215 research outputs found

### The gravity of light

A solution of the old problem raised by Tolman, Ehrenfest, Podolsky and
Wheeler, concerning the lack of attraction of two light pencils "moving
parallel", is proposed, considering that the light can be source of nonlinear
gravitational waves corresponding (in the would be quantum theory of gravity)
to spin-1 massless particles.Comment: Style is changed in standard latex, abstract has been reduced and the
order of sections has been change

### Liouville Integrability of the Schroedinger Equation

Canonical coordinates for both the Schroedinger and the nonlinear
Schroedinger equations are introduced, making more transparent their
Hamiltonian structures. It is shown that the Schroedinger equation, considered
as a classical field theory, shares with the nonlinear Schroedinger, and more
generally with Liouville completely integrable field theories, the existence of
a "recursion operator" which allows for the construction of infinitely many
conserved functionals pairwise commuting with respect to the corresponding
Poisson bracket. The approach may provide a good starting point to get a clear
interpretation of Quantum Mechanics in the general setting, provided by
Stone-von Neumann theorem, of Symplectic Mechanics. It may give new tools to
solve in the general case the inverse problem of Quantum Mechanics.Comment: 13 pages, Latex, no figure

### Noncommutative integrability and recursion operators

Geometric structures underlying commutative and non commutative integrable
dynamics are analyzed. They lead to a new characterization of noncommutative
integrability in terms of spectral properties and of Nijenhuis torsion of an
invariant (1,1) tensor field. The construction of compatible symplectic
structures is also discussed.Comment: 20 pages, LaTex, no figure

### Symplectic Structures and Quantum Mechanics

Canonical coordinates for the Schr\"odinger equation are introduced, making
more transparent its Hamiltonian structure. It is shown that the Schr\"odinger
equation, considered as a classical field theory, shares with Liouville
completely integrable field theories the existence of a {\sl recursion
operator} which allows for the infinitely many conserved functionals pairwise
commuting with respect to the corresponding Poisson bracket. The approach may
provide a good starting point to get a clear interpretation of Quantum
Mechanics in the general setting, provided by Stone-von Neumann theorem, of
Symplectic Mechanics. It may give new tools to solve in the general case the
inverse problem of quantum mechanics whose solution is given up to now only for
one-dimensional systems by the Gel'fand-Levitan-Marchenko formula.Comment: 11 pages, LaTex fil

### Vacuum Einstein metrics with bidimensional Killing leaves. I-Local aspects

The solutions of vacuum Einstein's field equations, for the class of
Riemannian metrics admitting a non Abelian bidimensional Lie algebra of Killing
fields, are explicitly described. They are parametrized either by solutions of
a transcendental equation (the tortoise equation), or by solutions of a linear
second order differential equation in two independent variables. Metrics,
corresponding to solutions of the tortoise equation, are characterized as those
that admit a 3-dimensional Lie algebra of Killing fields with bidimensional
leaves.Comment: LateX file, 33 pages, 2 figure

### Spin-1 gravitational waves. Theoretical and experimental aspects

Exact solutions of Einstein field equations invariant for a non-Abelian
2-dimensional Lie algebra of Killing fields are described. Physical properties
of these gravitational fields are studied, their wave character is checked by
making use of covariant criteria and the observable effects of such waves are
outlined. The possibility of detection of these waves with modern detectors,
spherical resonant antennas in particular, is sketched

### Superintegrability in the Manev Problem and its Real Form Dynamics

We report here the existence of Ermanno-Bernoulli type invariants for the
Manev model dynamics which may be viewed upon as remnants of Laplace-Runge-Lenz
vector whose conservation is characteristic of the Kepler model. If the orbits
are bounded these invariants exist only when a certain rationality condition is
met and thus we have superintegrability only on a subset of initial values. We
analyze real form dynamics of the Manev model and derive that it is always
superintegrable. We also discuss the symmetry algebras of the Manev model and
its real Hamiltonian form.Comment: 12 pages, LaTeX, In: Prof. G. Manev's Legacy in Contemporary
Astronomy, Theoretical and Gravitational Physics, V. Gerdjikov, M. Tsvetkov
(Eds), Heron Press, Sofia 2005, pp. 155-16

### Real Hamiltonian forms of Hamiltonian systems

We introduce the notion of a real form of a Hamiltonian dynamical system in
analogy with the notion of real forms for simple Lie algebras. This is done by
restricting the complexified initial dynamical system to the fixed point set of
a given involution. The resulting subspace is isomorphic (but not
symplectomorphic) to the initial phase space. Thus to each real Hamiltonian
system we are able to associate another nonequivalent (real) ones. A crucial
role in this construction is played by the assumed analyticity and the
invariance of the Hamiltonian under the involution. We show that if the initial
system is Liouville integrable, then its complexification and its real forms
will be integrable again and this provides a method of finding new integrable
systems starting from known ones. We demonstrate our construction by finding
real forms of dynamics for the Toda chain and a family of Calogero--Moser
models. For these models we also show that the involution of the complexified
phase space induces a Cartan-like involution of their Lax representations.Comment: 15 pages, No figures, EPJ-style (svjour.cls

### Recursion Operators for CBC system with reductions. Geometric theory

We discuss some recent developments of the geometric theory of the Recursion Operators (Generating Operators) for Caudrey-Beals-Coifman systems(CBC systems) on semisimple Lie algebras. As is well known the essence of this
interpretation is that the Recursion Operators could be considered as adjoint to Nijenhuis tensors on certain infinite-dimensional manifolds. In particular, we discuss
the case when there are Zp reductions of Mikhailov type

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