Confined dynamical systems with Carroll and Galilei symmetries (original) (raw)
Related papers
Two-dimensional dynamical systems which admit Lie and Noether symmetries
Journal of Physics A: Mathematical and Theoretical, 2011
We prove two theorems which relate the Lie point symmetries and the Noether symmetries of a dynamical system moving in a Riemannian space with the special projective group and the homothetic group of the space respectively. The theorems are applied to classify the two dimensional Newtonian dynamical systems, which admit a Lie point/Noether symmetry. Two cases are considered, the non-conservative and the conservative forces. The use of the results is demonstrated for the Kepler -Ermakov system, which in general is nonconservative and for potentials similar to the Hènon Heiles potential. Finally it is shown that in a FRW background with no matter present, the only scalar cosmological model which is integrable is the one for which 3-space is flat and the potential function of the scalar field is exponential. It is important to note that in all applications the generators of the symmetry vectors are found by reading the appropriate entry in the relevant tables.
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.
Acceleration-extended Galilean symmetries with central charges and their dynamical realizations
Physics Letters B, 2007
We add to Galilean symmetries the transformations describing constant accelerations. The corresponding extended Galilean algebra allows, in any dimension D = d + 1, the introduction of one central charge c while in D = 2 + 1 we can have three such charges: c, θ and θ . We present nonrelativistic classical mechanics models, with higher order time derivatives and show that they give dynamical realizations of our algebras. The presence of central charge c requires the acceleration square Lagrangian term. We show that the general Lagrangian with three central charges can be reinterpreted as describing an exotic planar particle coupled to a dynamical electric and a constant magnetic field. (P.C. Stichel), W.J.Zakrzewski@durham.ac.uk (W.J. Zakrzewski).
ON THE SYMMETRIES OF HAMILTONIAN SYSTEMS
International Journal of Modern Physics A, 1995
In this paper we show how the well-known local symmetries of Lagrangean systems, and in particular the diffeomorphism invariance, emerge in the Hamiltonian formulation. We show that only the constraints which are linear in the momenta generate transformations which correspond to symmetries of the corresponding Lagrangean system. The nonlinear constraints (which we have, for instance, in gravity, supergravity and string theory) rather generate the dynamics of the corresponding Lagrangean system. Only in a very special combination with "trivial" transformations proportional to the equations of motion do they lead to symmetry transformations. We reveal the importance of these special "trivial" transformations for the interconnection theorems which relate the symmetries of a system with its dynamics. We prove these theorems for general Hamiltonian systems. We apply the developed formalism to concrete physically relevant systems and in particular those which are diffeomorphism invariant. The connection between the parameters of the symmetry transformations in the Hamiltonian-and Lagrangean formalisms is found. The possible applications of our results are discussed.
Localization of the Galilean symmetry and dynamical realization of Newton–Cartan geometry
Classical and Quantum Gravity, 2015
Newtonian gravity was formulated as a geometrodynamic theory as far back in 1930s by Elie Cartan in what is named aptly as Newton Cartan space time. Though there are several approaches of realizing the algebraic structure of the Newton Cartan geometry from a contraction of the relativistic results, a dynamical (field theoretic) realization of it is lacking. In this paper we present such a realization from the localisation of the Galilean Symmetry of nonrelativistic matter field theories.
Group theoretical aspects of constants of motion and separable solutions in classical mechanics
Journal of Mathematical Analysis and Applications, 1979
The aim of this paper is to establish the group nature of all separable solutions and conserved quantities in classical mechanics by analyzing the group structure of the Hamilton-Jacobi equation. It is shown that consideration of only classical Lie point groups is insufficient. For this purpose the Lie-Bgcklund groups of tangent transformations, rigorously established by Ibragimov and Anderson, are used. It is also shown how these generalized groups induce Lie groups on Hamilton's equations. The generalization of the above results to any order partial differential equation, where the dependent variable does not appear explicitly, is obvious. In the second part of the paper we investigate a certain class of admissible operators of the time-independent Hamilton-Jacobi equation of any energy state including the zero state. It is shown that in the latter case additional symmetries may appear. Finally, some potentials of physical interest admitting higher symmetries are considered.
Generalized Galilei-Invariant Classical Mechanics
International Journal of Modern Physics A, 2005
To describe the "slow" motions of n interacting mass points, we give the most general four-dimensional (4D) noninstantaneous, nonparticle symmetric Galilei-invariant variational principle. It involves two-body invariants constructed from particle 4-positions and 4-velocities of the proper orthochronous inhomogeneous Galilei group. The resulting 4D equations of motion and multiple-time conserved quantities involve integrals over the worldlines of the other n-1 interacting particles. For a particular time-asymmetric retarded (advanced) interaction, we show the vanishing of all integrals over worldlines in the ten standard 4D multiple-time conserved quantities, thus yielding a Newtonian-like initial value problem. This interaction gives 3D noninstantaneous, nonparticle symmetric, coupled nonlinear second-order delay-differential equations of motion that involve only algebraic combinations of nonsimultaneous particle positions, velocities, and accelerations. The ten 3D noninst...
Dynamical Poincare Symmetry Realized by Field-dependent Diffeomorphisms
1998
We present several Galileo invariant Lagrangians, which are invariant against Poincare transformations defined in one higher (spatial) dimension. Thus these models, which arise in a variety of physical situations, provide a representation for a dynamical (hidden) Poincare symmetry. The action of this symmetry transformation on the dynamical variables is nonlinear, and in one case involves a peculiar field-dependent diffeomorphism. Some of our models are completely integrable, and we exhibit explicit solutions.
On the geometry of non-holonomic Lagrangian systems
Journal of Mathematical Physics, 1996
We present a geometric framework for non-holonomic Lagrangian systems in terms of distributions on the configuration manifold. If the constrained system is regular, an almost product structure on the phase space of velocities is constructed such that the constrained dynamics is obtained by projecting the free dynamics. If the constrained system is singular, we develop a constraint algorithm which is very similar to that developed by Dirac and Bergmann, and later globalized by Gotay and Nester. Special attention to the case of constrained systems given by connections is paid. In particular, we extend the results of Koiller for Č aplygin systems. An application to the so-called non-holonomic geometry is given.