Concatenating Variational Principles and the Kinetic Stress-Energy-Momentum Tensor

We show how to "concatenate" variational principles over dierent bases into one over a single base, thereby providing a unied Lagrangian treatment of interacting systems. As an example we study a Klein{ Gordon eld interacting with a mesically charged particle. We employ our method to give a novel group-theoretic derivation of the kinetic stress-energy-momentum tensor density corresponding to the particle

A meeting-point between mathematics and the theory of musical scales: well-formed scales.(Spanish: Un encuentro entre las matemáticas y la teoría de escalas musicales: Escalas bien formadas)

Depto. de Álgebra, Geometría y TopologíaInstituto de Matemática Interdisciplinar (IMI)Fac. de Ciencias MatemáticasTRUEpu

Homogeneous algebraic distributions

Let p:E→M be a vector bundle of dimension n+m and (xλ,yi), λ=1,…,n, i=1,…,m, be fibre coordinates. A vertical vector field X on E is said to be algebraic [respectively, algebraic homogeneous of degree d] if its coordinate expression is of the type X=∑mi=1Pi∂/∂yi, where Pi are polynomials [respectively, homogeneous polynomials of degree d] in coordinates yi. A vertical distribution over E is said to be algebraic [respectively, homogeneous algebraic of degree d] if all local generators are homogeneous algebraic [respectively, homogeneous algebraic of the same degree d] vector fields. It is proved that a vertical distribution locally spanned by vector fields X1,…,Xr is homogeneous algebraic of degree d if and only if an r×r matrix A=(aij), aij∈C∞(E), exists which is equal to d−1 times the identity matrix along the zero section of E, and such that [χ,Xj]=∑ri=1aijXi, j=1,…,r, where χ is the Liouville vector field

Hamiltonian structure of gauge-invariant variational problems

Let C→M be the bundle of connections of a principal bundle on M . The solutions to Hamilton–Cartan equations for a gauge-invariant Lagrangian density Λ on C satisfying a weak condition of regularity, are shown to admit an affine fibre-bundle structure over the set of solutions to Euler–Lagrange equations for Λ . This structure is also studied for the Jacobi fields and for the moduli space of extremals

Poisson-Poincar\'e reduction for Field Theories

Given a Hamiltonian system on a fiber bundle, there is a Poisson covariant formulation of the Hamilton equations. When a Lie group G acts freely, properly, preserving the fibers of the bundle and the Hamiltonian density is G-invariant, we study the reduction of this formulation to obtain an analogue of Poisson-Poincar\'e reduction for field theories. This procedure is related to the Lagrange-Poincar\'e reduction for field theories via a Legendre transformation. Finally, an application to a model of a charged strand evolving in an electric field is given.Comment: 34 page

Formas "gauge" invariantes en el fibrado de las conexiones de un fibrado principal

Tesis de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Departamento de Geometría y Topología, leída el 03-12-1999El objetivo principal de esta memoria ha sido el estudio y caracterización de las formas gauge invariantes en el fibrado de las conexiones CP de un fibrado principal. Aprovechando la identificación entre CP y el cociente del fibrado de jets J1P por la acciónd el grupo estructural, se resuelve y el problema en dos etapas: 1-Se estudia la caracterización con J1P 2-Se analiza qué formas proyectan a CP Se otienen así las siguientes resultados: a-Las formas gauge invariantes en J1P son básicamente sus formas de contactos. B- Las formas gauge en CP son las formas características, generadores de las clases características del fibrado principal. Finalmente se hace uso de esta caracterización para obtener dos resultados en la Teoría del Cálculo de Variaciones en CP * Se caracterizan las lagrangianas, en la línea del Teorema de Utiyama, invariantes por el grupod e automorfismo. * Se resuelve el problema de equivalencia variacional de lagrangianas gauge invariantesDepto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasTRUEpu

Disonancia, escalas y matemáticas

The goal of this article is to give a simple formulation of some aspects about the construction of musical scales and the classical laws of consonance