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

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

Differential Geometric Aspects of the Cartan Form: Symmetry Theory

This paper demonstrates that the classical Cartan form θ1L is not adequate for the determination of all the natural symmetries and conservation laws for a Lagrangian L. It is shown that the various extensions θ2L,..., θrL of the classical Cartan form, introduced in recent papers, give larger symmetry groups: G1⊆G2⊆⋅⋅⋅⊆Gr. This paper also introduces the notion of contact equivalent Lagrangians, which serves to clarify the idea that different Lagrangians can give rise to the same variational and symmetry theories

Kaluza-Klein Geometry

We formulate a Kaluza-Klein theory in terms of short exact sequences of vector bundles

Differential equations theory and applications: with Maple®

Includes bibliographical references and index

Trace Operators, Feynman Distributions, and Multiparameter White-Noise

Within the framework of white noise analysis on the probability space Omega = L*(R(d), R(M)), the recent work by Johnson and Kallianpur on the Hu-Meyer formula, traces, and natural extensions is generalized to the multiparameter case: d\u3e1. Besides providing a more general setting for these topics, the paper gives an alternative definition for the traces, a distributional version of the natural extension, and a generalized Kallianpur-Feynman distribution. The development illustrates how traces and natural extensions are intimately related to Wick products and the change of covariance formula from quantum field theory, as well as to the projective tenser product of Hilbert spaces from functional analysis