Related papers: Lorenz integrable system moves \`a la Poinsot
The hypothesis that the Lorentz transformations may be modified at Planck scale energies is further explored. We present a general formalism for theories which preserve the relativity of inertial frames with a non-linear action of the…
We consider transformations between uniformly accelerated systems, assuming that the Clock Hypothesis is false. We use the proper velocity-time description of events rather than the usual space-time description in order to obtain linear…
It is shown that the Lorentz invariant $f(T)$ gravity, defined by the coframe-connection-multiplier form of the Lagrangian, can be gauge-fixed to the pure coframe form. After clarifying basic aspects of the problem in the Lagrangian…
I tell about different mathematical tool that is important in general relativity. The text of the book includes definition of geometrical object, concept of reference frame, geometry of metric-affinne manifold. Using this concept I learn…
In this paper we consider the general setting for constructing Action Principles for three-dimensional first order autonomous equations. We present the results for some integrable and non-integrable cases of the Lotka-Volterra equation, and…
Making use of the SO(3,1) Lorentz algebra, we derive in this paper two series of Gauss-Bonnet type identities involving torsion, one being of the Pontryagin type and the other of the Euler type. Two of the six identities involve only…
It is shown that gravity on the line can be described by the two dimensional (2D) Hilbert-Einstein Lagrangian supplemented by a kinetic term for the coframe and a translational {\it boundary} term. The resulting model is equivalent to a…
Following the Poincare algebra for a free spinning particle and using the Casimirs of the algebra in the Hamiltonian approach, we construct systematically a set of Lagrangians for the relativistic spinning particle which includes the…
We here deduce Lorentz transformation (LT) as a member of a class of time-dependent coordinate transformations, complementary to those already known as spatial translations and rotations. This exercise validates the principle of physical…
The three-body problem is reexamined in the framework of general relativity. The Newtonian three-body problem admits Euler's collinear solution, where three bodies move around the common center of mass with the same orbital period and…
We study four problems in the dynamics of a body moving about a fixed point, providing a non-complex, analytical solution for all of them. For the first two, we will work on the motion first integrals. For the symmetrical heavy body, that…
While the Lorentz group serves as the basic language for Einstein's special theory of relativity, it is turning out to be the basic mathematical instrument in optical sciences, particularly in ray optics and polarization optics. The beam…
This paper aims at the most comprehensive and systematic construction and tabulation of mechanical systems that admit a second invariant, quadratic in velocities, other than the Hamiltonian. The configuration space is in general a 2D…
We consider Lagrangians in Hamilton's principle defined on the tangent space $TG$ of a Lie group $G$. Invariance of such a Lagrangian under the action of $G$ leads to the symmetry-reduced Euler-Lagrange equations called the Euler-Poincar\'e…
Matrix orthogonal Laurent polynomials in the unit circle and the theory of Toda-like integrable systems are connected using the Gauss--Borel factorization of two, left and a right, Cantero-Morales-Velazquez block moment matrices, which are…
A generalization of the Lorenz equations is proposed where the variables take values in a Lie algebra. The finite dimensionality of the representation encodes the quantum fluctuations, while the non-linear nature of the equations can…
We construct a normal form suited to {\it fast driven systems}. We call so systems including actions ${\rm I}$, angles {$\psi$}, and one fast coordinate $y$, moving under the action of a vector--field $N$ depending only on ${\rm I}$ and $y$…
In this article we discuss a weaker version of Liouville's theorem on the integrability of Hamiltonian systems. We show that in the case of Tonelli Hamiltonians the involution hypothesis on the integrals of motion can be completely dropped…
A gauge theory of the Lorentz group, based on the different behavior of spinors and vectors under local transformations, is formulated in a flat space-time and the role of the torsion field within the generalization to curved space-time is…
A heavy top with a fixed point and a rigid body in an ideal fluid are important examples of Hamiltonian systems on a dual to the semidirect product Lie algebra $e(n)=so(n)\ltimes\mathbb R^n$. We give a Lagrangian derivation of the…