17,231 research outputs found
Orthogonal Separation of the Hamilton-Jacobi Equation on Spaces of Constant Curvature
We review the theory of orthogonal separation of variables of the
Hamilton-Jacobi equation on spaces of constant curvature, highlighting key
contributions to the theory by Benenti. This theory revolves around a special
type of conformal Killing tensor, hereafter called a concircular tensor. First,
we show how to extend original results given by Benenti to intrinsically
characterize all (orthogonal) separable coordinates in spaces of constant
curvature using concircular tensors. This results in the construction of a
special class of separable coordinates known as Kalnins-Eisenhart-Miller
coordinates. Then we present the Benenti-Eisenhart-Kalnins-Miller separation
algorithm, which uses concircular tensors to intrinsically search for
Kalnins-Eisenhart-Miller coordinates which separate a given natural
Hamilton-Jacobi equation. As a new application of the theory, we show how to
obtain the separable coordinate systems in the two dimensional spaces of
constant curvature, Minkowski and (Anti-)de Sitter space. We also apply the
Benenti-Eisenhart-Kalnins-Miller separation algorithm to study the separability
of the three dimensional Calogero-Moser and Morosi-Tondo systems
Higher Structures in M-Theory
The key open problem of string theory remains its non-perturbative completion
to M-theory. A decisive hint to its inner workings comes from numerous
appearances of higher structures in the limits of M-theory that are already
understood, such as higher degree flux fields and their dualities, or the
higher algebraic structures governing closed string field theory. These are all
controlled by the higher homotopy theory of derived categories, generalised
cohomology theories, and -algebras. This is the introductory chapter
to the proceedings of the LMS/EPSRC Durham Symposium on Higher Structures in
M-Theory. We first review higher structures as well as their motivation in
string theory and beyond. Then we list the contributions in this volume,
putting them into context.Comment: 22 pages, Introductory Article to Proceedings of LMS/EPSRC Durham
Symposium Higher Structures in M-Theory, August 2018, references update
Nonlinear acoustic waves in channels with variable cross sections
The point symmetry group is studied for the generalized Webster-type equation
describing non-linear acoustic waves in lossy channels with variable cross
sections. It is shown that, for certain types of cross section profiles, the
admitted symmetry group is extended and the invariant solutions corresponding
to these profiles are obtained. Approximate analytic solutions to the
generalized Webster equation are derived for channels with smoothly varying
cross sections and arbitrary initial conditions.Comment: Revtex4, 10 pages, 2 figure. This is an enlarged contribution to
Acoustical Physics, 2012, v.58, No.3, p.269-276 with modest stylistic
corrections introduced mainly in the Introduction and References. Several
typos were also correcte
Geometrical classification of Killing tensors on bidimensional flat manifolds
Valence two Killing tensors in the Euclidean and Minkowski planes are
classified under the action of the group which preserves the type of the
corresponding Killing web. The classification is based on an analysis of the
system of determining partial differential equations for the group invariants
and is entirely algebraic. The approach allows to classify both characteristic
and non characteristic Killing tensors.Comment: 27 pages, 20 figures, pictures format changed to .eps, typos
correcte
The Rational Higher Structure of M-theory
We review how core structures of string/M-theory emerge as higher structures
in super homotopy theory; namely from systematic analysis of the brane bouquet
of universal invariant higher central extensions growing out of the superpoint.
Since super homotopy theory is immensely rich, to start with we consider this
in the rational/infinitesimal approximation which ignores torsion-subgroups in
brane charges and focuses on tangent spaces of super space-time. Already at
this level, super homotopy theory discovers all super -brane species, their
intersection laws, their M/IIA-, T- and S-duality relations, their black brane
avatars at ADE-singularities, including their instanton contributions, and,
last not least, Dirac charge quantization: for the D-branes it recovers twisted
K-theory, rationally, but for the M-branes it gives cohomotopy cohomology
theory. We close with an outlook on the lift of these results beyond the
rational/infinitesimal approximation to a candidate formalization of
microscopic M-theory in super homotopy theory.Comment: 32 pages, Contribution to Proceedings of LMS/EPSRC Durham Symposium
Higher Structures in M-Theory, August 201
Invariant classification of orthogonally separable Hamiltonian systems in Euclidean space
The problem of the invariant classification of the orthogonal coordinate webs
defined in Euclidean space is solved within the framework of Felix Klein's
Erlangen Program. The results are applied to the problem of integrability of
the Calogero-Moser model
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