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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
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