14 research outputs found
On a reduction of the generalized Darboux-Halphen system
The equations for the general Darboux-Halphen system obtained as a reduction
of the self-dual Yang-Mills can be transformed to a third-order system which
resembles the classical Darboux-Halphen system with a common additive terms. It
is shown that the transformed system can be further reduced to a constrained
non-autonomous, non-homogeneous dynamical system. This dynamical system becomes
homogeneous for the classical Darboux-Halphen case, and was studied in the
context of self-dual Einstein's equations for Bianchi IX metrics. A Lax pair
and Hamiltonian for this reduced system is derived and the solutions for the
system are prescribed in terms of hypergeometric functions.Comment: Some content of this article overlaps with content of another article
written previously by two of the authors: arXiv:1606.02910. We have also
removed the error, expanding and writing it into a completely new form,
several new results having been found. 14 page
Jacobi-Maupertuis metric of Lienard type equations and Jacobi Last Multiplier
We present a construction of the Jacobi-Maupertuis (JM) principle for an
equation of the Lienard type, viz \ddot{x} + f(x)x^2 + g(x) = 0 using Jacobi's
last multiplier. The JM metric allows us to reformulate the Newtonian equation
of motion for a variable mass as a geodesic equation for a Riemannian metric.
We illustrate the procedure with examples of Painleve-Gambier XXI, the Jacobi
equation and the Henon-Heiles system