We use a formula derived almost seventy years ago by Madhav Rao connecting the Jacobi Last Multiplier of a second-order ordinary differential equation and its Lagrangian and determine the Lagrangians of the Painlevé equations. Indeed this method yields the Lagrangians of many of the equations of the Painlevé-Gambier classification. Using the standard Legendre transformation we deduce the corresponding Hamiltonian functions. While such Hamiltonians are generally of non-standard form, they are found to be constants of motion. On the other hand for second-order equations of the Liénard class we employ a novel transformation to deduce their corresponding Lagrangians. We illustrate some particular cases and determine the conserved quantity (first integral) resulting from the associated Noetherian symmetry. Finally we consider a few systems of secondorder ordinary differential equations and deduce their Lagrangians by exploiting again the relation between the Jacobi Last Multiplier and the Lagrangian.
A metriplectic (or Leibniz) structure on a smooth manifold is a pair of skew-symmetric Poisson tensor P and symmetric metric tensor G. The dynamical system defined by the metriplectic structure can be expressed in terms of Leibniz bracket. This structure is used to model the geometry of the dissipative systems. The dynamics of purely dissipative systems are defined by the geometry induced on a phase space via a metric tensor. The notion of Leibniz brackets is extendable to infinite-dimensional spaces. We study metriplectic structure compatible with the Euler-Poincaré framework of the Burgers and WhithamBurgers equations. This means metriplectic structure can be constructed via Euler-Poincaré formalism. We also study the Euler-Poincaré frame work of the Holm-Staley equation, and this exhibits different type of metriplectic structure. Finally we study the 2D Navier-Stokes using metriplectic techniques.
Abstract. We derive the 2-component Camassa-Holm equation and corresponding N = 1 super generalization as geodesic flows with respect to the H 1 metric on the extended BottVirasoro and superconformal groups, respectively.
In this paper, the role of Jacobi’s last multiplier in mechanical systems with a position-dependent mass is unveiled. In particular, we map the Liénard II equation to a position-dependent mass system. The quantization of the Liénard II equation is then carried out using the point canonical transformation method together with the Von Roos ordering technique. Finally, we show how their eigenfunctions and eigenspectrum can be obtained in terms of associated Laguerre and exceptional Laguerre functions.
The Jacobi metric derived from the line element by one of the authors is shown to reduce to the standard formulation in the non-relativistic approximation. We obtain the Jacobi metric for various stationary metrics. Finally, the Jacobi-Maupertuis metric is formulated for time-dependent metrics by including the Eisenhart-Duval lift, known as the Jacobi-Eisenhart metric.
The construction of nonstandard Lagrangians and Hamiltonian structures for Liénard equations satisfying Chiellini condition is presented and their connection to time-dependent Hamiltonian formalism is shown. We also show that such nonstandard Lagrangians are deformations of simpler standard Lagrangians. We also exhibit their connection with contact Hamiltonian mechanics.
Karasu-Kalkani et al (2008 J. Math. Phys. 49 073516) recently derived a new sixth-order wave equation KdV6, which was shown by Kupershmidt (2008 Phys. Lett. 372A 2634) to have an infinite commuting hierarchy with a common infinite set of conserved densities. Incidentally, this equation was written for the first time by Calogero and is included in the book by Calogero and Degasperis (1982 Lecture Notes in Computer Science vol 144 (Amsterdam: North-Holland) p 516). In this paper, we give a geometric insight into the KdV6 equation. Using Kirillov's theory of coadjoint representation of the Virasoro algebra, we show how to obtain a large class of KdV6-type equations equivalent to the original equation. Using a semidirect product extension of the Virasoro algebra, , we propose the nonholonomic deformation of the Ito equation. We also show that the Adler–Kostant–Symes scheme provides a geometrical method for constructing nonholonomic deformed integrable systems. Applying the Adler–Kostant–Symes scheme to loop algebra, we construct a new nonholonomic deformation of the coupled KdV equation.
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