On the Riemann-Hilbert transformations for a Galilean invariant system (original) (raw)
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.
From the equations of motion to the canonical commutation relations
The problem of whether or not the equations of motion of a quantum system determine the commutation relations was posed by E.P.Wigner in 1950. A similar problem (known as "The Inverse Problem in the Calculus of Variations") was posed in a classical setting as back as in 1887 by H.Helmoltz and has received great attention also in recent times. The aim of this paper is to discuss how these two apparently unrelated problems can actually be discussed in a somewhat unified framework. After reviewing briefly the Inverse Problem and the existence of alternative structures for classical systems, we discuss the geometric structures that are intrinsically present in Quantum Mechanics, starting from finite-level systems and then moving to a more general setting by using the Weyl-Wigner approach, showing how this approach can accomodate in an almost natural way the existence of alternative structures in Quantum Mechanics as well.
Darboux Transformations for orthogonal differential systems and differential Galois Theory
2021
Darboux developed an algebraic mechanism to construct an infinite chain of “integrable" second order differential equations as well as their solutions. After a surprisingly long time, Darboux’s results had important features in the analytic context, for instance in quantum mechanics where it provides a convenient framework for Supersymmetric Quantum Mechanics. Today, there are a lot of papers regarding the use of Darboux transformations in various contexts, not only in mathematical physics. In this paper, we develop a generalization of the Darboux transformations for tensor product constructions on linear differential equations or systems. Moreover, we provide explicit Darboux transformations for sym(SL(2,C)) systems and, as a consequence, also for so(3, CK) systems, to construct an infinite chain of integrable (in Galois sense) linear differential systems. We introduce SUSY toy models for these tensor products, giving as an illustration the analysis of some shape invariant pot...
2000
A demonstration is given that the simplest model of quantum mechanics formulated on a plane non-commutative geometry endowed with a Galilean symmetry group in which the position and linear momentum-variable commutators are first order in the dynamical variables (and thus constitute a true Lie algebra) is incompatible with the hypothesis of spacial isotropy. "Civilization advances by extending the number of important operations which we can perform without thinking about them."
Symmetries of Non-Linear Systems: Group Approach to their Quantization
We report briefly on an approach to quantum theory entirely based on symmetry grounds which improves Geometric Quantization in some respects and provides an alternative to the canonical framework. The present scheme, being typically non-perturbative, is primarily intended for non-linear systems, although needless to say that finding the basic symmetry associated with a given (quantum) physical problem is in general a difficult task, which many times nearly emulates the complexity of finding the actual (classical) solutions. Apart from some interesting examples related to the electromagnetic and gravitational particle interactions, where an algebraic version of the equivalence principle naturally arises, we attempt to the quantum description of non-linear sigma models. In particular, we present the actual quantization of the partial-trace non-linear SU(2) sigma model as a representative case of non-linear quantum field theory.
ON THE CORRESPONDENCE BETWEEN DIFFERENTIAL EQUATIONS AND SYMMETRY ALGEBRAS
Symmetry and Perturbation Theory - Proceedings of the International Conference on SPT 2007, 2008
The theory of Lie remarkable equations, i.e., differential equations characterized by their Lie point symmetries, is reviewed and applied to ordinary differential equations. In particular, we consider some relevant Lie algebras of vector fields on R k and characterize Lie remarkable equations admitted by the considered Lie algebras.
Galilean invariance in 2+1 dimensions
1995
The Galilean invariance in three dimensional space-time is considered. It appears that the Galilei group in 2+1 dimensions posses a three-parameter family of projective representations. Their physical interpretation is discussed in some detail.
This study will explicitly demonstrate by example that an unrestricted infinite and forward recursive hierarchy of differential equations must be identified as an unclosed system of equations, despite the fact that to each unknown function in the hierarchy there exists a corresponding determined equation to which it can be bijectively mapped to. As a direct consequence, its admitted set of symmetry transformations must be identified as a weaker set of indeterminate equivalence transformations. The reason is that no unique general solution can be constructed, not even in principle. Instead, infinitely many disjoint and thus independent general solution manifolds exist. This is in clear contrast to a closed system of differential equations that only allows for a single and thus unique general solution manifold, which, by definition, covers all possible particular solutions this system can admit. Herein, different first order Riccati-ODEs serve as an example, but this analysis is not restricted to them. All conclusions drawn in this study will translate to any first order or higher order ODEs as well as to any PDEs.
A New Geometric Approach to Lie Systems and Physical Applications
Acta Applicandae Mathematicae, 2002
The characterization of systems of differential equations admitting a superposition function allowing us to write the general solution in terms of any fundamental set of particular solutions is discussed. These systems are shown to be related with equations on a Lie group and with some connections in fiber bundles. We develop two methods for dealing with such systems: the generalized Wei–Norman method and the reduction method, which are very useful when particular solutions of the original problem are known. The theory is illustrated with some applications in both classical and quantum mechanics.
Linear Transformations, Canonoid Transformations and BiHamiltonian Structures
We give a characterization of linear canonoid transformations on symplectic manifolds and we use it to generate biHamiltonian structures for some mechanical systems. Utilizing this characterization we also study the behavior of the harmonic oscillator under canonoid transformations. We present a description of canonoid transformations due to E.T. Whittaker, and we show that it leads, in a natural way, to the modern, coordinate-independent definition of canonoid transformations. We also generalize canonoid transformations to Poisson manifolds by introducing Poissonoid transformations. We give examples of such transformations for Euler's equations of the rigid body (on so * (3) and so * (4)) and for an integrable case of Kirchhoff's equations for the motion of a rigid body immersed in an ideal fluid. We study the relationship between biHamiltonian structures and Poissonoid transformations for these examples. We analyze the link between Poissonoid transformations, constants of motion, and symmetries.
Noncommutative mechanics and exotic Galilean symmetry
Theoretical and Mathematical Physics, 2011
In order to derive a large set of Hamiltonian dynamical systems, but with only first order Lagrangian, we resort to the formulation in terms of Lagrange-Souriau 2-form formalism. A wide class of systems derived in different phenomenological contexts are covered. The non-commutativity of the particle position coordinates are a natural consequence. Some explicit examples are considered.