White Noise Approach to Multiparameter Stochastic Integration.

In this dissertation we will set up the Hida theory of generalized Brownian functionals, or white noise analysis, on {\cal L}\sp{\*}(\IR\sp{\rm d}), the space of tempered distributions, and apply it to multiparameter stochastic integration. With the partial ordering on \IR\sbsp{+}{\rm d}: (s\sb1, dots,s\sb{\rm d}) $\u3c$ (t\sb1, dots,t\sb{\rm d}) if s\sb{\rm i} $\u3c$ t\sb{\rm i}, 1 $\leq$ i $\leq$ d, the Wiener process W((t\sb1, dots,t\sb{\rm d}),x) = $\langle$x,1\sb{\rm \lbrack 0,t\sb1)\times\cdots \times\lbrack 0,t\sb{d})}\rangle, x \in {\cal L}\sp{\*}(\IR\sp{\rm d}) is a generalization of a Brownian motion and there is the Wiener-Ito decomposition: L\sp2({\cal L}\sp{\*}(\IR\sp{\rm d})) = \sum\sbsp{\rm n = 0}{\infty}\timesK\sb{\rm n}, where K\sb{\rm n} is the space of n-tuple Weiner integrals. As in the one-dimensional case, there are the continuous inclusions (L\sp2)\sp+ \subset L\sp2({\cal L}\sp{\*}(\IR\sp{\rm d})) \subset (L\sp2)\sp-, and (L\sp2)\sp- is considered the space of generalized Wiener functionals. We will define the differentiation operator \partial\sb{\rm (t\sb1,\dots,t\sb{d}}) and its adjoint \partial\sp{\*}\sb{\rm (t\sb1,\dots,t\sb{d})} and give some properties. We prove that the multidimensional time Ito stochastic integral is a special case of a white noise integral and give conditions for its existence. For d = 2 the Ito integral is not sufficient for representing elements of L\sp2({\cal L}\sp{\*}(\IR\sp2)). We show that the other integral involved can also be realized in the white noise setting. For F \in {\cal L}\sp{\*}(\IR\sp{\rm d}) we will then define F(W((s,t),x) as an element of (L\sp2)\sp- and obtain a generalized Ito formula

Solar sail dynamics in the three-body problem: homoclinic paths of points and orbits

In this paper we consider the orbital previous termdynamicsnext term of a previous termsolar sailnext term in the Earth-Sun circular restricted three-body problem. The equations of motion of the previous termsailnext term are given by a set of non-linear autonomous ordinary differential equations, which are non-conservative due to the non-central nature of the force on the previous termsail.next term We consider first the equilibria and linearisation of the system, then examine the non-linear system paying particular attention to its periodic solutions and invariant manifolds. Interestingly, we find there are equilibria admitting homoclinic paths where the stable and unstable invariant manifolds are identical. What is more, we find that periodic orbits about these equilibria also admit homoclinic paths; in fact the entire unstable invariant manifold winds off the periodic orbit, only to wind back onto it in the future. This unexpected result shows that periodic orbits may inherit the homoclinic nature of the point about which they are described

Local symmetries in gauge theories in a finite-dimensional setting

It is shown that the correct mathematical implementation of symmetry in the geometric formulation of classical field theory leads naturally beyond the concept of Lie groups and their actions on manifolds, out into the realm of Lie group bundles and, more generally, of Lie groupoids and their actions on fiber bundles. This applies not only to local symmetries, which lie at the heart of gauge theories, but is already true even for global symmetries when one allows for fields that are sections of bundles with (possibly) non-trivial topology or, even when these are topologically trivial, in the absence of a preferred trivialization.Comment: 23 page

On geometrical representation of the Jacobian in a path integral reduction problem

The geometrical representation of the Jacobian in the path integral reduction problem which describes a motion of the scalar particle on a smooth compact Riemannian manifold with the given free isometric action of the compact semisimple Lie group is obtained. By using the formula for the scalar curvature of the manifold with the Kaluza--Klein metric, we present the Jacobian as difference of the scalar curvature of the total space of the principal fibre bundle and the terms that are the scalar curvature of the orbit space, the scalar curvature of the orbit, the second fundamental form of the orbit and the square of the principle fibre bundle curvature.Comment: 8 page

Symmetries in Classical Field Theory

The multisymplectic description of Classical Field Theories is revisited, including its relation with the presymplectic formalism on the space of Cauchy data. Both descriptions allow us to give a complete scheme of classification of infinitesimal symmetries, and to obtain the corresponding conservation laws.Comment: 70S05; 70H33; 55R10; 58A2

Dependent coordinates in path integral measure factorization

The transformation of the path integral measure under the reduction procedure in the dynamical systems with a symmetry is considered. The investigation is carried out in the case of the Wiener--type path integrals that are used for description of the diffusion on a smooth compact Riemannian manifold with the given free isometric action of the compact semisimple unimodular Lie group. The transformation of the path integral, which factorizes the path integral measure, is based on the application of the optimal nonlinear filtering equation from the stochastic theory. The integral relation between the kernels of the original and reduced semigroup are obtained.Comment: LaTeX2e, 28 page

Nonclassical Fields With Singularities On a Moving Surface

Fields with singularities on a moving surface S with boundary ∂S can be represented as distributions which have their support concentrated on S and ∂S. This paper considers such fields of the form F={ f }+λδS̃, where { f } is the distribution determined by a field f and λδS̃ is a Dirac delta distribution with density λ concentrated on the tube S̃ swept out by the moving surface. A straightforward calculation of the distributional gradient, curl, divergence, and time derivative of such fields yields fields of the following general form: G={ g } +αδS̃ +βδ∂S̃ +γ∇n(⋅)δS̃. The density α is shown to contain all the information which is customarily presented in the jump conditions for fields with singularities at a moving interface. Examples from electromagnetic field theory are presented to show the significance of the other terms { g }, βδ∂S̃, and γ∇n(⋅)δS̃

The Kinematical Aspect of the Fundamental Theorem of Calculus

Stokes’s theorem and Poincaré’s lemma are two well‐known and remarkable aspects of the fundamental theorem of calculus in several variables. With the fanfare accorded to these two aspects, the kinematical aspect of the fundamental theorem of calculus has been unjustly neglected. This aspect, appropriately labeled the transport theorem, has special cases which are well known to workers in continuum mechanics. This paper serves to emphasize the importance of the transport theorem and to briefly survey the other aspects of the fundamental theorem of calculus

The Geometry of Gauge-Particle Field Interaction: A Generalization of Utiyama\u27s Theorem

The paper classifies the locally gauge invariant Lagrangians on the jet bundle J1 (E Θ C), for interacting particle and gauge fields. This serves to clarify the global nature of the Utiyama extension process (Yang-Mills trick) for arbitrary principal bundles P and gives the classical (local) results when P is trivial: P = M × G. The emphasis of the paper is a formulation of the results in terms of geometric objects on associated bundles over M rather than on bundles over P