We present a direct approach to the construction of Lagrangians for a large class of one-dimensional dynamical systems with a simple dependence (monomial or polynomial) on the velocity. We rederive and generalize some recent results and find Lagrangian formulations which seem to be new. Some of the considered systems (e.g., motions with the friction proportional to the velocity and to the square of the velocity) admit infinite families of different Lagrangian formulations.
A time dependent generalization of the Ginzburg -Landau Lagrangian is proposed. It contains two terms determining the time dependence and the four arbitrary scalar functions. Relevant equations, which coincide with equations following from the suitable Hamiltonian, are derived by a standard variational technique. These equations determine the energy conservation law and admit twofold time dependence which leads either to first or to second order time derivatives in Ginzburg -Landau equations. By introducing the gauge invariant potentials and choosing the gauge which differs slightly from the classical Lorentz one, the theory simplifies significantly. The results gained are discussed and compared to some earlier propositions. The presented approach, when reduced to a static one, is found to be in perfect agreement with that reported recently by the Koláček group. This indicates indirectly, that the equation with the first order time derivative seems to be more justified.
We formulate and discuss integrable analogue of the sine-Gordon equation on arbitrary time scales. This unification contains the sineGordon equation, discrete sine-Gordon equation and the Hirota equation (doubly discrete sine-Gordon equation) as special cases. We present the Lax pair, check compatibility conditions and construct the Darboux-Bäcklund transformation. Finally, we obtain a soliton solution on arbitrary time scale. The solution is expressed by the so called Cayley exponential function.
The classical and quantum model of high spin particles within the manifestly covariant framework. The internal (spin) degrees of freedom are described by two C(3, 1) Clifford algebra spinors. The covariant quantization leads to PCT invariant spectrum of particles with spin dependent masses. The quantum model contains elementary particles and the cluster states generating infinite degeneracy of the mass spectrum.
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