91,223 research outputs found
On the gradient of the Green tensor in two-dimensional elastodynamic problems, and related integrals: Distributional approach and regularization, with application to nonuniformly moving sources
The two-dimensional elastodynamic Green tensor is the primary building block
of solutions of linear elasticity problems dealing with nonuniformly moving
rectilinear line sources, such as dislocations. Elastodynamic solutions for
these problems involve derivatives of this Green tensor, which stand as
hypersingular kernels. These objects, well defined as distributions, prove
cumbersome to handle in practice. This paper, restricted to isotropic media,
examines some of their representations in the framework of distribution theory.
A particularly convenient regularization of the Green tensor is introduced,
that amounts to considering line sources of finite width. Technically, it is
implemented by an analytic continuation of the Green tensor to complex times.
It is applied to the computation of regularized forms of certain integrals of
tensor character that involve the gradient of the Green tensor. These integrals
are fundamental to the computation of the elastodynamic fields in the problem
of nonuniformly moving dislocations. The obtained expressions indifferently
cover cases of subsonic, transonic, or supersonic motion. We observe that for
faster-than-wave motion, one of the two branches of the Mach cone(s) displayed
by the Cartesian components of these tensor integrals is extinguished for some
particular orientations of source velocity vector.Comment: 25 pages, 6 figure
The Adiabatic Invariance of the Action Variable in Classical Dynamics
We consider one-dimensional classical time-dependent Hamiltonian systems with
quasi-periodic orbits. It is well-known that such systems possess an adiabatic
invariant which coincides with the action variable of the Hamiltonian
formalism. We present a new proof of the adiabatic invariance of this quantity
and illustrate our arguments by means of explicit calculations for the harmonic
oscillator.
The new proof makes essential use of the Hamiltonian formalism. The key step
is the introduction of a slowly-varying quantity closely related to the action
variable. This new quantity arises naturally within the Hamiltonian framework
as follows: a canonical transformation is first performed to convert the system
to action-angle coordinates; then the new quantity is constructed as an action
integral (effectively a new action variable) using the new coordinates. The
integration required for this construction provides, in a natural way, the
averaging procedure introduced in other proofs, though here it is an average in
phase space rather than over time.Comment: 8 page
Theory of Semiclassical Transition Probabilities for Inelastic and Reactive Collisions. V. Uniform Approximation in Multidimensional Systems
An integral semiclassical expression for the S matrix of inelastic and reactive collisions was formulated
earlier in this series. In the present paper a uniform approximation for the expression is derived for the
case of multidimensional systems. The method is an extension of that employed in Part II for the case
of one internal coordinate. The final result, Eq. (2), is highly symmetrical, thus making some of its properties
immediately clear
Decay of Solutions of the Teukolsky Equation for Higher Spin in the Schwarzschild Geometry
We prove that the Schwarzschild black hole is linearly stable under
electromagnetic and gravitational perturbations. Our method is to show that for
spin or , solutions of the Teukolsky equation with smooth, compactly
supported initial data outside the event horizon, decay in .Comment: 32 pages, LaTeX, 2 figures, error in expression for energy density of
gravitational waves correcte
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