We present a method, based on the Routh stability criterion, for finding the complex roots of polynomials with real coefficients. As in Gauss’s 1815 proof of the Fundamental Theorem of Algebra, we reduce the problem to that of finding real roots and square roots of certain associated polynomials. Unlike Gauss’s proof, our method generates these associated polynomials in an algorithmic way.

Based on an inversion of the Routh table construction, a unimodular characterization of all Hurwitz polynomials is obtained. In parameter space, the Hurwitz polynomials of degree n correspond to the positive 2-tant. The method is then used to construct classes of stable continuous-time delay-difference equations and delay differential equations of neutral type, by a suitable limiting process.

We study the dynamics of the relative motion of satellites in the gravitational field of the Earth, including the effects of the bulge of the Earth (the J2 effect). Using Routh reduction and dynamical systems ideas, a method is found that locates orbits such that a cluster of satellites remains close with very little dispersing, even with no controls.

Sushmitha Baskar, Ramanathan Baskar and Joyanto Routh, Speleothems from Sahastradhara Caves in Siwalik Himalaya, India: Possible Biogenic Inputs, 2014, Geomicrobiology Journal, (31), 8, 664-681. Geomicrobiology Journal is available online at informaworldTM: http://dx.doi.org/10.1080/01490451.2013.871087 Copyright: Taylor & Francis: STM, Behavioural Science and Public Health Titles http://www.ta...

We study reductions of the Hamiltonian flows restricted to their invariant submanifolds. As examples, we consider partial Lagrange-Routh reductions of the natural mechanical systems such as geodesic flows on compact Lie groups and n-dimensional variants of the classical Hess-Appel’rot case of a heavy rigid body motion about a fixed point. MSC: 37J35, 37J15, 53D20

Point vortex ows of a steady, two dimensional, inviscid, and incompressible uid are studied for doubly connected geometries. The Routh function is explicitly constructed, and equilibrium conngurations of vortices are found by determining critical points numerically. The numerical computations make use of an analogue of the Schwarz-Christooel transformation for doubly connected regions.