Development of Regular & Singular Perturbation Methods Anshu Narang-Siddarth

Field of Celestial Mechanics Concerned with producing ephemeris data All stars have theoretically the same center Least square circle fits (leaving radius and center as free parameters to be estimated) provide an estimate of the Earth’s spin vector direction …

Search for Common Principles Pre – 1700s 1700s

1800’s By 1800, motion of a celestial body could be described by: This vector eqn can be written as 3 scalar 2 nd order differential eqns. These eqns are nonlinear, Can they still be analytically solved? Newton’s 2 nd Law:

Newton’s conjecture Newton conjectured this force law to be consistent with Kepler’s laws, his calculus, differential equations, and to make the Earth-Moon dynamics ( ) become consistent with Newton’s corrected version of Kepler’s Laws.

N-Body Problem

Comparison of Relative Acceleration (In G’s for an Earth Satellite) PlanetAcceleration on a satellite Earth0.89 Sun Mercury Venus Jupiter Saturn Uranus

Idea of Perturbations Rewrite the perturbing effects as perturbations of the dominant force From here on the symbol will be small perturbation quantity

1830; Poisson Look for a solution as a series of the perturbation quantity See a similarity with Taylor’s series

Reduced Problem Substitute series solution in the original problem To get: Need to solve these reduced problems! In 1887: King of Sweden announced a prize for anyone who could find the solution to the problem. Announcement said: Given a system of arbitrarily many mass points that attract each according to Newton's law, under the assumption that no two points ever collide, try to find a representation of the coordinates of each point as a series in a variable that is some known function of time and for all of whose values the series converges uniformly.

Foundation of Perturbation Methods --Poincaré When is the series convergent? How many terms in the series do we need? Change focus from to Concept of Asymptotic Analysis

New Era: Fluid Mechanics Navier-Stokes equations (1822; 1845) accounts for flow over objects (Newton’s second law) Following Poincaré: (1/Re) was considered small perturbation quantity and set to zero. The results obtained concluded airplanes cannot fly! Perturbation methods had failed Re: ratio of inertial and viscous forces

Singular Perturbation Methods; 1904

Singular Perturbation Problem simple straightforward series approximation does not give an accurate solution throughout the domain Leads to different approximations being valid in different domains Singular Perturbation Methods: aim to find useful, approximate solutions by solving either Finding an approximate solution of set of equations An approximate set of equations and/or

Role in Numerical Analysis Solving a linear system (C. Lanczos)

Singular Perturbations in the 21 st Century

References Robert O’ Malley “Development in Singular Perturbations”, 2013 K. G Lamb, “Course Notes for AMATH 732”, 2010 John. L. Junkins “Two Body Fundamentals”: Lecture notes, 2012 John D. Anderson Jr, “Ludwig Prandtl’s Boundary Layer”, American Physical Society, 2005 Roger Bate, Donald Mueller and Jerry White “Fundamentals of Astrodynamics”, Dover Publications