165 research outputs found
A nonlinear superposition rule for solutions of the Milne--Pinney equation
A superposition rule for two solutions of a Milne--Pinney equation is
derived.Comment: 14 pages, 2 figure
Superposition rules, Lie theorem and partial differential equations
A rigorous geometric proof of the Lie's Theorem on nonlinear superposition
rules for solutions of non-autonomous ordinary differential equations is given
filling in all the gaps present in the existing literature. The proof is based
on an alternative but equivalent definition of a superposition rule: it is
considered as a foliation with some suitable properties. The problem of
uniqueness of the superposition function is solved, the key point being the
codimension of the foliation constructed from the given Lie algebra of vector
fields. Finally, as a more convincing argument supporting the use of this
alternative definition of superposition rule, it is shown that this definition
allows an immediate generalization of Lie's Theorem for the case of systems of
partial differential equations.Comment: 22 page
Hamiltonian versus Lagrangian formulations of supermechanic
We take advantage of different generalizations of the tangent manifold to the
context of graded manifolds, together with the notion of super section along a
morphism of graded manifolds, to obtain intrinsic definitions of the main
objects in supermechanics such as, the vertical endomorphism, the canonical and
the Cartan's graded forms, the total time derivative operator and the
super--Legendre transformation. In this way, we obtain a correspondence between
the Lagrangian and the Hamiltonian formulations of supermechanics.Comment: Plain TeX, 24 pages. Submitted to J. Phys.
Quantum Lie systems and integrability conditions
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 we use both developments to obtain a geometric theory of
integrability in Quantum Mechanics and we use it to provide a series of
non-trivial integrable quantum mechanical models and to recover some known
results from our unifying point of view
Integrability of Lie systems and some of its applications in physics
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
literature will be analyzed from this new perspective and some applications in
physics will be given.Comment: 16 page
The quantum free particle on spherical and hyperbolic spaces: A curvature dependent approach
The quantum free particle on the sphere () and on the
hyperbolic plane () is studied using a formalism that
considers the curvature \k as a parameter. The first part is mainly concerned
with the analysis of some geometric formalisms appropriate for the description
of the dynamics on the spaces (, \IR^2, ) and with
the the transition from the classical -dependent system to the quantum
one using the quantization of the Noether momenta. The Schr\"odinger
separability and the quantum superintegrability are also discussed. The second
part is devoted to the resolution of the -dependent Schr\"odinger
equation. First the characterization of the -dependent `curved' plane
waves is analyzed and then the specific properties of the spherical case are
studied with great detail. It is proved that if then a discrete
spectrum is obtained. The wavefunctions, that are related with a
-dependent family of orthogonal polynomials, are explicitly obtained
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