The central force problem in n dimensions
ABSTRACT :
The motion of a particle moving under the influence of a central force is a fundamental paradigm in dynamics. The problem of planetary motion, specifically the derivation of Keplers laws, motivated Newtons monumental work, effectively signalling the start of modern physics. Today, the central force problem stands as a basic lesson in dynamics. In this talk, we shall discuss the classical central force problem in a general number (n) of spatial dimensions as an instructive illustration of important aspects of Hamiltonian dynamics such as integrability, invariance, super-integrability and dynamical symmetry. This is also in line with the usefulness of treating the number of dimensions as a variable parameter in physical problems. The dependence of various quantities on the spatial dimensionality leads to a proper perspective of the problems concerned. We shall consider the orbital angular momentum in n dimensions, and discuss in some detail the role it plays in the integrability of the central force problem. We then consider an important superintegrable case, the Kepler problem, in n dimensions. The existence of an additional vector constant of the motion over and above the angular momentum makes this problem maximally superintegrable. We shall discuss the significance of these constants of the motion as generators of the dynamical symmetry group of the Hamiltonian. This group, which is larger than the kinematical symmetry group for a general central force, is identified for closed as well as open orbits in the Kepler problem. The dynamical symmetry group in the case of the n-dimensional isotropic oscillator will also be identified and discussed briefly. The role of constants of the motion as generators of the Lie algebras (realised via Poisson brackets) of the relevant symmetry groups will be brought out.
ABOUT THE SPEAKER:
Professor V. Balakrishnan earned his B.Sc. (Honours) and M. Sc.degrees in physics from the University of Delhi, and his doctoral degree in theoretical high energy physics from Brandeis University, USA in 1970. After a decade at TIFR (Mumbai) and RRC (Kalpakkam), he joined IIT Madras as Professor of physics in December 1980, and retired as Professor Emeritus in December 2013. He is currently Adjunct Professor in the Dept. of Physics at IIT Madras. His research interests have spanned many areas over the years, including particle physics, many-body theory, condensed matter physics, stochastic processes, quantum dynamics, nonlinear dynamics and chaos. He has done significant work on anelastic relaxation, extreme value distributions and recurrence statistics in dynamical systems, non-Markovian random walks and anomalous diffusion. He has authored four books including a comprehensive book on Mathematical Physics. Over four decades, he has taught a wide variety of extremely popular courses at the undergraduate, masters and doctoral levels. He has received high accolades for his teaching at IIT Madras and for his lectures at numerous courses sponsored by the UGC, DST (SERC) and ICTP (Trieste, Italy). He has given seven video courses, under the auspices of the NPTEL programme of the MHRD. These have received very high acclaim. He is a Fellow of the Indian Academy of Sciences. He continues to be academically active, giving lectures, writing books, and publishing research papers.