On the Riemann-Hilbert transformations for a Galilean invariant system (original) (raw)
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On the Galilean-invariant equations for particles with arbitrary spin
1976
In our preceding paper [1] the equations of motion which are invariant under the Galilei group G have been obtained starting from the assumption that the Hamiltonian of a nonrelativistic particle has positive eigenvalues and negative ones. These nonrelativistic equations as well as the relativistic Dirac equation lead to the spin-orbit and to the Darwin interactions by the standard replacement p µ → π µ = p µ − eA µ . Previously it was generally accepted the hypothesis that the spin-orbit and the Darwin interactions are truly relativistic effects .
Lie–Hamilton systems on the plane: Properties, classification and applications
Journal of Differential Equations, 2015
We study Lie-Hamilton systems on the plane, i.e. systems of first-order differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of planar Hamiltonian vector fields with respect to a Poisson structure. We start with the local classification of finite-dimensional real Lie algebras of vector fields on the plane obtained in [A. González-López, N. Kamran and P.J. Olver, Proc. London Math. Soc. 64, 339 (1992)] and we interpret their results as a local classification of Lie systems. Moreover, by determining which of these real Lie algebras consist of Hamiltonian vector fields with respect to a Poisson structure, we provide the complete local classification of Lie-Hamilton systems on the plane. We present and study through our results new Lie-Hamilton systems of interest which are used to investigate relevant non-autonomous differential equations, e.g. we get explicit local diffeomorphisms between such systems. In particular, the Milne-Pinney, second-order Kummer-Schwarz, complex Riccati and Buchdahl equations as well as some Lotka-Volterra and nonlinear biomathematical models are analysed from this Lie-Hamilton approach.
2021
We comprehensively study admissible transformations between normal linear systems of second-order ordinary differential equations with an arbitrary number of dependent variables under several appropriate gauges of the arbitrary elements parameterizing these systems. For each class from the constructed chain of nested gauged classes of such systems, we single out its singular subclass, which appears to consist of systems being similar to the elementary (free particle) system whereas the regular subclass is the complement of the singular one. This allows us to exhaustively describe the equivalence groupoids of the above classes as well as of their singular and regular subclasses. Applying various algebraic techniques, we establish principal properties of Lie symmetries of the systems under consideration and outline ways for completely classifying these symmetries. In particular, we compute the sharp lower and upper bounds for the dimensions of the maximal Lie invariance algebras posse...
Lie systems: theory, generalisations, and applications
2011
Lie systems form a class of systems of first-order ordinary differential equations whose general solutions can be described in terms of certain finite families of particular solutions and a set of constants, by means of a particular type of mapping: the so-called superposition rule. Apart from this fundamental property, Lie systems enjoy many other geometrical features and they appear in multiple branches of Mathematics and Physics, which strongly motivates their study. These facts, together with the authors' recent findings in the theory of Lie systems, led to the redaction of this essay, which aims to describe such new achievements within a self-contained guide to the whole theory of Lie systems, their generalisations, and applications.
Galilei-invariant equations for massive fields
Journal of Physics A: Mathematical and Theoretical, 2009
Galilei-invariant equations for massive fields with various spins have been found and classified. They have been derived directly, i.e., by using requirement of the Galilei invariance and various facts on representations of the Galilei group deduced in the paper written by de Montigny M, Niederle J and Nikitin A G, J. Phys. A 39, 1-21, 2006. A completed list of non-equivalent Galileiinvariant wave equations for vector and scalar fields is presented. It shows two things. First that the collection of such equations is very broad and describes many physically consistent systems. In particular it is possible to describe spin-orbit and Darwin couplings in frames of Galilei-invariant approach. Second, these Galilei-invariant equations can be obtained either via contraction of known relativistic equations or via contractions of quite new relativistic wave equations.
Classical and Quantum Gravity, 1996
The possible external couplings of an extended non-relativistic classical system are characterized by gauging its maximal dynamical symmetry group at the center-of-mass. The Galilean one-time and two-times harmonic oscillators are exploited as models. The following remarkable results are then obtained: 1) a peculiar form of interaction of the system as a whole with the external gauge fields; 2) a modification of the dynamical part of the symmetry transformations, which is needed to take into account the alteration of the dynamics itself, induced by the gauge fields. In particular, the Yang-Mills fields associated to the internal rotations have the effect of modifying the time derivative of the internal variables in a scheme of minimal coupling (introduction of an internal covariant derivative); 3) given their dynamical effect, the Yang-Mills fields associated to the internal rotations apparently define a sort of Galilean spin connection, while the Yang-Mills fields associated to the quadrupole momentum and to the internal energy have the effect of introducing a sort of dynamically induced internal metric in the relative space.
Transformation theory for anti-self-dual equations
Publications of the Research Institute for Mathematical Sciences
An infinite-dimensional Lie algebra acting on solutions to the anti-self-dual equations on a four-dimensional Euclidean space is derived by means of the Riemann-Hilbert problem. Three types of Backlund transformations are considered in the framework of the Riemann-Hilbert problem. § 1. Introduction In recent years remarkable progress has been made in studies on nonlinear field equations, and two possible approaches have been proposed. The first one is the symmetry theory which includes soliton theory and Backlund transformations. The second one has arisen from algebraic geometry. In investigations of the symmetries, certain two-dimensional field equations have been found to admit infinite-dimensional Lie algebras: Kinnersley and Chitre [11] revealed that the Kac-Moody algebra §1(2,1?) (X)JR[C, C" 1 ] acts on solutions to the stationary axially symmetric gravitational field equations. This Lie algebra originates in certain symmetries of the field equations. Hauser and Ernst [10] exponentiated all of these infinitesimal actions and constructed the transformation theory by means of the Riemann-Hilbert problem. Following their method, recently Ueno [15] has shown that the Kac-Moody algebras 8u(ri)®R[£, C" 1 ], and go(rc) 01? [C, C" 1 ] flfl °n solutions to SU(n), SO (n) chiral fields, respectively.