5,864 research outputs found
Discrete Routh Reduction
This paper develops the theory of abelian Routh reduction for discrete
mechanical systems and applies it to the variational integration of mechanical
systems with abelian symmetry. The reduction of variational Runge-Kutta
discretizations is considered, as well as the extent to which symmetry
reduction and discretization commute. These reduced methods allow the direct
simulation of dynamical features such as relative equilibria and relative
periodic orbits that can be obscured or difficult to identify in the unreduced
dynamics. The methods are demonstrated for the dynamics of an Earth orbiting
satellite with a non-spherical correction, as well as the double
spherical pendulum. The problem is interesting because in the unreduced
picture, geometric phases inherent in the model and those due to numerical
discretization can be hard to distinguish, but this issue does not appear in
the reduced algorithm, where one can directly observe interesting dynamical
structures in the reduced phase space (the cotangent bundle of shape space), in
which the geometric phases have been removed. The main feature of the double
spherical pendulum example is that it has a nontrivial magnetic term in its
reduced symplectic form. Our method is still efficient as it can directly
handle the essential non-canonical nature of the symplectic structure. In
contrast, a traditional symplectic method for canonical systems could require
repeated coordinate changes if one is evoking Darboux' theorem to transform the
symplectic structure into canonical form, thereby incurring additional
computational cost. Our method allows one to design reduced symplectic
integrators in a natural way, despite the noncanonical nature of the symplectic
structure.Comment: 24 pages, 7 figures, numerous minor improvements, references added,
fixed typo
Invariant Lagrangians, mechanical connections and the Lagrange-Poincare equations
We deal with Lagrangian systems that are invariant under the action of a
symmetry group. The mechanical connection is a principal connection that is
associated to Lagrangians which have a kinetic energy function that is defined
by a Riemannian metric. In this paper we extend this notion to arbitrary
Lagrangians. We then derive the reduced Lagrange-Poincare equations in a new
fashion and we show how solutions of the Euler-Lagrange equations can be
reconstructed with the help of the mechanical connection. Illustrative examples
confirm the theory.Comment: 22 pages, to appear in J. Phys. A: Math. Theor., D2HFest special
issu
Dirac method and symplectic submanifolds in the cotangent bundle of a factorizable Lie group
In this work we study some symplectic submanifolds in the cotangent bundle of
a factorizable Lie group defined by second class constraints. By applying the
Dirac method, we study many issues of these spaces as fundamental Dirac
brackets, symmetries, and collective dynamics. This last item allows to study
integrability as inherited from a system on the whole cotangent bundle, leading
in a natural way to the AKS theory for integrable systems
Routhian reduction for quasi-invariant Lagrangians
In this paper we describe Routhian reduction as a special case of standard
symplectic reduction, also called Marsden-Weinstein reduction. We use this
correspondence to present a generalization of Routhian reduction for
quasi-invariant Lagrangians, i.e. Lagrangians that are invariant up to a total
time derivative. We show how functional Routhian reduction can be seen as a
particular instance of reduction of a quasi-invariant Lagrangian, and we
exhibit a Routhian reduction procedure for the special case of Lagrangians with
quasi-cyclic coordinates. As an application we consider the dynamics of a
charged particle in a magnetic field.Comment: 24 pages, 3 figure
Friction force microscopy : a simple technique for identifying graphene on rough substrates and mapping the orientation of graphene grains on copper
At a single atom thick, it is challenging to distinguish graphene from its substrate using conventional techniques. In this paper we show that friction force microscopy (FFM) is a simple and quick technique for identifying graphene on a range of samples, from growth substrates to rough insulators. We show that FFM is particularly effective for characterizing graphene grown on copper where it can correlate the graphene growth to the three-dimensional surface topography. Atomic lattice stickâslip friction is readily resolved and enables the crystallographic orientation of the graphene to be mapped nondestructively, reproducibly and at high resolution. We expect FFM to be similarly effective for studying graphene growth on other metal/locally crystalline substrates, including SiC, and for studying growth of other two-dimensional materials such as molybdenum disulfide and hexagonal boron nitride
Point vortices on the sphere: a case with opposite vorticities
We study systems formed of 2N point vortices on a sphere with N vortices of
strength +1 and N vortices of strength -1. In this case, the Hamiltonian is
conserved by the symmetry which exchanges the positive vortices with the
negative vortices. We prove the existence of some fixed and relative
equilibria, and then study their stability with the ``Energy Momentum Method''.
Most of the results obtained are nonlinear stability results. To end, some
bifurcations are described.Comment: 35 pages, 9 figure
Routh's procedure for non-Abelian symmetry groups
We extend Routh's reduction procedure to an arbitrary Lagrangian system (that
is, one whose Lagrangian is not necessarily the difference of kinetic and
potential energies) with a symmetry group which is not necessarily Abelian. To
do so we analyse the restriction of the Euler-Lagrange field to a level set of
momentum in velocity phase space. We present a new method of analysis based on
the use of quasi-velocities. We discuss the reconstruction of solutions of the
full Euler-Lagrange equations from those of the reduced equations.Comment: 30 pages, to appear in J Math Phy
Symmetry Reduction by Lifting for Maps
We study diffeomorphisms that have one-parameter families of continuous
symmetries. For general maps, in contrast to the symplectic case, existence of
a symmetry no longer implies existence of an invariant. Conversely, a map with
an invariant need not have a symmetry. We show that when a symmetry flow has a
global Poincar\'{e} section there are coordinates in which the map takes a
reduced, skew-product form, and hence allows for reduction of dimensionality.
We show that the reduction of a volume-preserving map again is volume
preserving. Finally we sharpen the Noether theorem for symplectic maps. A
number of illustrative examples are discussed and the method is compared with
traditional reduction techniques.Comment: laTeX, 31 pages, 5 figure
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