Searcharxiv⌕ Search

arXiv subjects

Alexei A. Deriglazov

Publications and source records attributed to Alexei A. Deriglazov.

At least 19 recordsLinked to original sources
arXiv2026

Dynamics of a relativistic discrete body: rigidity conditions, and covariant equations of motion

Rigidity conditions for a body considered as a discrete system of relativistic particles are proposed. They by themselves do not yet determine an evolution of the system, and some second-order equations must be added to them. Poincaré-covariant equations of motion compatible with these rigidity conditions are proposed and discussed. The resulting theory has the expected six dynamical degrees of freedom, allowing for more general motions than in Born's theory. Therefore, treating a relativistic body as a discrete system of particles could be a promising alternative to the standard approach based on Born's rigidity conditions.

gr-qc↗
arXiv2025

Variational problem and Hamiltonian formulation of the Lagrange-d'Alembert equations with nonlinear nonholonomic constraints

Any given system of ordinary differential equations in $n$-dimensional configuration space can be obtained from a peculiar variational problem with one local symmetry. The obtained action functional leads to the Hamiltonian formulation in $(4n+2)$-dimensional phase space. As concrete examples, we discuss the cases of Lagrange-d'Alembert equations with nonlinear nonholonomic constraints, as well as the equations of motion with dissipative (frictional) forces.

math-ph↗
arXiv2025

An alternative interpretation of the Grioli gyroscope suspension points

We present a detailed analysis of all possible regular precessions of a heavy asymmetric body with a fixed point not coinciding with the center of mass. The calculations are done in terms of the rotation matrix, by writing the Euler-Poisson equations with all involved vectors parameterized in the Laboratory frame. It is shown that a regular precession is possible if the suspension point is chosen on the straight lines (lying in the principal plane) which are frontiers of the regions where, as the distance from the center of mass increases, the interchange of the intermediate and largest moments of inertia occurs. Like the spin of an electron in quantum mechanics, the frequency of regular precession in classical mechanics turns out to be rigidly fixed by two values, i.e., quantized.

physics.class-ph↗
arXiv2024

An asymmetrical body: example of analytical solution for the rotation matrix in elementary functions and Dzhanibekov effect

We solved the Poisson equations, obtaining their exact solution in elementary functions for the rotation matrix of a free asymmetrical body with angular velocity vector lying on separatrices. This allows us to discuss the temporal evolution of Dzhanibekov's nut directly in the Laboratory system, where it is observed. The rotation matrix depends on two parameters with clear physical interpretation as a frequency and a damping factor of the solution. Qualitative analysis of the solution shows that it properly describes a single-jump Dzhanibekov effect.

nlin.SI↗
arXiv2024

Lagrange top: integrability according to Liouville and examples of analytic solutions

Equations of a heavy rotating body with one fixed point can be deduced starting from a variational problem with holonomic constraints. When applying this formalism to the particular case of a Lagrange top, in the formulation with a diagonal inertia tensor the potential energy has more complicated form as compared with that assumed in the literature on dynamics of a rigid body. This implies the corresponding improvements in equations of motion. Therefore, we revised this case, presenting several examples of analytical solutions to the improved equations. The case of precession without nutation has a surprisingly rich relationship between the rotation and precession rates, and this is discussed in detail.

physics.class-ph↗
arXiv2024

Rotation matrix of a charged symmetrical body: one-parameter family of solutions in elementary functions

Equations of motion of a charged symmetrical body in external constant and homogeneous electric and magnetic fields are deduced starting from the variational problem, where the body is considered as a system of charged point particles subject to holonomic constraints. The final equations are written for the center-of mass-coordinate, rotation matrix and angular velocity. General solution to the equations of motion is obtained for the case of a charged ball. For the case of a symmetrical charged body (solenoid), the task of obtaining the general solution is reduced to the problem of a one-dimensional cubic pseudo-oscillator. Besides, we present a one-parametric family of solutions to the problem in elementary functions.

physics.class-ph↗
arXiv2024

Dynamics on a submanifold: intermediate formalism versus Hamiltonian reduction of Dirac bracket, and integrability

We consider Hamiltonian formulation of a dynamical system forced to move on a submanifold $G_α(q^A)=0$. If for some reasons we are interested in knowing the dynamics of all original variables $q^A(t)$, the most economical would be a Hamiltonian formulation on the intermediate phase-space submanifold spanned by reducible variables $q^A$ and an irreducible set of momenta $p_i$, $[i]=[A]-[α]$. We describe and compare two different possibilities for establishing the Poisson structure and Hamiltonian dynamics on an intermediate submanifold: Hamiltonian reduction of the Dirac bracket and intermediate formalism. As an example of the application of intermediate formalism, we deduce on this basis the Euler-Poisson equations of a spinning body, establish the underlying Poisson structure, and write their general solution in terms of the exponential of the Hamiltonian vector field.

math-ph↗
arXiv2023

Geodesic motion on the symplectic leaf of $SO(3)$ with distorted $e(3)$ algebra and Liouville integrability of a free rigid body

The solutions to the Euler-Poisson equations are geodesic lines of $SO(3)$ manifold with the metric determined by the inertia tensor. However, the Poisson structure on the corresponding symplectic leaf does not depend on the inertia tensor. We calculate its explicit form and confirm that it differs from the algebra $e(3)$. The obtained Poisson brackets are used to demonstrate the Liouville integrability of a free rigid body. The general solution to the Euler-Poisson equations is written in terms of exponential of the Hamiltonian vector field.

math-ph↗
arXiv2023

Poincaré-Chetaev equations in the Dirac's formalism of constrained systems

We single out a class of Lagrangians on a group manifold, for which one can introduce non-canonical coordinates in the phase space, which simplify the construction of the Poisson structure without explicitly calculating the Dirac bracket. In the case of $SO(3)$\,- manifold, the application of this formalism leads to the Poincaré-Chetaev equations. The general solution to these equations is written in terms of exponential of the Hamiltonian vector field.

math-ph↗
arXiv2023

Has the problem of the motion of a heavy symmetric top been solved in quadratures?

We have revised the problem of the motion of a heavy symmetric top. When formulating equations of the Lagrange top with the diagonal inertia tensor, the potential energy has more complicated form as compared with that assumed in the literature on dynamics of a rotating body. This implies the corresponding improvements in equations of motion. Using the Liouville's theorem, we solve the improved equations in quadratures and present the explicit expressions for the resulting elliptic integrals.

physics.class-ph↗
arXiv2023

Lagrangian and Hamiltonian formulations of asymmetric rigid body, considered as a constrained system

This work is devoted to a systematic exposition of the dynamics of a rigid body, considered as a system with kinematic constraints. Having accepted the variational problem in accordance with this, we no longer need any additional postulates or assumptions about the behavior of the rigid body. All the basic quantities and characteristics of a rigid body, as well as the equations of motion and integrals of motion, are obtained from the variational problem by direct and unequivocal calculations within the framework of standard methods of classical mechanics. Several equivalent forms for the equations of motion of rotational degrees of freedom are deduced and discussed on this basis. Using the resulting formulation, we revise some cases of integrability, and discuss a number of peculiar properties, that are not always taken into account when formulating the laws of motion of a rigid body.

physics.class-ph↗
arXiv2023

Euler-Poisson equations of a dancing spinning top, integrability and examples of analytical solutions

Equations of a rotating body with one point constrained to move freely on a plane (dancing top) are deduced from the Lagrangian variational problem. They formally look like the Euler-Poisson equations of a heavy body with fixed point, immersed in a fictitious gravity field. Using this analogy, we have found examples of analytical solutions for the case of a heavy symmetrical dancing top. They describe the motions with center of mass keeping its height fixed above the supporting plane. General solution to equations of a dancing top in terms of exponential of Hamiltonian field is given. An extra constraint, that take into account the reaction of supporting plane, leads to modification of the canonical Poisson structure and therefore the integrability according to Liouville is under the question.

math-ph↗
arXiv2022

Basic notions of Poisson and symplectic geometry in local coordinates, with applications to Hamiltonian systems

This work contains a brief and elementary exposition of the foundations of Poisson and symplectic geometries, with an emphasis on applications for Hamiltonian systems with second-class constraints. In particular, we clarify the geometric meaning of the Dirac bracket on a symplectic manifold and provide a proof of the Jacobi identity on a Poisson manifold. A number of applications of the Dirac bracket are described: applications for the proof of the compatibility of a system consisting of differential and algebraic equations, as well as applications for the problem of reduction of a Hamiltonian system with known integrals of motion.

math.SG↗
arXiv2022

An angular rainbow of light from curved spacetime

We try to go beyond the geometrical optics approximation, by showing that a massless polarized particle allows a wide class of non minimal interactions with an arbitrary gravitational field. One specific example of a curvature-dependent interaction is presented, that results in a frequency-dependent Faraday effect. Even in a Schwarzschild spacetime, this leads to the angular dispersion of polarization planes for a linearly-polarized beam of waves with different frequencies, propagating along the same ray.

gr-qc↗
arXiv2021

Spinning gravimagnetic particles in Schwarzschild-like black holes

We study the motion of a spinning particle with gravimagnetic moment in Schwarzschild-like spacetimes with a metric $ds^2=-f(r) dt^2 + f^{-1}(r) dr^2 + r^2 dΩ^2$, specifically we deal with Schwarzschild, Reissner-Nordstrom black holes as well as Ayon-Beato-Garcia and Bardeen regular spacetimes. First, we introduce the Hamiltonian system of equations which describes such kind of particles. In the case of null gravimagnetic moment, the equations are equivalent to the Mathisson-Papapetrou-Tulczyjew-Dixon (MPTD) equations. Working in the equatorial plane, using the constants of motion generated by the symmetries of the considered spacetimes and the Spin Supplementary Conditions (SSC), we change the problem of solving six differential equations for the momenta and the non-vanishing spin-tensor components to solving six algebraic equations. We show that the equation for the $P_0(r)$ component totally decouples, $P_0(r)$ can be found by solving a 6th order polynomial. We analyze the conditions for existence of solutions of this algebraic system for the relevant cases of gravimagnetic moment equal to unit, which corresponds to a gravimagnetic particle, and zero which corresponds to the MTPD system. A numerical algorithm to generate solutions of the momenta $P_μ$ is provided and some solutions are generated.

gr-qc↗