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.
