On Maxwell Electrodynamics in Multi-Dimensional Spaces

The governing equations of Maxwell electrodynamics in multi-dimensional spaces are derived from the variational principle of least action, which is applied to the action function of the electromagnetic field. The Hamiltonian approach for the electromagnetic field in multi-dimensional pseudo-Euclidean (flat) spaces has also been developed and investigated. Based on the two arising first-class constraints, we have generalized to multi-dimensional spaces a number of different gauges known for the three-dimensional electromagnetic field. For multi-dimensional spaces of non-zero curvature the governing equations for the multi-dimensional electromagnetic field are written in a manifestly covariant form. Multi-dimensional Einstein’s equations of metric gravity in the presence of an electromagnetic field have been re-written in the true tensor form. Methods of scalar electrodynamics are applied to analyze Maxwell equations in the two and one-dimensional spaces.

Effective action of string theory at order $$\alpha '$$ in the presence of boundary

Recently, using the assumption that the string theory effective action at the critical dimension is background independent, the classical on-shell effective action of the bosonic string theory at order $$\alpha '$$ α ′ in a spacetime manifold without boundary has been reproduced, up to an overall parameter, by imposing the O(1, 1) symmetry when the background has a circle. In the presence of the boundary, we consider a background which has boundary and a circle such that the unit normal vector of the boundary is independent of the circle. Then the O(1, 1) symmetry can fix the bulk action without using the lowest order equation of motion. Moreover, the above constraints and the constraint from the principle of the least action in the presence of boundary can fix the boundary action, up to five boundary parameters. In the least action principle, we assume that not only the values of the massless fields but also the values of their first derivatives are arbitrary on the boundary. We have also observed that the cosmological reduction of the leading order action in the presence of the Hawking–Gibbons boundary term, produces zero cosmological boundary action. Imposing this as another constraint on the boundary couplings at order $$\alpha '$$ α ′ , we find the boundary action up to two parameters. For a specific value for these two parameters, the gravity couplings in the boundary become the Chern–Simons gravity plus another term which has the Laplacian of the extrinsic curvature.

The Principle of Maximal Simplicity for Modular Inorganic Crystal Structures

Modularity is an important construction principle of many inorganic crystal structures that has been used for the analysis of structural relations, classification, structure description and structure prediction. The principle of maximal simplicity for modular inorganic crystal structures can be formulated as follows: in a modular series of inorganic crystal structures, the most common and abundant in nature and experiments are those arrangements that possess maximal simplicity and minimal structural information. The latter can be quantitatively estimated using information-based structural complexity parameters. The principle is applied for the modular series based upon 0D (lovozerite family), 1D (biopyriboles) and 2D (spinelloids and kurchatovite family) modules. This principle is empirical and is valid for those cases only, where there are no factors that may lead to the destabilization of simplest structural arrangements. The physical basis of the principle is in the relations between structural complexity and configurational entropy sensu stricto (which should be distinguished from the entropy of mixing). It can also be seen as an analogy of the principle of least action in physics.

Wave Functional of the Universe and Time

A version of the quantum theory of gravity based on the concept of the wave functional of the universe is proposed. To determine the physical wave functional, the quantum principle of least action is formulated as a secular equation for the corresponding action operator. Its solution, the wave functional, is an invariant of general covariant transformations of spacetime. In the new formulation, the history of the evolution of the universe is described in terms of coordinate time together with arbitrary lapse and shift functions, which makes this description close to the formulation of the principle of general covariance in the classical theory of Einstein’s gravity. In the new formulation of quantum theory, an invariant parameter of the evolutionary time of the universe is defined, which is a generalization of the classical geodesic time measured by a standard clock along time-like geodesics.

Null shells and double layers in quadratic gravity

For a singular hypersurface of arbitrary type in quadratic gravity motion equations were obtained using only the least action principle. It turned out that the coefficients in the motion equations are zeroed with a combination corresponding to the Gauss-Bonnet term. Therefore it does not create neither double layers nor thin shells. It has been demonstrated that there is no “external pressure” for any type of null singular hypersurface. It turned out that null spherically symmetric singular hupersurfaces in quadratic gravity cannot be a double layer, and only thin shells are possible. The system of motion equations in this case is reduced to one which is expressed through the invariants of spherical geometry along with the Lichnerowicz conditions. Spherically symmetric null thin shells were investigated for spherically symmetric solutions of conformal gravity as applications, in particular, for various vacua and Vaidya-type solutions.

OBTAINING THE EQUATIONS OF MOTION OF A MECHANICAL BODY THROUGH ANALYTICAL METHODS

The aim of this paper is to obtain the equations of motion in n-dimensional space for the case where no external forces act on a mechanical system using analytical methods. One such method is known as Lagrangian Mechanics. Lagrangian Mechanics is founded on the principle of least action which states that the spontaneous change from one configuration to another of a dynamical system has a minimum action value if the law of conservation of energy holds.

Understanding single-slit experiments on the basis of Galton board experiments

In a single-slit experiment conducted for microparticles, the well-aligned rough structure of the slit wall can be viewed as a Galton board. Thus, when microparticles pass through the single slit, both the particle probability density (PPD) and particle direction of motion have a normal distribution. Therefore, when the distance between the slit and the receiving film becomes large, particles with different directions of motion will separate into different particle groups. By the nature of a normal distribution, the PPD for any particle group should also be normal-distributed. Obviously, between any two neighboring particle groups, there should be a valley in the PPD and thus the particle groups are observed as discrete fringes. All phenomena observed in the single-slit experiment can be explained reasonably well from the above viewpoint. In particular, analysis shows that the PPD can be described by the square of the modulus of the average least action of particles at a given location.

Causality in Discrete Time Physics Derived from Maupertuis Reduced Action Principle

Causality describes the process and consequences from an action: a cause has an effect. Causality is preserved in classical physics as well as in special and general theories of relativity. Surprisingly, causality as a relationship between the cause and its effect is in neither of these theories considered a law or a principle. Its existence in physics has even been challenged by prominent opponents in part due to the time symmetric nature of the physical laws. With the use of the reduced action and the least action principle of Maupertuis along with a discrete dynamical time physics yielding an arrow of time, causality is defined as the partial spatial derivative of the reduced action and as such is position- and momentum-dependent and requests the presence of space. With this definition the system evolves from one step to the next without the need of time, while (discrete) time can be reconstructed.