Related papers: Ortho-normal quaternion frames, Lagrangian evoluti…
The Lagrangian formalism is used to derive covariant equations that are suitable for use in continuously distributed matter in curved spacetime. Special attention is given to theoretical representation, in which the Lagrangian and its…
Predicting particle transport in complex flows is traditionally achieved by solving the Navier-Stokes equations. While various numerical and experimental methods exist, they typically require deep physical insights and incur high…
A new perspective on the classical mechanical formulation of particle trajectories in lorentz-violating theories is presented. Using the extended hamiltonian formalism, a Legendre Transformation between the associated covariant Lagrangian…
A modified quantum kinetic equation which takes account of the noninertial features of rotating frame is proposed. The vector and axial-vector field components of the Wigner function for chiral fluids are worked out in a semiclassical…
We investigate the global dynamics of the field equations of (pure) quadratic theories of gravity which generalise Einstein's theory in spatially flat homogeneous and isotropic cosmological models with a perfect fluid. We introduce global…
It is well known that a rigid motion of the Euclidean plane can be written as the composition of at most three reflections. It is perhaps not so widely known that a similar result holds for Euclidean space in any number of dimensions. The…
We present fast and accurate ways to normalize two and three dimensional vectors and quaternions and compute their length. Our approach is an adaptation of ideas used in the linear algebra library LAPACK, and we believe that the…
One of the important ways development takes place in mathematics is via a process of generalization. On the basis of a recent characterization of this process we propose a principle that generalizations of mathematical structures that are…
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…
This work proposes a quaternion-based sliding variable that describes exponentially convergent error dynamics for any forward complete desired attitude trajectory. The proposed sliding variable directly operates on the non-Euclidean space…
A novel single-frame quaternion estimator processing two vector observations is introduced. The singular cases are examined, and appropriate rotational solutions are provided. Additionally, an alternative method involving sequential…
Rigid body dynamics on the rotation group have typically been represented in terms of rotation matrices, unit quaternions, or local coordinates, such as Euler angles. Due to the coordinate singularities associated with local coordinate…
Patterns arise spontaneously in a range of systems spanning the sciences, and their study typically focuses on mechanisms to understand their evolution in space-time. Increasingly, there has been a transition towards controlling these…
The formation and development of oscillations is an important traffic flow phenomenon. Recent studies found that along a vehicle platoon described in the Lagrangian specification, traffic oscillations grow in a concave way. Since stationary…
We investigate the purely spatial Lagrangian coordinate transformation from the Lagrangian to the basic Eulerian frame. We demonstrate three techniques for extracting the relativistic displacement field from a given solution in the…
We consider a general approach to the theory of continuous media starting from Lagrangian formalism. This formalism which uses the trajectories if constituents of media is very convenient for taking into account different types of…
In the last one and a half centuries, the analysis of quaternions has not only led to further developments in mathematics but has also been and remains an important catalyst for the further development of theories in physics. At the same…
In this article we investigate a system of geometric evolution equations describing a curvature driven motion of a family of 3D curves in the normal and binormal directions. Evolving curves may be subject of mutual interactions having both…
The dynamics of self-gravitating fluid bodies is described by the Euler-Einstein system of partial differential equations. The break-down of well-posedness on the fluid-vacuum interface remains a challenging open problem, which is…
In 1966, Arnold [1] showed that the Lagrangian flow of ideal incompressible fluids (described by Euler equations) coincide with the geodesic flow on the manifold of volume preserving diffeomorphisms of the fluid domain. Arnold's proof and…