944 research outputs found
Approximate renormalization for the break-up of invariant tori with three frequencies
We construct an approximate renormalization transformation for Hamiltonian
systems with three degrees of freedom in order to study the break-up of
invariant tori with three incommensurate frequencies which belong to the cubic
field , where . This renormalization has two
fixed points~: a stable one and a hyperbolic one with a codimension one stable
manifold. We compute the associated critical exponents that characterize the
universality class for the break-up of the invariant tori we consider.Comment: 5 pages, REVTe
Determination of the threshold of the break-up of invariant tori in a class of three frequency Hamiltonian systems
We consider a class of Hamiltonians with three degrees of freedom that can be
mapped into quasi-periodically driven pendulums. The purpose of this paper is
to determine the threshold of the break-up of invariant tori with a specific
frequency vector. We apply two techniques: the frequency map analysis and
renormalization-group methods. The renormalization transformation acting on a
Hamiltonian is a canonical change of coordinates which is a combination of a
partial elimination of the irrelevant modes of the Hamiltonian and a rescaling
of phase space around the considered torus. We give numerical evidence that the
critical coupling at which the renormalization transformation starts to diverge
is the same as the value given by the frequency map analysis for the break-up
of invariant tori. Furthermore, we obtain by these methods numerical values of
the threshold of the break-up of the last invariant torus.Comment: 18 pages, 4 figure
Incomplete Dirac reduction of constrained Hamiltonian systems
First-class constraints constitute a potential obstacle to the computation of
a Poisson bracket in Dirac's theory of constrained Hamiltonian systems. Using
the pseudoinverse instead of the inverse of the matrix defined by the Poisson
brackets between the constraints, we show that a Dirac-Poisson bracket can be
constructed, even if it corresponds to an incomplete reduction of the original
Hamiltonian system. The uniqueness of Dirac brackets is discussed
Conservative dissipation: How important is the Jacobi identity in the dynamics?
Hamiltonian dynamics are characterized by a function, called the Hamiltonian,
and a Poisson bracket. The Hamiltonian is a conserved quantity due to the
anti-symmetry of the Poisson bracket. The Poisson bracket satisfies the Jacobi
identity which is usually more intricate and more complex to comprehend than
the conservation of the Hamiltonian. Here we investigate the importance of the
Jacobi identity in the dynamics by considering three different types of
conservative flows in R3 : Hamiltonian, almost-Poisson and metriplectic. The
comparison of their dynamics reveals the importance of the Jacobi identity in
structuring the resulting phase space
Second order nonlinear gyrokinetic theory : From the particle to the gyrocenter
A gyrokinetic reduction is based on a specific ordering of the different
small parameters characterizing the background magnetic field and the
fluctuating electromagnetic fields. In this tutorial, we consider the following
ordering of the small parameters: where
is the small parameter associated with spatial inhomogeneities of
the background magnetic field and characterizes the small
amplitude of the fluctuating fields. In particular, we do not make any
assumption on the amplitude of the background magnetic field. Given this choice
of ordering, we describe a self-contained and systematic derivation which is
particularly well suited for the gyrokinetic reduction, following a two-step
procedure. We follow the approach developed in [Sugama, Physics of Plasmas 7,
466 (2000)]:In a first step, using a translation in velocity, we embed the
transformation performed on the symplectic part of the gyrocentre reduction in
the guiding-centre one. In a second step, using a canonical Lie transform, we
eliminate the gyroangle dependence from the Hamiltonian. As a consequence, we
explicitly derive the fully electromagnetic gyrokinetic equations at the second
order in
Photon plasma--wave interaction via Compton scattering
The Kompaneets theory of photon kinetic evolution due to the Compton effect
in the absence of absorption and emission is extended to the case of the Vlasov
plasma wave oscillations. Under the assumption that the electron distribution
function at equilibrium is perturbed by a solution of the linearised Vlasov
equation in the long-wavelength limit, a solution of the Kompaneets kinetic
equation for the photon distribution function is found and discussed
Kinematics of fluid particles on the sea surface. Hamiltonian theory
We derive the John-Sclavounos equations describing the motion of a fluid
particle on the sea surface from first principles using Lagrangian and
Hamiltonian formalisms applied to the motion of a frictionless particle
constrained on an unsteady surface. The main result is that vorticity generated
on a stress-free surface vanishes at a wave crest when the horizontal particle
velocity equals the crest propagation speed, which is the kinematic criterion
for wave breaking. If this holds for the largest crest, then the symplectic
two-form associated with the Hamiltonian dynamics reduces instantaneously to
that associated with the motion of a particle in free flight, as if the surface
did not exist. Further, exploiting the conservation of the Hamiltonian function
for steady surfaces and traveling waves we show that particle velocities remain
bounded at all times, ruling out the possibility of the finite-time blowup of
solutions
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