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Authors = Alexei A. Deriglazov ORCID = 0000-0001-7464-1415

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17 pages, 477 KiB  
Article
Improved Equations of the Lagrange Top and Examples of Analytical Solutions
by Alexei A. Deriglazov
Particles 2024, 7(3), 543-559; https://doi.org/10.3390/particles7030030 - 24 Jun 2024
Cited by 2 | Viewed by 1482
Abstract
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 [...] Read more.
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 a 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, which is discussed in detail. Full article
(This article belongs to the Special Issue Feature Papers for Particles 2023)
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18 pages, 450 KiB  
Article
Rotation Matrix of a Charged Symmetrical Body: One-Parameter Family of Solutions in Elementary Functions
by Alexei A. Deriglazov
Universe 2024, 10(6), 250; https://doi.org/10.3390/universe10060250 - 3 Jun 2024
Cited by 3 | Viewed by 871
Abstract
Euler–Poisson equations 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 [...] Read more.
Euler–Poisson equations 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. A 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. In addition, we present a one-parametric family of solutions to the problem in elementary functions. Full article
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10 pages, 296 KiB  
Brief Report
Poincaré–Chetaev Equations in Dirac’s Formalism of Constrained Systems
by Alexei A. Deriglazov
Particles 2023, 6(4), 913-922; https://doi.org/10.3390/particles6040059 - 13 Oct 2023
Cited by 5 | Viewed by 1328
Abstract
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 the [...] Read more.
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 the 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 an exponential of the Hamiltonian vector field. Full article
43 pages, 541 KiB  
Review
Basic Notions of Poisson and Symplectic Geometry in Local Coordinates, with Applications to Hamiltonian Systems
by Alexei A. Deriglazov
Universe 2022, 8(10), 536; https://doi.org/10.3390/universe8100536 - 17 Oct 2022
Cited by 7 | Viewed by 1874
Abstract
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 [...] Read more.
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 proof of the compatibility of a system consisting of differential and algebraic equations, as well as applications for the problem of the reduction of a Hamiltonian system with known integrals of motion. Full article
(This article belongs to the Section Mathematical Physics)
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