One-parameter Symmetry Groups Research Articles

Determination of the symmetry groups of radiation hydrodynamics equations and the compatible equations of state and opacities

The purpose of this study was to find all the symmetry groups of the radiation hydrodynamics equations with no a priori assumptions on the equations of state (EOS) and opacities. As shown in earlier works, the application of the Lie group technique to such a system of equations leads to invariance conditions in the form of linear differential equations, which, up until now, were only partially solved. In this paper, using the same technique and under the same assumptions, but with a simpler formulation, we show that these equations can be entirely solved analytically. This result enables us to list all the one-parameter groups that may be symmetry groups of the system. To be actually so, they must be associated with suitable EOS and opacities whose general expressions are also given. The interesting point is that some of them can be chosen so as to fit realistic data for EOS and opacities. Using this property, we propose a method to design low-scale experiments to simulate radiative processes, which would involve too much energy to have experimented with at their full scale. In addition, we derive the reduced systems associated with the one-parameter symmetry groups found. We show that some classical self-similar problems can be extended to more general EOS and opacities, and we treat in detail the self-similar expansion of a semi-infinite medium submitted to an internal source of energy.

Absence of Nontrivial Symmetries to the Heat Equation in Goursat Groups of Dimension at Least 4

Using the extension method, we study the one-parameter symmetry groups of the heat equation ∂tp = Δp, where $$\Delta=X_1^2+X_2^2$$ is the sub-Laplacian constructed by a Goursat distribution span({X1, X2}) in ℝn, where the vector fields X1 and X2 satisfy the commutation relations [X1, Xj] = Xj+1 (where Xn+1 = 0) and [Xj, Xk] = 0 for j ≥ 1 and k ≥ 1. We show that there are no such groups for n ≥ 4 (with exception of the linear transformations of solutions which are admitted by every linear equation).

The zoo of solitons for curve shortening in Note **On the occasion of Richard Hamilton's nth birthday.

We provide a detailed description and classification of solutions to the curve shortening equation in that are invariant under one-parameter symmetry groups of the equation. We pay particular attention to geometric properties of the curves and asymptotic properties of their ends. We find generalized helices, and a connection with curve shortening on the unit sphere . Expanding–rotating solitons turn out to be asymptotic to generalized logarithmic spirals and the rotating–shrinking solitons are most complicated in terms of asymptotic properties of their ends. We find that almost all of the rotating–shrinking solitons are asymptotic to circles.Many of the curve shortening solitons we classify are either space curves, or evolving space curves. In addition to the figures in this paper, we have prepared a number of animations of the solitons, which can be viewed in the online supplementary data (stacks.iop.org/Non/26/1189/mmedia).

On Lagrangian and Hamiltonian systems with homogeneous trajectories

Motivated by various results on homogeneous geodesics of Riemannian spaces, we study homogeneous trajectories, i.e. trajectories which are orbits of a one-parameter symmetry group, of Lagrangian and Hamiltonian systems. We present criteria under which an orbit of a one-parameter subgroup of a symmetry group G is a solution of the Euler–Lagrange or Hamiltonian equations. In particular, we generalize the ‘geodesic lemma’ known in Riemannian geometry to Lagrangian and Hamiltonian systems. We present results on the existence of homogeneous trajectories of Lagrangian systems. We study Hamiltonian and Lagrangian geodesic orbit (g.o.) spaces, i.e. homogeneous spaces G/H with G-invariant Lagrangian or Hamiltonian functions on which every solution of the equations of motion is homogeneous. We show that the Hamiltonian g.o. spaces are related to the functions that are invariant under the coadjoint action of G. Riemannian g.o. spaces thus correspond to special Ad*(G)-invariant functions. An Ad*(G)-invariant function that is related to a g.o. space also serves as a potential for the mapping called the ‘geodesic graph’. As an illustration we discuss the Riemannian g.o. metrics on SU(3)/SU(2).

Symmetries and exact solutions of discrete nonconservative systems

Based on the property of the discrete model entirely inheriting the symmetry of the continuous system, we present a method to construct exact solutions with continuous groups of transformations in discrete nonconservative systems. The Noether’s identity of the discrete nonconservative system is obtained. The symmetric discrete Lagrangian and symmetric discrete nonconservative forces are defined for the system. Generalized quasi-extremal equations of discrete nonconservative systems are presented. Discrete conserved quantities are derived with symmetries associated with the continuous system. We have also found that the existence of the one-parameter symmetry group provides a reduction to a conserved quantity; but the existence of a two-parameter symmetry group makes it possible to obtain an exact solution for a discrete nonconservative system. Several examples are discussed to illustrate these results.

Conservation laws for equations of mixed elliptic-hyperbolic and degenerate types

For partial differential equations of mixed elliptic-hyperbolic and degenerate types which are the Euler-Lagrange equations for an associated Lagrangian, invariance with respect to changes in independent and dependent variables is investigated, as are results in the classification of continuous one-parameter symmetry groups. For the variational and divergence symmetries, conservation laws are derived via the method of multipliers. The conservation laws resulting from anisotropic dilations are applied to prove uniqueness theorems for linear and nonlinear problems, and the invariance under dilations of the linear part is used to derive critical exponent phenomena and to obtain localized energy estimates for supercritical problems.

Hypersurfaces in P n (C) with one-parameter symmetry groups

Hypersurfaces in Pn(C) with a oneparameter symmetry group can be characterized by failure of versality of a certain unfolding. We suppose the hypersurface has isolated singularities. If the symmetr...

Cyclic symmetries of one-dimensional non-linear wave equations

Symmetries corresponding to point transformations of four common classes of one-dimensional non-linear wave equations: utt=[F(u)ux]x;utt=F(ux)uxx;utt=F(uxx) and uxt=F(u) are here classified. It is found that in addition to the well-known continuous one-parameter symmetry groups arising from infinitesimal Lie point transformations there also exist discrete symmetries forming cyclic groups of finite order.

Divergence-free vector-field and reduction

By the Lie symmetry group, the reduction for divergence-free vector-fields (DFVs) is studied, and the following results are found. A n-dimensional DFV can be locally reduced to a ( n − 1)-dimensional DFV if it admits a one-parameter symmetry group that is spatial and divergenceless. More generally, a n-dimensional DFV admitting a r-parameter, spatial, divergenceless Abelian (commutable) symmetry group can be locally reduced to a ( n − r)-dimensional DFV.

The structure of infinitesimal symmetries of geodesic flows on a two-dimensional torus

The problem of geodesic lines on a two-dimensional torus is considered. One-parameter symmetry groups in the four-dimensional phase space that are generated by vector fields commuting with the initial Hamiltonian vector field are studied. As proved by Kozlov and Bolotin, a geodesic flow on a two-dimensional torus admitting a non-trivial infinitesimal symmetry of degree n has a many-valued integral that is a polynomial of degree at most n in the momentum variables. Kozlov and the present author proved earlier that first- and second-order infinitesimal symmetries are related to hidden cyclic coordinates and separated variables. In the present paper the structure of polynomial infinitesimal symmetries of degree at most four is described under the assumption that these symmetry fields are non-Hamiltonian.

Invariant properties of Timoshenko beam equations

The Lie groups theory is applied to study the invariant properties of the Timoshenko beam equations. A group classification with respect to the external load is performed and the full symmetry groups are determined. Then, the most general form of the solutions to the Timoshenko beam equations, invariant under an arbitrary one-parameter symmetry group is obtained. As an example, invariant solutions, representing travelling waves along the beam are discussed. Next, for each case of external loads, specified during the group classification procedure, the variational and divergence symmetries of the governing equations are determined. Finally, eight conservation laws for the solutions to the Timoshenko beam equations are derived via the Noether theorem, each being valid for a particular choice of the external load.

On the integrability and perturbation of three-dimensional fluid flows with symmetry

The purpose of this paper is to develop analytical methods for studyingparticle paths in a class of three-dimensional incompressible fluid flows. In this paper we study three-dimensionalvolume preserving vector fields that are invariant under the action of a one-parameter symmetry group whose infinitesimal generator is autonomous and volume-preserving. We show that there exists a coordinate system in which the vector field assumes a simple form. In particular, the evolution of two of the coordinates is governed by a time-dependent, one-degree-of-freedom Hamiltonian system with the evolution of the remaining coordinate being governed by a first-order differential equation that depends only on the other two coordinates and time. The new coordinates depend only on the symmetry group of the vector field. Therefore they arefield-independent. The coordinate transformation is constructive. If the vector field is time-independent, then it possesses an integral of motion. Moreover, we show that the system can be further reduced toaction-angle-angle coordinates. These are analogous to the familiar action-angle variables from Hamiltonian mechanics and are quite useful for perturbative studies of the class of systems we consider. In fact, we show how our coordinate transformation puts us in a position to apply recent extensions of the Kolmogorov-Arnold-Moser (KAM) theorem for three-dimensional, volume-preserving maps as well as three-dimensional versions of Melnikov's method. We discuss the integrability of the class of flows considered, and draw an analogy with Clebsch variables in fluid mechanics.

Scale invariance and the Bohr–Wilson–Sommerfeld (BWS) quantization for power law one-dimensional potential wells

The classical periods of motion τ(E) are computed for a particle under the influence of a potential well of the form U(x)=α‖x‖ν, with both ν and α positive real constants. Assuming the reflection convention at the origin, these results can be extended to the cases where both ν and α are negative real constants. Also, the scale invariance exhibited by these potentials is analyzed using dimensional arguments directly on the classical equations of motion as well as the more powerful Lie method, appropriate for studying one-parameter symmetry groups of differential equations. The action variables J(E) are obtained from τ(E) and the Bohr–Wilson–Sommerfeld (BWS) quantization rule for the energy spectrum of all the above potentials is reobtained. An interpretation of the results is given in the light of semiclassical arguments.

On continuity of mean values of unbounded observables

An unbounded observable whose mean value at some state changes discontinuously under the action of a *-weakly continuous one-parameter symmetry group is exhibited. Also, a sufficient condition for differentiability of a mean value of a not necessarily bounded observable that evolves under the action of a *-weakly continuous one-parameter automorphism group is given.

On the hierarchy of the Landau-Lifshitz equation

The Landau-Lifshitz equation (LL-equation) is shown to have infinitely many commuting one-parameter symmetry groups and constants of motion in involution. The infinitesimal generators of these one-parameter symmetry groups define the hierarchy of the LL-equation. We give explicit formulas for these generators and for the conserved densities. Furthermore, infinitely many time-dependent symmetry groups and conservation laws are constructed. All results are obtained via a simple direct Lie algebra approach and without making any use of the quantum inverse scattering method based on the ingenious results of R.J. Baxter.

On the hamiltonian structures of the Korteweg-de Vries and sine-Gordon theories

We construct a Miura-like transformation which transforms symplectic forms, Poisson brackets and conservation laws of KdV into those of SG, and then, transforms the hamiltonian flow of KdV into a one-parameter symmetry group of SG.

Lie theory and the wave equation in space–time. 3. Semisubgroup coordinates

We classify and study those coordinate systems which permit R separation of variables for the wave equation in four-dimensional space–time and such that at least one of the variables corresponds to a one-parameter symmetry group of the wave equation. We discuss over 100 such systems and relate them to orbits of triplets of commuting operators in the enveloping algebra of the conformal group SO(4,2).

Lie theory and separation of variables. 8. Semisubgroup coordinates for Ψt t − Δ2Ψ = 0

We classify and study all coordinate systems which permit R-separation of variables for the wave equation in three space–time variables and such that at least one of the variables corresponds to a one-parameter symmetry group of the wave equation. We discuss 33 such systems and relate them to orbits of commuting operators in the enveloping algebra of the conformal group SO (3,2).