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  • 1
    Electronic Resource
    Electronic Resource
    Dynamical symmetries, bi-Hamiltonian structures, and superintegrable n=2 systems (2000)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 41 (2000), S. 2121-2134 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The theory of dynamical but non-Cartan (or non-Noether) symmetries and the existence of bi-Hamiltonian structures is studied using the symplectic formalism approach. The results are applied to the study of superintegrable systems. It is shown that certain families of n=2 superintegrable systems related with the harmonic oscillator (as, e.g., the so-called Smorodinsky–Winternitz system) are bi-Hamiltonian systems endowed with dynamical symmetries of non-Cartan class. © 2000 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.533230
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  • 2
    Electronic Resource
    Electronic Resource
    Superintegrability of the Calogero–Moser system: Constants of motion, master symmetries, and time-dependent symmetries (1999)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 40 (1999), S. 236-247 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The classical n-dimensional Calogero–Moser system is a maximally superintegrable system endowed with a rich variety of symmetries and constants of motion. In the first part of the article some properties related with the existence of several families of constants of motion are analyzed. In the second part, the master symmetries and the time-dependent symmetries of this system are studied. © 1999 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532770
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  • 3
    Electronic Resource
    Electronic Resource
    Superintegrable n=2 systems, quadratic constants of motion, and potentials of Drach (1997)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 38 (1997), S. 4165-4178 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The properties of superintegrable systems in two degrees of freedom, possessing three independent globally defined constants of motion, are studied using as an approach, the existence of hidden symmetries and the generalized Noether's theorem. The potentials are obtained as solution of a system of two partial differential equations. First the case of standard Lagrangians is studied and then the method is applied to the case of Lagrangians with a pseudo-Euclidean kinetic term. Finally, the results are related with other approaches and with a family of potentials admitting a second integral of motion cubic in the velocities obtained by Drach. © 1997 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532089
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  • 4
    Electronic Resource
    Electronic Resource
    Extended tangent bundle formalism for time-dependent Lagrangian systems (1991)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 32 (1991), S. 500-505 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The Lagrangian formalism for time-dependent systems is developed using vector fields defined in the extended tangent bundle T(Q×R). The definition of the time-independent extended Lagrangian function L associated to a time-dependent Lagrangian L is given and then the techniques of symplectic mechanics are used to prove some properties of the equations determining the Euler–Lagrange vector field. The relations between the time-independent and the time-dependent Helmholtz conditions are analyzed from a geometric perspective and, finally, the Noether theorem for the extended Lagrangian L is considered.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529442
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  • 5
    Electronic Resource
    Electronic Resource
    On harmonic oscillators on the two-dimensional sphere S2 and the hyperbolic plane H2 (2002)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 43 (2002), S. 431-451 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Two Harmonic Oscillators (isotropic and nonisotropic 2:1) are studied on the two-dimensional sphere S2 and the hyperbolic plane H2. Both systems are integrable and super-integrable with constants of motion quadratic in the momenta. These properties are shown to derive from a complex factorization for the constants of motion, which holds for arbitrary values of the curvature κ, and the dynamics of the Euclidean harmonic 1:1 and 2:1 oscillators is directly recovered for κ=0. The harmonic oscillators on either the standard unit sphere (radius R=1) or the unit Lobachewski plane ("radius" R=1) appear as the particular values of the κ-dependent potentials for the values κ=1 and κ=−1. Finally a particular potential is proposed for representing the general spherical (hyperbolic) n:1 anisotropic harmonic oscillator on a two-dimensional manifold of constant curvature. © 2002 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.1423402
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  • 6
    Electronic Resource
    Electronic Resource
    Superintegrable systems on the two-dimensional sphere S2 and the hyperbolic plane H2 (1999)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 40 (1999), S. 5026-5057 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The existence of superintegrable systems with n=2 degrees of freedom possessing three independent globally defined constants of motion which are quadratic in the velocities is studied on the two-dimensional sphere S2 and on the hyperbolic plane H2. The approach used is based on enforcing the conditions for the existence of two independent integrals (further than the energy). This is done in a way which allows us to discuss at once the cases of the sphere S2 and the hyperbolical plane H2, by considering the curvature κ as a parameter. Different superintegrable potentials are obtained as the solutions of certain systems of two κ-dependent second order partial differential equations. The Euclidean results are directly recovered for κ=0, and the superintegrable potentials on either the standard unit sphere (radius R=1) or the unit Lobachewski plane ("radius" R=1) appear as the particular values of the κ-dependent superintegrable potentials for the values κ=1 and κ=−1. Some new superintegrable potentials are found, both on S2 and H2. The correspondence between superintegrable systems in spaces of zero and nonzero curvature is discussed. © 1999 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.533014
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  • 7
    Electronic Resource
    Electronic Resource
    Integrable three-particle systems, hidden symmetries and deformations of the Calogero–Moser system (1995)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 36 (1995), S. 3541-3558 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The properties of some three-particle systems are studied using as an approach the theory of generalized or hidden symmetries. It is proven that the existence of a family that possesses, in addition to the Energy function, two nonlinear constants of motion in involution. This family of integrable systems is obtained by considering deformations of the Lagrangian of the Calogero–Moser system. Finally, the generalization of the results to the general case of n particles is discussed. © 1995 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.530980
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  • 8
    Electronic Resource
    Electronic Resource
    An integrable three-particle system (1994)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 1219-1232 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A study of the existence of some integrable systems with nonlinear constants of motion is presented using the approach of the theory of generalized (dynamical or hidden) symmetries. Two Lagrangians are considered, both obtained by modifying the Toda Lagrangian. First a two-particle system is studied and then the results are generalized to a three-particle system. It is shown that in both cases the Lagrangians possess nonlinear constants of motion in involution and, thus, they are integrable.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.530585
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  • 9
    Electronic Resource
    Electronic Resource
    Canonoid transformations from a geometric perspective (1988)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 29 (1988), S. 2181-2186 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The concept of canonoid transformation for a locally Hamiltonian vector field is introduced, and its relation with the existence of non-Noether constants of the motion is shown from a geometrical viewpoint. The equations determining generating functions for such canonoid transformations are obtained and applications to some particular problems given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528146
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  • 10
    Electronic Resource
    Electronic Resource
    Time-dependent Lagrangian systems: A geometric approach to the theory of systems with constraints (1994)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 748-758 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A geometric approach to the theory of time-dependent regular Lagrangian systems with constraints is presented using the framework of the exact contact manifold (TQ×R,aitch-thetaL). The main subject of the article concerns the properties of the time-dependent nonholonomic constraints and the geometric approach to the method of the Lagrange multipliers. It is shown that every constraint determines a contact one-form, a vertical vector field, and a nonvertical vector field. The explicit form of the vector field representing the constrained dynamics is obtained and, finally, the properties of all these one-forms and vector fields are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.530664
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