7 research outputs found
On scattering of solitons for the Klein-Gordon equation coupled to a particle
We establish the long time soliton asymptotics for the translation invariant
nonlinear system consisting of the Klein-Gordon equation coupled to a charged
relativistic particle. The coupled system has a six dimensional invariant
manifold of the soliton solutions. We show that in the large time approximation
any finite energy solution, with the initial state close to the solitary
manifold, is a sum of a soliton and a dispersive wave which is a solution of
the free Klein-Gordon equation. It is assumed that the charge density satisfies
the Wiener condition which is a version of the ``Fermi Golden Rule''. The proof
is based on an extension of the general strategy introduced by Soffer and
Weinstein, Buslaev and Perelman, and others: symplectic projection in Hilbert
space onto the solitary manifold, modulation equations for the parameters of
the projection, and decay of the transversal component.Comment: 47 pages, 2 figure
Convergence to equilibrium distribution. The Klein-Gordon equation coupled to a particle
We consider the Hamiltonian system consisting of a Klein-Gordon vector field
and a particle in . The initial date of the system is a random function
with a finite mean density of energy which also satisfies a Rosenblatt- or
Ibragimov-type mixing condition. Moreover, initial correlation functions are
translation-invariant. We study the distribution of the solution at
time . The main result is the convergence of to a Gaussian
measure as , where is translation-invariant.Comment: 22 page
Relativistic dynamics of accelerating particles derived from field equations
In relativistic mechanics the energy-momentum of a free point mass moving
without acceleration forms a four-vector. Einstein's celebrated energy-mass
relation E=mc^2 is commonly derived from that fact. By contrast, in Newtonian
mechanics the mass is introduced for an accelerated motion as a measure of
inertia. In this paper we rigorously derive the relativistic point mechanics
and Einstein's energy-mass relation using our recently introduced neoclassical
field theory where a charge is not a point but a distribution. We show that
both the approaches to the definition of mass are complementary within the
framework of our field theory. This theory also predicts a small difference
between the electron rest mass relevant to the Penning trap experiments and its
mass relevant to spectroscopic measurements.Comment: A few typos were correcte