Integrable and superintegrable Hamiltonian systems in magnetic fields

McSween, Eric; Winternitz, P.

doi:10.1063/1.533283

Cited by 58 publications

(71 citation statements)

References 25 publications

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“…This property leads to a substantial reduction in the usually very complicated set of equations. Much less is known in the case of Hamiltonians with velocity dependent potentials (Dorizzi et al, 1985;McSween and Winternitz, 2000), even in the simplest case of a linear dependence. This is very unfortunate in view of several physical applications of such Hamiltonians.…”

Section: Introductionmentioning

confidence: 98%

On Integrable Hamiltonians with Velocity Dependent Potentials

Pucacco

2004

Celestial Mech Dyn Astr

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We study strongly and weakly integrable 2-dimensional Hamiltonian systems with velocity dependent potentials. We determine the set of conditions which must be satisfied in order to allow the existence of an independent second invariant polynomial in the momenta. We then investigate the linear case for which a complete solution of the problem can be obtained. We recover the classical set of linear strongly integrable systems and provide several new examples of weakly integrable systems whose equations of motion can be explicitly solved at a fixed value of the energy.

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Section: Introductionmentioning

confidence: 98%

On Integrable Hamiltonians with Velocity Dependent Potentials

Pucacco

2004

Celestial Mech Dyn Astr

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“…In fact, the concept of superintegrability has also been extended to the case of Hamiltonians describing systems in magnetic fields (velocity-dependent terms in the potential) [31]- [34] and to quantum systems which include spin interactions [35]- [38]. We note that if we consider velocity-dependent potentials then quadratic integrability no longer implies the separation of variables and the constants of the motion have both even and odd powers.…”

Section: Introductionmentioning

confidence: 99%

Higher order superintegrability of separable potentials with a new approach to the Post–Winternitz system

Rañada¹

2013

J. Phys. A: Math. Theor.

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The higher-order superintegrability of separable potentials is studied. It is proved that these potentials possess (in addition to the two quadratic integrals) a third integral of higher-order in the momenta that can be obtained as the product of powers of two particular rather simple complex functions. Some systems related with the harmonic oscillator, as the generalized SW system and the TTW system, were studied in previous papers; now a similar analysis is presented for superintegrable systems related with the Kepler problem. In this way, a new proof of the superintegrability of the Post-Winternitz system is presented and the explicit expression of the integral is obtained. Finally, the relations between the superintegrable systems with quadratic constants of the motion (separable in several different coordinate systems) and the superintegrable systems with higher-order constants of the motion are analyzed.

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“…By now, superintegrable physical systems can be regarded as one of the most intensively developed and significant fields of mathematical physics. The problem of classifying superintegrable stationary Schrödinger equations with scalar potential has been solved by Winternitz with co-workers [25] and Evans [26] for space dimensions n = 2 and n = 3 (see also [4]). They have found all potentials that allow for separability of the corresponding Schrödinger equation in more than one coordinate system.…”

Section: Separation Of Variables In the Pauli-maxwell Systemmentioning

confidence: 99%

“…Integrable Hamiltonian systems with velocity-dependent potentials have been studied for the case n = 2, i.e., in a Euclidean plane by Winternitz with co-authors [3,4]. Recently Benenti with co-authors [5] studied the problem of separation of variables in the stationary Hamilton-Jacobi equation with vector-potential from a geometrical point of view.…”

Section: Introductionmentioning

confidence: 99%

On separable Pauli equations

Zhalij

2002

Journal of Mathematical Physics

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We classify (1+3)-dimensional Pauli equations for a spin-1 2 particle interacting with the electro-magnetic field, that are solvable by the method of separation of variables. As a result, we obtain the eleven classes of vector-potentials of the electro-magnetic field A(t, x) = (A 0 (t, x), A(t, x)) providing separability of the corresponding Pauli equations. It is established, in particular, that the necessary condition for the Pauli equation to be separable into second-order matrix ordinary differential equations is its equivalence to the system of two uncoupled Schrödinger equations. In addition, the magnetic field has to be independent of spatial variables. We prove that coordinate systems and the vector-potentials of the electro-magnetic field providing the separability of the corresponding Pauli equations coincide with those for the Schrödinger equations. Furthermore, an efficient algorithm for constructing all coordinate systems providing the separability of Pauli equation with a fixed vector-potential of the electro-magnetic field is developed. Finally, we describe all vector-potentials A(t, x) that (a) provide the separability of Pauli equation, (b) satisfy vacuum Maxwell equations without currents, and (c) describe non-zero magnetic field.

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Integrable and superintegrable Hamiltonian systems in magnetic fields

Cited by 58 publications

References 25 publications

On Integrable Hamiltonians with Velocity Dependent Potentials

On Integrable Hamiltonians with Velocity Dependent Potentials

Higher order superintegrability of separable potentials with a new approach to the Post–Winternitz system

On separable Pauli equations

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