Dynamical symmetries and the Ermakov invariant (original) (raw)

2001, Physics Letters A

https://doi.org/10.1016/S0375-9601(00)00835-5

checkGet notified about relevant papers

checkSave papers to use in your research

checkJoin the discussion with peers

checkTrack your impact

Abstract

Ermakov systems possessing Noether point symmetry are identified among the Ermakov systems that derive from a Lagrangian formalism and, the Ermakov invariant is shown to result from an associated symmetry of dynamical character. The Ermakov invariant and the associated Noether invariant, are sufficient to reduce these systems to quadratures.

On the Hamiltonian structure of Ermakov systems

Journal of Physics A: Mathematical and General, 1996

A canonical Hamiltonian formalism is derived for a class of Ermakov systems specified by several different frequency functions. This class of systems comprises all known cases of Hamiltonian Ermakov systems and can always be reduced to quadratures. The Hamiltonian structure is explored to find exact solutions for the Calogero system and for a noncentral potential with dynamic symmetry. Some generalizations of these systems possessing exact solutions are also identified and solved.

The symmetries of Lagrangian systems and constants of motion

This work was intended as an attempt to pose a better definition for Lagrangian systems and their symmetries. Symmetries and infinitesimal symmetries of these sys- tems are defined and then, we proceed with the study of constants of motion in these systems and find their relations to the symmetries of the system. All these definitions are done in the cases time-independent and time-dependent Lagrangians.

Generalized Hamiltonian structures for Ermakov systems

Journal of Physics A: Mathematical and General, 2002

We construct Poisson structures for Ermakov systems, using the Ermakov invariant as the Hamiltonian. Two classes of Poisson structures are obtained, one of them degenerate, in which case we derive the Casimir functions. In some situations, the existence of Casimir functions can give rise to superintegrable Ermakov systems. Finally, we characterize the cases where linearization of the equations of motion is possible.

Ermakov-Lewis dynamic invariants with some applications

Arxiv preprint math-ph/0002005, 2000

Contents: Introduction(3).The method of Ermakov(4).The method of Milne(7). Pinney's result(8).Lewis' results(8). The interpretation of Eliezer and Gray(14). The connection of the Ermakov invariant with N\"other's theorem(17). Possible generalizations of Ermakov's method(20). Geometrical angles and phases in Ermakov's problem(22). Application to the minisuperspace cosmology(26). Application to physical optics(42).Conclusions(47). Appendix A: Calculation of the integral of I(48). Appendix B: Calculation of <\hat{H}>

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.

Symmetries in Non-Linear Mechanics

In this paper we exploit the use of symmetries of a physical system so as to characterize the corresponding solution manifold by means of Noether invariants. This constitutes a necessary preliminary step towards the correct quantisation in non-linear cases, where the success of Canonical Quantisation is not guaranteed in general. To achieve this task "point symmetries" of the Lagrangian are generally not enough, and the notion of contact transformations is in order. The use of the Poincaré-Cartan form permits finding both the symplectic structure on the solution manifold, through the Hamilton-Jacobi transformation, and the required symmetries, realized as Hamiltonian vector fields, associated with functions on the solution manifold (thus constituting an inverse of the Noether Theorem), lifted back to the evolution space through the inverse of this Hamilton-Jacobi mapping. In this framework, solutions and symmetries are somehow identified and this correspondence is also kept at a perturbative level. We present simple non-trivial examples of this interplay between symmetries and solutions pointing out the usefulness of this mechanism in approaching the corresponding quantisation.

POINT SYMMETRIES OF LAGRANGIANS

We give an elementary exposition of a method to obtain the infinitesimal point symmetries of Lagrangians. Besides, we exhibit the Lanczos approach to Noether’s theorem to construct the first integral associated with each symmetry.

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.

Lie and Noether point symmetries for a class of nonautonomous dynamical systems

Journal of Mathematical Physics, 2017

We prove two general theorems that determine the Lie and the Noether point symmetries for the equations of motion of a dynamical system that moves in a general Riemannian space under the action of a time dependent potential W(t,x)=ω(t)V(x). We apply the theorems to the case of a time dependent central potential and the harmonic oscillator and determine all Lie and Noether point symmetries. Finally we prove that these theorems also apply to the case of a dynamical system with linear dumping and study two examples.

Translational symmetries of quadratic Lagrangians

International Journal of Modern Physics A, 2021

In this paper, we show that a quadratic Lagrangian, with no constraints, containing ordinary time derivatives up to the order [Formula: see text] of [Formula: see text] dynamical variables, has [Formula: see text] symmetries consisting in the translation of the variables with solutions of the equations of motion. We construct explicitly the generators of these transformations and prove that they satisfy the Heisenberg algebra. We also analyze other specific cases which are not included in our previous statement: the Klein–Gordon Lagrangian, [Formula: see text] Fermi oscillators and the Dirac Lagrangian. In the first case, the system is described by an equation involving partial derivatives, the second case is described by Grassmann variables and the third shows both features. Furthermore, the Fermi oscillator and the Dirac field Lagrangians lead to second class constraints. We prove that also in these last two cases there are translational symmetries and we construct the algebra of ...

Loading Preview

Sorry, preview is currently unavailable. You can download the paper by clicking the button above.

References (27)

R. S. Kaushal, Int. J. Theor. Phys. 37 (1998) 1793.
W. Sarlet and F. Cantrijn, SIAM Rev. 23 (1981) 467.
F. Haas and J. Goedert, J. Phys. A: Math. Gen. 32 (1999) 6837.
V. P. Ermakov, Univ. Isv. Kiev 20 (1880) 1.
J. R. Ray and J. L. Reid, J. Math. Phys. 20 (1979) 2054.
J. R. Ray, Prog. Theor. Phys. 65 (1981) 877.
J. R. Ray, J. Phys. A: Math. Gen. 13 (1980) 1969.
R. S. Kaushal and H. J. Korsch, J. Math. Phys. 22 (1981) 1904.
H. Kanasugi and H. Okada, Prog. Theor. Phys. 93 (1995) 949.
S. S. Simic, J. Phys. A: Math. Gen. 33 (2000) 5435.
V. I. Arnold, Mathematical methods of classical mechanics. New York: Springer-Verlag, 1978.
J. Goedert, Phys. Lett. A 136 (1989) 391.
Cerveró and J. D. Lejarreta J. M. , Phys. Lett. A 156 (1991) 201.
F. Haas and J. Goedert, J. Phys. A: Math. Gen. 29 (1996) 4083.
J. L. Reid and J. R. Ray, J. Math. Phys. 21 (1980) 1583.
K. Saermark, Phys. Lett. A 90 (1982) 5.
J. Goedert and F. Haas, Phys. Lett. A 239 (1998) 348.
C. Athorne, J. Phys. A: Math. Gen. 31 (1998) 6605.
C. Athorne, C. Rogers, U. Ramgulam and A. Osbaldestin, Phys. Lett. A 143 (1990) 207.
F. Haas and J. Goedert, J. Phys. A: Math. Gen. 32 (1999) 2835.
C. Athorne, J. Phys. A: Math. Gen. 24 (1991) L1385.
A. Munier, J. R. Burgan, M. R. Feix and E. Fijalkow, Astron. Astrophys. 94 (1981) 373.
P. G. L. Leach, Phys. Lett. A 158 (1991) 102.
B. Grammaticos and B. Dorizzi, J. Math. Phys. 25 (1984) 2194.
A. K. Dhara and S. V. Lawande, J. Math. Phys. 27 (1986) 1331.
P. G. L. Leach, H. R. Lewis and W. Sarlet, J. Math. Phys. 25 (1984) 486.
C. Athorne, J. Phys. A: Math. Gen. 24 (1991) 945.

On the Lie symmetries of a class of generalized Ermakov systems

Physics Letters A, 1998

The symmetry analysis of Ermakov systems is extended to the generalized case where the frequency depends on the dynamical variables besides time. In this extended framework, a whole class of nonlinearly coupled oscillators are viewed as Hamiltonian Ermakov system and exactly solved in closed form.

Three classes of Ermakov systems and nonlocal symmetries

Eprint Arxiv 0904 2605, 2009

Ermakov systems have attracted enormous treatments in recent times particularly in symmetry analysis. In this paper we consider three classes of the Ermakov systems by using a simple algebraic reduction process with imposed conditions on the magnitude of the angular momentum of each system class to obtain new generalized symmetries. We note that this imposed condition transforms the Kepler-Ermakov systems to the generalized Ermakov systems.

On the Ermakov systems and nonlocal symmetries

2009

Symmetry analysis of the Ermakov systems has attracted enormous treatments in recent times. In this paper we consider three classes of the Ermakov systems and obtain their nonlocal symmetries using a simple algebraic reduction process. We observed that these nonlocal symmetries are new to the literature.

Lie point symmetries for reduced Ermakov systems

Physics Letters A, 2004

Reduced Ermakov systems are defined as Ermakov systems restricted to the level surfaces of the Ermakov invariant. The condition for Lie point symmetries for reduced Ermakov systems is solved yielding four infinite families of systems. It is shown that SL(2, R) always is a group of point symmetries for the reduced Ermakov systems. The theory is applied to a model example and to the equations of motion of an ion under a generalized Paul trap.

A summary on symmetries and conserved quantities of autonomous Hamiltonian systems

Journal of Geometric Mechanics, 2020

A complete geometric classification of symmetries of autonomous Hamiltonian mechanical systems is established; explaining how to obtain their associated conserved quantities in all cases. In particular, first we review well-known results and properties about the symmetries of the Hamiltonian and of the symplectic form and then some new kinds of non-symplectic symmetries and their conserved quantities are introduced and studied.

Notes on Lagrangean and Hamiltonian Symmetries

1995

The Hamiltonization of local symmetries of the form δq A = ǫ a Ra A (q,q) or δq A =ǫ a Ra A (q,q) for arbitrary Lagrangean model L(q A ,q A) is considered. We show as the initial symmetries are transformed in the transition from L to first order action, and then to the Hamiltonian action SH = dτ (pAq A − H0 − v α Φα), where Φα are the all (first and second class) primary constraints. An exact formulae for local symmetries of SH in terms of the initial generators Ra A and all primary constraints Φα are obtained.

Recent Applications of the Theory of Lie Systems in Ermakov Systems?

2008

We review some recent results of the theory of Lie systems in order to apply such results to study Ermakov systems. The fundamental properties of Ermakov systems, i.e. their superposition rules, the Lewis-Ermakov invariants, etc., are found from this new perspective. We also obtain new results, such as a new superposition rule for the Pinney equation in terms of three solutions of a related Riccati equation.

Ermakov–Lewis invariants and Reid systems

Physics Letters A, 2014

Reid's mth-order generalized Ermakov systems of nonlinear coupling constant α are equivalent to an integrable Emden-Fowler equation. We obtain a general formula for the Ermakov-Lewis invariant from this perspective and use the Bessel solutions of a parabolic frequency parametric oscillator for an illustrative calculation in this case (m = 2). Next, for higher-order Reid systems (m ≥ 3) we obtain a closed formula for the invariant and also discuss the parametric solutions of these systems through the integration of the Emden-Fowler equation.

On Symmetry and Conserved Quantities in Classical Mechanics

The Western Ontario Series in Philosophy of Science, 2000

This paper expounds the relations between continuous symmetries and conserved quantities, i.e. Noether's "first theorem", in both the Lagrangian and Hamiltonian frameworks for classical mechanics. This illustrates one of mechanics' grand themes: exploiting a symmetry so as to reduce the number of variables needed to treat a problem.

Cited by

Amplitude and phase representation of quantum invariants for the time-dependent harmonic oscillator

Physical Review A, 2003

The correspondence between classical and quantum invariants is established. The Ermakov Lewis quantum invariant of the time dependent harmonic oscillator is translated from the coordinate and momentum operators into amplitude and phase operators. In doing so, Turski's phase operator as well as Susskind-Glogower operators are generalized to the time dependent harmonic oscillator case. A quantum derivation of the Manley-Rowe relations is shown as an example.

Geometrization of Lie and Noether symmetries with applications in Cosmology

Journal of Physics: Conference Series, 2013

We derive the Lie and the Noether conditions for the equations of motion of a dynamical system in a n−dimensional Riemannian space. We solve these conditions in the sense that we express the symmetry generating vectors in terms of the special projective and the homothetic vectors of the space. Therefore the Lie and the Noether symmetries for these equations are geometric symmetries or, equivalently, the geometry of the space is modulating the motion of dynamical systems in that space. We give two theorems which contain all the necessary conditions which allow one to determine the Lie and the Noether symmetries of a specific dynamical system in a given Riemannian space. We apply the theorems to various interesting situations covering Newtonian 2d and 3d systems as well as dynamical systems in cosmology.

Generalized Hamiltonian structures for Ermakov systems

Journal of Physics A: Mathematical and General, 2002

Anisotropic Bose-Einstein condensates and completely integrable dynamical systems

Physical Review A, 2002

A Gaussian ansatz for the wave function of two-dimensional harmonically trapped anisotropic Bose-Einstein condensates is shown to lead, via a variational procedure, to a coupled system of two second-order, nonlinear ordinary differential equations. This dynamical system is shown to be in the general class of Ermakov systems. Complete integrability of the resulting Ermakov system is proven. Using the exact solution, collapse of the condensate is analyzed in detail. Timedependence of the trapping potential is allowed.

Geometrization of Lie and Noether symmetries and applications

International Journal of Modern Physics: Conference Series, 2015

We derive the Lie and the Noether conditions for the equations of motion of a dynamical system in an n-dimensional Riemannian space. We solve these conditions in the sense that we express the symmetry generating vectors in terms of the collineation vectors of the space. More specifically we give two theorems which contain all the necessary conditions which allow one to determine the Lie and the Noether point symmetries of a specific dynamical system evolving in a given Riemannian space in terms of the projective and the homothetic collineations of space. We apply these theorems to various interesting situations in Newtonian Physics.

Dynamical symmetries and the Ermakov invariant (original) (raw)

Sign up for access to the world's latest research

Abstract

Related papers

References (27)

Related papers

Related topics

Cited by