1,527 research outputs found
The Energy-Momentum Method
This paper develops the energy momentum methodJor studying
stability and bifurcation of Lagrangian and Hamiltonian systems with symmetry. The method was specifically designed to deal with the stability of rotating structures. The relation with the energy-Casimir method is given and the energy-momentum method is shown to be more general. Stability of rigid body motion is given 10 illustrate the method. Some discussion of its applicability to general rotating systems and block diagonalization is also given
Stability Analysis of a Rigid Body with Attached Geometrically Nonlinear Rod by the Energy-Momentum Method
This paper applies the energy-momentum method to the problem of nonlinear stability of relative equilibria of a rigid body with attached flexible appendage in a uniformly rotating state. The appendage is modeled as a geometrically exact rod which allows for finite bending, shearing and twist in three dimensions. Application of the energy-momentum method to this example depends crucially on a
special choice of variables in terms of which the second variation block diagonalizes into blocks associated with rigid body modes and internal vibration modes respectively. The analysis yields a nonlinear stability result which states that relative equilibria are nonlinearly stable provided that; (i) the angular velocity is bounded above by the square root of the minimum eigenvalue of an associated
linear operator and, (ii) the whole assemblage is rotating about the minimum axis of inertia
Global superscaling analysis of quasielastic electron scattering with relativistic effective mass
We present a global analysis of the inclusive quasielastic electron
scattering data with a superscaling approach with relativistic effective mass.
The SuSAM* model exploits the approximation of factorization of the scaling
function out of the cross section under quasifree conditions. Our
approach is based on the relativistic mean field theory of nuclear matter where
a relativistic effective mass for the nucleon encodes the dynamics of nucleons
moving in presence of scalar and vector potentials. Both the scaling variable
and the single nucleon cross sections include the effective mass as a
parameter to be fitted to the data alongside the Fermi momentum . Several
methods to extract the scaling function and its uncertainty from the data are
proposed and compared. The model predictions for the quasielastic cross section
and the theoretical error bands are presented and discussed for nuclei along
the periodic table from to : H, H, He, He,
C, Li, Be, Mg, Ni,
Y, Sn, Ta, W, Au, O, Al,
Ca, Ca, Fe, Pb, and U.
We find that more than 9000 of the total data fall within the
quasielastic theoretical bands. Predictions for Ti and Ar are
also provided for the kinematics of interest to neutrino experiments.Comment: 26 pages, 20 figures and 4 table
Normalizing connections and the energy-momentum method
The block diagonalization method for determining the stability of relative equilibria is discussed from
the point of view of connections. We construct connections whose horizontal and vertical decompositions simultaneosly put the second variation of the augmented Hamiltonian and the symplectic structure into normal form. The cotangent bundle reduction theorem provides the setting in which the results are obtained
A block diagonalization theorem in the energy-momentum method
We prove a geometric generalization of a block diagonalization theorem first found by the authors for
rotating elastic rods. The result here is given in the general context of simple mechanical systems with a
symmetry group acting by isometries on a configuration manifold. The result provides a choice of
variables for linearized dynamics at a relative equilibrium which block diagonalizes the second variation of
an augmented energy these variables effectively separate the rotational and internal vibrational modes. The
second variation of the effective Hamiltonian is block diagonal. separating the modes completely. while the
symplectic form has an off diagonal term which represents the dynamic interaction between these modes.
Otherwise, the symplectic form is in a type of normal form. The result sets the stage for the development
of useful criteria for bifurcation as well as the stability criteria found here. In addition, the techniques
should apply to other systems as well, such as rotating fluid masses
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