[en] In this paper, for any given positive masses we prove that the constrained variational minimization solutions of the 3-body problem with a fixed energy or a fixed mean potential energy in R³ or R² are precisely the planar equilateral triangle circular solutions found by J. Lagrange in 1772. (author)

[en] In the paper the necessity to consider physical processes at nonlinear interaction of soliton waves during processes of forced combination scattering is stated. The nonlinear equations in partial derivative describing such process are deduced. The numerical algorithm of solution is constructed, the program in the PASCAL language is written and the results are received as diagrams. (author)

[en] The authors discuss the consequences of the introduction of a quantum of time τ₀ in the formalism of non-relativistic quantum mechanics, by referring themselves in particular to the theory of the chronon as proposed by P. Caldirola. Such an interesting 'finite difference' theory, forwards -at the classical level- a solution for the motion of a particle endowed with a non-negligible charge in an external electromagnetic field, overcoming all the known difficulties met by Abraham-Lorentz's and Dirac's approaches (and even allowing a clear answer to the question whether a free falling charged particle does or not emit radiation), and -at the quantum level- yields a remarkable mass spectrum for leptons. After having briefly reviewed Caldirola's approach, the first aim of the authors is to work out, discuss, and compare one another the mew representations of Quantum Mechanics (QM) resulting from it, in the Schroedinger, Heisenberg and density-operator (Liouville-von Neumann) pictures, respectively.The authors also obtain the (retarded) finite-difference Schroedinger equation within the Feynman path integral approach, and study some of its relevant solutions. They, then, derive the time-evolution operators of this discrete theory, and use them to get the finite-difference Heisenberg equations. At last, the density matrix formalism is applied to the solution of the measurement problem in QM, with very interesting results, so as a natural explication of 'decoherence', which reveal the power of dicretized (in particular, retarded) QM

[en] The authors have introduced an extended formulation of dynamics which allows a clear distinction between temporal behavior of stable systems on one hand, chaotic and unstable systems on the other. For stable systems, the evolution is deterministic and time reversible. For chaotic and unstable systems it is probabilistic and breaks time-symmetry. The new formulation of dynamics incorporates the second law of thermodynamics and leads therefore to a great unification of physics. More precisely: (1) the authors found that unstable dynamical systems have new spectral decompositions in extended functional spaces. These spectral decompositions incorporate new characteristics of the time evolution associated with irreversibility; (2) these new spectral decompositions can be in general obtained iteratively through a well-defined algorithm; and (3) the method provides spectral decompositions of the time evolutions operator in the Liouville space of density matrices or distribution functions which cannot be implemented in the Hilbert space (wave functions) or by trajectories. In summary, they obtain an extension of classical and quantum mechanics. This extension leads to the inclusion of irreversibility into the framework of dynamics. Moreover, it also provides the theoretical basis for probabilistic predictions and control methods for complex systems, where the conventional deterministic predictions and control methods do not work

[en] We discuss continuous duality transformations and the properties of classical theories with invariant interactions between electromagnetic fields and matter. The case of scalar fields is treated in some detail. Special discrete elements of the continuous group are shown to be related to the Legendre transformation with respect to the field strengths

[en] Lie algebra contractions from o(n + 1) to e(n) are used to obtain asymptotic limits of interbases expansions between bases corresponding to different subgroup chains for the group O(n + 1). The contractions lead to interbases expansions for different subgroup chains of the Euclidean group E(n). The article is restricted to the low dimensional cases n = 2 and n = 3

[en] A semiclassical formula is calculated for the tunneling current of electrons trapped in a potential well which can tunnel into and across a wide quantum well. The tunneling current is measured at the second interface of this well and the calculations idealized an experimental situation where a strong magnetic field tilted with respect to an electric field was used. It is shown that the contribution to the tunneling current, due to trajectories which begin at the first interface and end on the second, is dominant for periodic orbits which hit both walls of the quantum well. (author)

[en] The bi-harmonic equation of flexural vibrations of elastic plates is studied by a semiclassical method which can easily be generalized for other models of wave propagation. The surface and perimeter terms of the asymptotic number of levels are derived exactly. The next constant term is also derived. A semiclassical approximation of the quantization condition is obtained. A Berry-Tabor formula and a Gutzwiller trace formula are deduced for the integrable and chaotic cases respectively. From 600 eigenvalues of a clamped stadium plate obtained by a specially developed numerical algorithm, the trace formula is assessed, looking at its Fourier transform compared with the membrane case. (author)

No abstract available

[en] We study bifurcations of localized stationary solitons of the externally driven, damped nonlinear Schroedinger equation iΨ_t + Ψ_xx + 2|Ψ|²Ψ=-iγΨ-h e^iΩt, in the region of large γ (γ>1/2). For each pair of h and γ, there are two coexisting solitons, Ψ₊ and Ψ_-. As the driver's strength h increases for the fixed γ, the Ψ₊ soliton merges with the flat background while the Ψ_- forms a stationary collective state with two 'psi-pluses': Ψ_- → Ψ₍₊-₊₎. We obtain other stationary solutions and identify them as multisoliton complexes Ψ₍₊₊₎, Ψ_(--), Ψ_(-+), Ψ_(---), Ψ_(-+-) etc. The corresponding intersoliton separations are compared to predictions of a variational approximation