Abstract: The problem of symmetries in field theory has been analyzed using geometric frameworks, such as the multisymplectic models by using in particular the multivector field formalism. In this paper, we expand the vector fields associated to infinitesimal symmetries which give rise to invariant quantities as Noether currents for classical field theories and relativistic mechanic using the multisymplectic geometry where the Poincaré-Cartan form has thus been greatly simplified using the Second Order Partial Differential Equation (SOPDE) for multi-vector fields verifying Euler equations. These symmetries have been classified naturally according to the construction of the fiber bundle used. In this work, unlike other works using the analytical method, our geometric model has allowed us firstly to distinguish the angular moments of the gauge field obtained during different transformations while these moments are gathered in a single expression and are obtained during a rotation in the Minkowsky space. Secondly, no conditions are imposed on the Lagrangian of the mechanics with respect to its dependence in time and in qi, the currents obtained naturally from the transformations are respectively the energy and the momentum of the system.
Abstract: In this paper, we present a simple effective numerical geometric method to estimate the divergence of a vector field over a curved surface. The conservation law is an important principle in physics and mathematics. However, many well-known numerical methods for solving diffusion equations do not obey conservation laws. Our presented method in this paper combines the divergence theorem with a generalized finite difference method and obeys the conservation law on discrete closed surfaces. We use the similar method to solve the Cahn-Hilliard equations on evolving spherical surfaces and observe stability results in our numerical simulations.
Abstract: The paper deals with a mathematical model for fluid dynamic flows on road networks which is based on conservation laws. This nonlinear framework is based on the conservation of cars. We focus on traffic circle, which is a finite number of roads that meet at some junctions. The traffic circle with junctions having either one incoming and two outgoing or two incoming and one outgoing roads. We describe the numerical schemes with the particular boundary conditions used to produce approximated solutions of the problem.
Abstract: The solvated electron is self-trapped (polaron) owing
to strong interaction with the quantum polarization field. If the
electron and quantum field are strongly coupled then the collective
localized state of the field and quasi-particle is formed. In such a
formation the electron motion is rather intricate. On the one hand the
electron oscillated within a rather deep polarization potential well
and undergoes the optical transitions, and on the other, it moves
together with the center of inertia of the system and participates in
the thermal random walk. The problem is to separate these motions
correctly, rigorously taking into account the conservation laws. This
can be conveniently done using Bogolyubov-Tyablikov method of
canonical transformation to the collective coordinates. This
transformation removes the translational degeneracy and allows one
to develop the successive approximation algorithm for the energy and
wave function while simultaneously fulfilling the law of conservation
of total momentum of the system. The resulting equations determine
the electron transitions and depend explicitly on the translational
velocity of the quasi-particle as whole. The frequency of optical
transition is calculated for the solvated electron in ammonia, and an
estimate is made for the thermal-induced spectral bandwidth.