Development ofRegular & 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’sspin vector direction …

Search for Common Principles

Pre – 1700s 1700s

Copernicus

Brahe

Kepler

Galileo

&

calculusNewton

differential equations

&

variational calculus PDEs

rigid body dynamicsEuler

celestial mechanics

equation for inviscid flow

&

probability theory

rigid body fluid dynGauss

systems of eqns

celestial mechanics

1800’s

By 1800, motion of a celestial body could be described by:

m r&&F

( , ), , , , , ...t gravity atmospheric density attitude &F F r r

This vector eqn can be written as 3 scalar 2nd

order differential eqns.These eqns are nonlinear,Can they still beanalytically solved?

Newton’s 2nd Law:

( ) ( )Nd

dt g

Newton’s conjecture

12r1 212 21 2

12

Gm mf f

r

21f

1 212 12 12 213

12r̂

Gm mf

r f i r f

21f

1m

2m12

12r̂ r

ri

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

1 280 80earth moonm M M m

N-Body Problem

3 3 3

1

nj moon j earthearth moon

earth moon earth moon jjearth moon j moon j earth

perturbing effects

m mGm

r r r

G r rrr

1444444444442444444444443

&&

31,

nj earth

earth jj moon j earth

Gmr

rr&&

31,earth

nj moon

moon jj j moon

Gmr

rr&&

Comparison of Relative Acceleration(In G’s for an Earth Satellite)

Planet Acceleration on a satellite

Earth 0.89

Sun 0.0006

Mercury 0.00000000026

Venus 0.000000019

Jupiter 0.000000032

Saturn 0.0000000023

Uranus 0.00000000008

Idea of Perturbations

Rewrite the perturbing effects as perturbations of the dominant force

From here on the symbol will be small perturbation quantity

3 3 3

1

nj moon j earthearth moon

earth moon earth moon jjearth moon j moon j earth

DOMINANT FORCE perturbing effects

m mGm

r r r

G r rrr

1444444444442444444444443 1444444444442444444444443

&&

(0) (1) 2 (2) ... earth moonperturbing effects

F F Fr 14444444244444443&&

1830; Poisson

Look for a solution as a series

of the perturbation quantity

(0) (1) 2 (2), ... t t t tr r r r See a similarity

with Taylor’s series

(0) (1) 2 (2) (0) (1) 2 (2) ... perturbing effects

F F Fr r r 14444444244444443&& && &&

(0) (0) (1) 2 (2)

(0) (0) (0) (0) (1) 2 (2)

...

F . ..

F t t t

F t t t t

r r r

r r r r

Reduced ProblemSubstitute series solution in the original problem

To get:

(0) (1) 2 (2) (0) (0) (0) (0) (1) 2 (2)

(1) (0) (0) (0) (1) 2 (2) 2 (2)

F . ..

F . .. ...

perturbing effects

F t t t t

F t t t t F

r r r r r r r

r r r r144444444444444444444444444444424444444444444444444444444444 3

&& && &&

44

(0) (0) (0)F tr r&&

(1) (0) (0) (1) (1) (0)F . t t F tr r r r&&

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 --PoincaréWhen is the series

convergent?

How many terms in the series do we need?

Change focus from

to

(0) (1) 2 (2), ... t t t tr r r r

(n)

1

as N

N

n

n

tr

(n)

1

as 0or t

N

n

n

tr

Concept of Asymptotic Analysis

New Era: Fluid MechanicsNavier-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

20

1

Re

Du

p uDt

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)

2.00001

1.00001 2.00002

x y

x y

Singular Perturbations in the21st 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