensive applicability for analytically solving both ordinary and partial differential equations.
The geometric theory of Lie systems is used to establish integrability conditions for several systems of differential equations, in particular some Riccati equations and Ermakov systems. Many different integrability criteria in the…
We analyze families of non-autonomous systems of first-order ordinary differential equations admitting a common time-dependent superposition rule, i.e., a time-dependent map expressing any solution of each of these systems in terms of a…
A Lie system is a system of differential equations admitting a superposition rule, i.e., a function describing its general solution in terms of any generic set of particular solutions and some constants. Following ideas going back to the…
We explain that general differential calculus and Lie theory have a common foundation: Lie Calculus is differential calculus, seen from the point of view of Lie theory, by making use of the groupoid concept as link between them. Higher…
We give an account of the construction of exterior differential systems based on the notion of tableaux over Lie algebras as developed in [Comm. Anal. Geom 14 (2006), 475-496; math.DG/0412169]. The definition of a tableau over a Lie algebra…
A theorem of Maurer-Cartan type for Lie algebroids is presented. Suppose that any vector subbundle of a Lie algebroid is called interior differential system (IDS) for that Lie algebroid. A theorem of Cartan type is obtained. Extending the…
Doctoral Thesis, year 2002, about Lie systems and applications in Physics and Control Theory. The text is in English. Advisor: Jos\'e F. Cari\~nena
A new category of Lie algebras, called generalized Lie algebras, is presented such that classical Lie algebras and Lie-Rinehart algebras are objects of this new category. A new philosophy over generalized Lie algebroids theory is presented…
The main purpose of this work is to introduce and analyse some generalizations of diverse superposition rules for first-order differential equations to the setting of second-order differential equations. As a result, we find a way to apply…
The notion of an exterior differential system (on a manifold) has recently been extended to the setting of a Lie algebroid. Here, we further develop the theory and we present two versions of the Cartan-K\"ahler theorem in the case where the…
The theory of Lie systems has recently been applied to Quantum Mechanics and additionally some integrability conditions for Lie systems of differential equations have also recently been analysed from a geometric perspective. In this paper…
These are expository notes from the 2008 Srni Winter School. They have two purposes: (1) to give a quick introduction to exterior differential systems (EDS), which is a collection of techniques for determining local existence to systems of…
Our paper illustrates how the theory of Lie systems allows recovering known results and provide new examples of piecewise deterministic processes with phase-type jumps for which the corresponding first-time passage problems may be solved…
The geometric theory of Lie systems will be used to establish integrability conditions for several systems of differential equations, in particular Riccati equations and Ermakov systems. Many different integrability criteria in the…