Random integral equation formulation of a generalized Langevin equation

In this paper, the generalized Langevin equation introduced by Kubo and Mori is formulated as a random integral equation. We consider (1) the existence and uniqueness of the solution, (2) moments of the solution process, (3) a comparison theorem for solution processes, and (4) the Cauchy polygonal approximation to the solution.     

Lagrangian reduction of generalized nonholonomic systems

In this paper we study the Lagrangian reduction of generalized nonholonomic systems (GNHS) with symmetry. We restrict ourselves to those GNHS, defined on a configuration space Q$Q$, with kinematic constraints given by a general submanifold _CK⊂TQ$C_K\subset TQ$, and variational constraints given by a distribution _CV$C_V$ on Q$Q$. We develop a reduction procedure that is similar to that for nonholonomic systems satisfying d’Alembert’s principle, i.e. with _CK$C_K$ a distribution and _CV=_CK$C_V=C_K$. Special care is taken in identifying the geometrical structures and mappings involved. We illustrate the general theory with an example.     

Nonperturbative solutions of the Lane-Emden equation

A new, iterative approach to solving the Lane-Emden equation leads to approximate solutions which display the coupling constant as a persistent interaction. The approach generalizes to nonlinear field equations.     

On a Lagrangian formulation of gravitoelectromagnetism

We examine a Lagrangian formulation of gravity based on an approach analogous to electromagnetism, called Gravitoelectromagnetism (GEM). The gravitational analogue of the electromagnetic field tensor is a three-index tensor, \({\mathcal {F}_{\mu\nu\lambda}}\), defined in terms of a two-index gravitoelectromagnetic potential, \({\mathcal {A}_{\mu\nu}}\). The energy-momentum tensor is derived and is symmetric. We construct a Lagrangian which allows us to describe interactions between fermions, photons and gravitons. We calculate transition amplitudes of various processes involving gravitons: gravitational Møller scattering, gravitational Compton scattering, and the graviton photoproduction.     

Analytic and numerical solutions to the Lane-Emden equation

In this Letter, we present analytical solutions to the Lane-Emden equation describing the thermal behavior of a spherical cloud of gas acting under the mutual attraction of its molecules. Solutions are obtained by using the traditional power series approach and by using the Homotopy Analysis Method (HAM). We show that the series solutions obtained by the Homotopy Analysis Method converge in a larger interval than in the case of the corresponding traditional series solutions. Furthermore, we obtained numerical solutions (using Runge-Kutta-Fehlberg 4-5 technique) which are used to validate the analytical solutions.     

Convergent iterative solutions of Schroedinger equation for a generalized double well potential

We present an explicit convergent iterative solution for the lowest energy state of the Schroedinger equation with a generalized double well potential . The condition for the convergence of the iteration procedure and the dependence of the shape of the groundstate wave function on the parameter a are discussed.     

Geometrical construction of a generalized Skyrme Lagrangian

A geometrical scheme is proposed that leads to a straightforward generalization of a Skyrme Lagrangian, comprising higher-order terms up to the eighth power in pion fields.Boursière DGRST.     

Palatini variation of a generalized Einstein Lagrangian

The Palatini variation of the Einstein Lagiangian is analyzed for a symmetric metric and for a more general sesquilinear (Hermitian) form. In the latter case the Lagrangian is no longer projectively invariant, and the connection is determined to bemetrical up to a real transformation. The Einstein-Straus equations emerge as a naturalgeometrical generalization of general relativity.Supported by the Natural Sciences and Engineering Research Council of Canada.     

Lagrangian formulation of a geometrical scalar-tensor theory of gravitation

The author's geometrical theory of the scalar-tensor gravitational field is extended by formulating it in terms of a Lagrangian. An exact solution of the coupled nonlinear field equations for a static point mass is also presented. This theory which is conformally equivalent to the empty spaceEinstein equations predicts the same results for experiments as the usual theory of Brans and Dicke which has a non-zero energy momentum tensor.     

Lagrangian formulation of a general Hamiltonian theory with constraints

A Lagrangian formulation is constructed for a general Hamiltonian theory with constraints. A modification is proposed of the standard procedure of the Hamiltonianization of a Lagrangian theory in the case when the Lagrangian theory has primary constraints. The obtained results are used to establish the Lagrangian form of infinitesimally small canonical transformations in the Hamiltonian formalism.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 104–112, March, 1986.     

A complete Lagrangian formulation of superconductivity

We present here a complete Lagrangian formulation of superconductivity. Only the boson degrees of freedom are explicitly introduced. The BCS solution is obtained as classical approximation. The physical interpretation of the Goldstone boson comes up naturally from this theory.     

On the equivalence of generalized Langevin equation and generalized master equation

It is shown that the generalized Langevin equation derived by Mori is equivalent to the generalized master equation of Nakajima and Zwanzig. Both equations of motion are related, just in the same way as Heisenberg and Schrödinger picture are related.     

Behavior of viscous solutions in Lagrangian formulation

In this paper, the behavior of shock-capturing methods in Lagrangian coordinate is investigated. The relation between viscous shock and inviscid one is analyzed quantitatively, and the procedure of a viscous shock formation and propagation with a jump type initial data is described. In general, a viscous shock profile and a discontinuous one include different energy and momentum, and these discrepancies result in the generation of waves in all families when a single wave Riemann problem (shock or rarefaction) is solved. Employing this method, some anomalous behavior, such as, viscous shock interaction, shock passing through ununiform grids, postshock oscillations and lower density phenomenon is explained well. Using some classical schemes to solve the inviscid flow in Lagrangian coordinate may be not adequate enough to correctly describe flow motion in the discretized space. Partial discrepancies between von Neumann artificial viscosity method and Godunov method are exhibited. Some reviews are given to those methods which can ameliorate even eliminate entropy errors. A hybrid scheme based on the understanding to the behavior of viscous solution is proposed to suppress the overheating error.     

Heat Fluctuations and a generalized Gibbs equation

Starting from a generalized Gibbs equation and from the Einstein formula for the probability of fluctuations, we obtain the correlation function of heat fluctuations.     

Derivation of an effective Lagrangian from a generalized scalar curvature

A spinor Lagrangian invariant under global coordinate, local Lorentz and local chiral SU(n) × SU(n) gauge transformations is presented. The invariance requirement necessitates the introduction of boson fields, and a theory for these fields is then developed by relating them to generalizations of the vector connections in general relativity and utilizing an expanded scalar curvature as a boson Lagrangian. In implementing this plan, the local Lorentz group is found to greatly facilitate the correlation of the boson fields occurring in the spinor Lagrangian with the generalized vector connections.The independent boson fields of the theory are assumed to be the inhomogeneously transforming irreducible parts of the connections. It turns out that no homogeneously transforming parts are necessary to reproduce the chiral Lagrangian usually used as a basis for phenomenological field theories. The Lagrangian in question appears when the gravitational interaction is turned off. It includes pseudoscalar, spinor, vector, and axial vector fields, and the vector fields carry mass in spite of the fact that the theory is locally gauge invariant.