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The logical line is traced of formulation of theory of mechanics founded on the basic correlations of mathematics of hypercomplex numbers and associated geometric images. Namely, it is shown that the physical equations of quantum, classical…
We consider the canonical ensemble of $N$ particles admitting a strange Hamiltonian description. Each of the particles obeys a set of Newtonian equation of motion, which can also be described by the standard canonical Hamiltonian mechanics.…
The Lagrangian formulation of classical mechanics is widely applicable in solving a vast array of physics problems encountered in the undergraduate and graduate physics curriculum. Unfortunately, many treatments of the topic lack…
The conventional method of generating initial conditions for cosmological N-body simulations introduces a significant error in the real-space statistical properties of the particles. More specifically, the finite box size leads to a…
The pattern of deviations from Standard Model predictions and couplings is different for theories of new physics based on a non-linear realization of the $SU(2)_L\times U(1)_Y$ gauge symmetry breaking and those assuming a linear…
The concept of definability of quantum fields in a set-theoretical foundation is introduced. We propose an axiomatic set theory and then derive a nonlinear sigma model and the Schroedinger equation in a Lagrangian form; this follows…
The Hamiltonian formulation for the mechanical systems with reparametrization-invariant Lagrangians, depending on the worldline external curvatures is given, which is based on the use of moving frame. A complete sets of constraints are…
A generalized theory of gauge transformations is presented on the basis of the covariant Hamiltonian formalism of field theory, for which the covariant canonical field equations are equivalent to the Euler-Lagrange field equations. Similar…
We establish a Lagrangian variational framework for general relativistic continuum theories that permits the development of the process of Lagrangian reduction by symmetry in the relativistic context. Starting with a continuum version of…
Background fields of electromagnetic and gravitational type emerge in the low kinetic energy limit of any regular Lagrangian system and, in particular, in the corresponding limit of any spacetime theory in which the free motion of test…
The Heisenberg, interaction, and Schr\"odinger pictures of motion are considered in Lagrangian (canonical) quantum field theory. The equations of motion (for state vectors and field operators) are derived for arbitrary Lagrangians which are…
We examine the problem of defining Lagrangian and Hamiltonian mechanics for a particle moving on a quantum plane $Q_{q,p}$. For Lagrangian mechanics, we first define a tangent quantum plane $TQ_{q,p}$ spanned by noncommuting particle…
We derive the equations of motion of an action-dependent version of the Einstein-Hilbert Lagrangian, as a specific instance of the Herglotz variational problem. Action-dependent Lagrangians lead to dissipative dynamics, which cannot be…
The imaginary-time formulation is investigated in the field theory. The Dirac and photon's Lagrangians are treated in the imaginary-time and space and are invariant under scale transformations, which are motivated by the Tolman-Ehrenfest…
We examine the class of initial conditions which give rise to inflation. Our analysis is carried out for several popular models including: Higgs inflation, Starobinsky inflation, chaotic inflation, axion monodromy inflation and…
We start from classical general relativity coupled to matter fields. Each configuration variable and its conjugate momentum, as also space-time points, are raised to the status of matrices [equivalently operators]. These matrices obey a…
Many advances in physics have in common that some idea which was previously accepted as fundamental, general, and inescapable was subsequently seen to be consequent, special, and dispensable. The idea was not truly a general feature of the…
Effects of a Bohmian type quantum-relativistic theory are explored. The model is obtained by introducing a new and independent time parameter whose relative motions are not directly observable and cause the quantum uncertainties of the…
The four (electro-magnetic, weak, strong and gravitational) interactions are described by singular Lagrangians and by Dirac-Bergmann theory of Hamiltonian constraints. As a consequence a subset of the original configuration variables are…
We derive the matching conditions for cosmological perturbations in a Friedmann Universe where the equation of state undergoes a sharp jump, for instance as a result of a phase transition. The physics of the transition which is needed to…