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
calculus &

Newton 

differential
equations


Copernicus
Brahe
Kepler
Galileo
Pre – 1700s
variational calculus & PDEs 


rigid body dynamics


Euler 

celestial
mechanics


 equation for inviscid flow 
 probability theory 


rigid body fluid & dyn 
Gauss 

systems
of
eqns


 celestial mechanics 
1700s
1800’s
By 1800, motion of a celestial body could be
described by:
Newton’s 2nd Law:
F  m&
r&
g
() 
N
d
()
dt
This vector eqn can be
written as 3 scalar 2nd
order differential eqns.
These eqns are nonlinear,
Can they still be
analytically solved?
F  F (t , r , r&
, gravity, atmospheric density, attitude, ...)
Newton’s conjecture
m2
f 21
f12  f 21 
m1
Gm1m2
r122
r12
r
iˆr  12
r12
f 21
Gm1m2
ˆ
f12  f12 ir 
r12   f 21
3
r12
Newton conjectured this force law to be consistent with Kepler’s laws,
his calculus, differential equations, and to make the Earth-Moon
dynamics (m1  M earth  80M moon  80m2) become consistent with Newton’s
corrected version of Kepler’s Laws.
N-Body Problem
&
r&
earth  
 r j earth
Gm j  3

r
j 1, moon
 j earth
n



 r j  moon 
&
r&

moon    Gm j 
3

j 1,earth
 rj  moon 
n
&
r&
earth  moon  
G  mearth  mmoon 
r
3
earth  moon
 r j  moon
r j earth 
  Gm j  3
 3


r
r
j 1
j  moon
j  earth 
 42
14444444444
4444444444
43
n
rearth  moon
perturbing effects
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
&
r&
earth  moon  
G  mearth  mmoon 
 r j  moon
r j earth 
rearth  moon   Gm j  3
 3

3

r earth  moon
rj earth 
j 1
j  moon
 r 42
14444444444
42 444444444443 14444444444
4444444444
43
DOMINANT FORCE
n
perturbing effects
(0)
(1)
2
(2)
&
r&

F


F


F
 ...
144444442 444444
43
earth  moon
From here on the symbol
quantity

perturbing effects
will be small perturbation
1830; Poisson
Look for a solution as a series
of the perturbation quantity 
r  t ,    r (0)  t    r (1)  t    2r (2)  t   ... See a similarity
with Taylor’s
series
(0)
(1)
(2)
&
r&
 &
r&
  2&
r&
 F (0)  1444444
F (1) 42
 2 444444
F (2)  ...
43
perturbing effects
F (0)  r (0)  t    r (1)  t    2r (2)  t   ... 
F (0)  r (0)  t     F(0)  r (0)  t   .  r (1)  t    2r (2)  t   ..
Reduced Problem
Substitute series solution in the original problem
(0)
(1)
(2)
&
&
&
r&
 &
r&
  2r
 F (0)  r (0)  t     F(0)  r (0)  t   .  r (1)  t    2r (2)  t   ..
  F (1)  r (0)  t     F(0)  r (0)  t   .  r (1)  t    2r (2)  t   ..   2 F (2)  ...
14444444444444444444444444444442 4444444444444444444444444444443
perturbing effects
To get:
(0)
(0)
(0)
&
&
r  F r t 
Need to solve
these reduced
problems!
(1)
&
r&
  F(0)  r (0)  t   . r (1)  t    F (1)  r (0)  t  
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
r  t ,    r (0)  t    r (1)  t    2r (2)  t   ...
convergent?
How many terms in the series do we need?
Change focus from
N
n (n)

 r  t  as N  
n 1
to
N
n (n)

 r  t  as   0 or t  
n 1
Concept of
Asymptotic
Analysis
New Era:
Fluid Mechanics
Navier-Stokes equations (1822; 1845) accounts for
flow over objects (Newton’s second law)
1 2
 Du 


p

 u

Re
 Dt 
0 
Re: ratio of
inertial and
viscous forces
Following Poincaré: (1/Re) was considered small perturbation
quantity and set to zero. The results obtained concluded
airplanes cannot fly! Perturbation methods had failed
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)
x  y  2.00001
x  1.00001 y  2.00002
Singular
Perturbations
in the
21st 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