We present a variational framework for analyzing planar elastic rods subjected to distributed magnetic and gravitational loads, systematically reducing nested integrals in the energy functional to single-integral expressions. By exploiting Fubini's theorem and introducing cumulative field functions, the approach simplifies the derivation of the governing equations while clearly separating contributions from bending, magnetic torques, and gravity. Inspired by studies on threedimensional hard-magnetic rods, the proposed framework not only enhances the clarity of energy-functional analysis but also provides a concise and systematic mathematical formulation for the theoretical study of planar rods under combined magnetic and gravitational effects.
The geometric study of invariant manifolds in completely integrable Hamiltonian systems remains a fundamental topic in mathematical physics. This paper investigates the Sawada-Kotera case of the generalized H & eacute;non-Heiles system on a fourdimensional Poisson manifold. We focus on the metric properties of the Liouville tori that foliate the phase space. By analyzing the momentum map and its critical values, we explicitly compute the components of the Gaussian torsion. Our main result demonstrates that the Gaussian torsion vanishes identically across all regular chambers of the bifurcation diagram, identifying these tori as a geometrically distinguished class of surfaces.
Square roots of real and complex (complexified) quaternions, namely, the Hamilton's quaternion, coquaternion, nectorine, and conectorine are investigated. The isomorphism between the quaternions and multivectors in Clifford algebras is employed for this purpose. Root examples for all named quaternions are presented from which follows that the real and complex quaternionic roots may assume multiple discrete or continuous forms, or there may be no roots at all.
In this article we will show the correspondence between eigenvalue problems defining some Lie algebra and the possibility to generate a compatible Lie brackets. Compatible Lie brackets give rise to a bi-Hamiltonian structure, which can be effectively used to construct integrable systems related to this Lie algebra. We will describe this construction from the perspective of eigenvalue problems and illustrate it on selected example.
We investigate hypersurfaces in the Euclidean four-space R4 that are foliated by portions of spheres and satisfy a Weingarten relation of the form aK1 + bK2 + cK3 = d, where a, b, c, and dare constants. In this relation, K1, K2, and K3 denote the mean curvature, the second curvature, and the Gauss-Kronecker curvature, respectively.
This paper presents a structural reformulation of the polynomial root-finding problem from the perspective of representation theory of finite abelian groups and the Discrete Fourier Transform (DFT). By embedding polynomials into cyclic number algebras, we demonstrate that the global configuration of polynomial roots can be understood as the decomposition of the regular representation of cyclic groups in the frequency domain, with DFT serving as the natural tool for this decomposition. This yields a structural method for analyzing root distributions and constructing numerical initial values, serving as a beneficial supplement to traditional numerical root-finding algorithms. The framework is extended to multivariate polynomial systems, demonstrating potential adaptability in parallel computing environments. Numerical experiments demonstrate positive effects on initial value selection and iteration stability.
We consider the Newton equations and the Euler-Lagrange equations that admit reduction of order via the conservation of energy law. Passing to the first-order equation of the constant energy, "extraneous solutions" that are not solutions of the initial second-order equation can appear. Analysis of "extraneous solutions" leads to a number of interesting results. For example, it helps to understand the behaviour of geodesics on surfaces of revolution.
The theory considered interprets gravity as a pressure force. Thus, the scalar gravitational field defines the gravity acceleration field. However, it also determines the relation between the flat “background metric” and a curved “physical metric”. Here we derive the equations of motion of the mass centers of a system of weakly gravitating bodies in the second version of that theory. We use the framework which was built and used for the first version. Namely, we use an asymptotic scheme of post-Newtonian (PN) approximation to derive the local (field) PN equations, and by integration inside the bodies we deduce from those local equations the equations of motion of the mass centers, using also an asymptotic framework for the good separation between the different bodies.
We study the generalized Fisher metric on the Lie groups SO(2) and SO(3) via the Souriau thermodynamics Lie group theories. Then we give the effect of twococycle on the integrability of gradient systems due to the Fisher metric and the Souriau-Fisher metric. In addition, we show how the cocycle can locally modify the Fisher metric on a coadjoint orbit, in explicit terms of brackets and central extensions on the Lie groups SO(2) and SO(3).
We present a plethora of explicitly parameterized real symplectic matrices in dimension four which were missing up to now. Structurally, these matrices depend on a set of ten real parameters and split naturally as an union of squeezing, rotational and boost transformations, and their fundamental representations have been derived relying on exponential, Cayley and Fedorov-like maps. Explicit formulas relating the Lie algebra and the Lie group elements in both directions were found. Besides, an algorithmic procedure for factorization of an arbitrary symplectic matrix as a product of three matrices with clear mathematical/physical interpretations was found and exemplified via numerous matrices. Surprisingly, as a side effect of the decomposition, we have arrived at alternative realizations of the real ortho-symplectic matrices in four dimensions.
Ferapontov and Fordy considered the system of hydrodynamic type with Hamiltonian, which is equipped with the pseudo Riemannian metric tensor, the involutive function was defined on the phase space. This system is applied the construction of 6-dimensional symplectic Haantjes manifolds and we construct some examples of the symplectic Haantjes structure, which contain the Haantjes chains.
In this work, we examine the conditions under which a compact gradient Ricci-Yamabe soliton is Einstein, that is, when the soliton is trivial. We then proceed to establish that a potential vector field is Killing whenever it is solenoidal. Furthermore, we investigate that a radially flat gradient Ricci-Yamabe soliton is rigid. Finally, we prove that a gradient Ricci-Yamabe soliton admits non-parallel, closed homothetic vector fields under certain conditions.
This paper starts with a short historical review of the appearance of the rotational matrices in science which principle purpose is to display them explicitly in their original form and notation. Then we continue with reviewing the symplectic matrices in four-dimensional real space with emphasis on a special class of them - the so-called ortho-symplectic matrices. Further on, we present a plethora of uniformly generated rotational matrices in 3D following the scheme described in a This scheme is based on the Hopf map which presents the Euclidean space as a shadow of the four-dimensional real space. Viewed in this way, it is easy to be seen that the rotational motions in the total space is transmitted as rotational motions in the base space and this is used to built up any three-dimensional rotational matrix (including the symmetrical ones). Additionally, a theorem is proved which says that all 3D rotational matrices can be obtained in this manner. On the way it is pointed out also that this opens the possibility to describe easily their composition and to generate new 3D rotational matrices.
Functions parametrized by translations and rotations are associated to unitary operators, which act on square-integrable functions on spheres. This Weyl correspondence allows for phase space representations of quantum operators over the Euclidean motion group, as well as providing Wigner distribution functions on the group. The star-product of a pair of phase-space functions will be defined via the composition of two Weyl operators and properties of this star-product analogous to standard properties of algebras of quantum observables will be inferred from the Weyl calculus of operators. A fundamental formula relating the Weyl operator and the Wigner distribution function will also be proved.
The main subject of this paper are the second-order differential invariants of submersions with respect to the group of conformal transformations of Euclidean spaces. In particular, it is proved that the ratio of principal surface curvatures is a second-order differential invariant with respect to the group of conformal transformations.
In this paper, we define a statistical Ricci-soliton vector field and then we show that if structural vector field of compact Hopf statistical hypersurface of complex space form is a statistical Ricci-soliton vector field, it has at most three constant principal curvatures.
In this paper we present a straightforward algorithm that reconstructs the entire three-dimensional rotational matrix from the knowledge of any five of its elements. Additionally, we have demonstrated the application of the proposed scheme via various concrete examples of incomplete matrices.
We introduce a novel four-dimensional spinor representation of the Lorentz group in which both Dirac and Weyl spinors are realized as four-component objects living in a common vector space. Furthermore, Dirac spinors can be expressed as vector sum - rather than a direct sum - of left- and right-chiral four-component Weyl spinors. In this representation, Dirac spinors and their left and right components transform under the same spinor space, permitting an unambiguous identification of their chiral constituents. This formalism provides a symmetric and geometrically transparent reinterpretation of Weyl and Dirac spinors and may offer new insights into extended spinor models and relativistic field theories.
A generalization of the Euler's elastic problem, i.e., finding a stationary configuration (planar elastica) of the Bernoulli's thin ideal elastic rod with boundary conditions defined through fixed endpoints and/or tangents at the endpoints, for the chosen nonlocal differential constitutive stress-strain relation (i.e., nonlocal theory of elasticity) is considered. In the classical (local) Euler-Bernoulli's beam model, the general solutions of the governing equations (that are inhomogeneous but linear) for bending moments and shear forces in the case of large deformations can be obtained using the Jacobi elliptic functions and incomplete elliptic integrals. For the discussed nonlocal toy differential model, the general solutions of the governing equations (that are this time nonlinear) can also be expressed in the parametric form through the linear combinations of all three incomplete elliptic integrals. As further research, we plan to apply some boundary conditions (clamped, simply supported, etc.) for the obtained nonlocal general solutions in order to compare them to the local solutions for the corresponding boundary value problems.
The description of the Z2 x Z2-graded special linear Lie superalgebra is carried out via a set of generators that satisfy triple relations and are called creation and annihilation operators. With respect to these generators, a class of Fock type representations of the algebra is constructed. The properties of the underlying statistics are discussed and its Pauli principle is formulated.
