Lagrangian formalism for nonlinear second-order Riccati systems: One-dimensional integrability and two-dimensional superintegrability

Abstract

The existence of a Lagrangian description for the second-order Riccati equation is analyzed and the results are applied to the study of two different nonlinear systems both related with the generalized Riccati equation. The Lagrangians are nonnatural and the forces are not derivable from a potential. The constant value E of a preserved energy function can be used as an appropriate parameter for characterizing the behaviour of the solutions of these two systems. In the second part the existence of two-dimensional versions endowed with superintegrability is proved. The explicit expressions of the additional integrals are obtained in both cases. Finally it is proved that the orbits of the second system, that represents a nonlinear oscillator, can be considered as nonlinear Lissajous figures

Solutions of some second order ODEs by the extended Prelle-Singer method and symmetries

“…which coincides with the results in [5]. In fact it has been shown by Cariñena et al that this first integral plays the role of the Hamiltonian for (5.18).…”

From Classical to Discrete Gravity through Exponential Non-Standard Lagrangians in General Relativity

“…The solution x2 = π/2 defines a plane through the origin of space intersecting the unit sphere in a great circle. The 1st solution x1 = λ corresponds for a motion around the great circle with unit speed whereas the 2nd solution 1 1 x    corresponds for a motion around the great circle with a velocity 1   . This may suggest that the geodesic dynamics in the ENSL approach are not characterized just by a universal speed, but possibly by other universal velocities, which are proportional to 1   .…”

Two-dimensional isochronous nonstandard Hamiltonian systems

“…Certain nonstandard Lagrangian and Hamiltonian dynamical systems [1][2][3] encompass very interesting classes of nonlinear oscillators and admit fascinating dynamical properties [4][5][6] such as isochronous oscillations, linearization, nonlocal transformations and so on [7][8][9][10][11][12][13]. In particular, the Liénard class of oscillators appear in the study of a wide range of fields such as seismology [14], biological regulatory systems [15], in the study of a self graviting stellar gas cloud [16], optoelectronics, fluid mechanics [17].…”