-1-
Y. Zarmi
PART II: PERTURBATION METHODS
4. How to assess the quality of an approximation
4.1 Preliminaries
We will be mainly interested in weakly perturbed harmonic motion:
d 2x
dx


2

,t; 
2   0 x   f x,


d
d
(4. 1)
Change the time variable into a dimensionless one:
t  0 
(4. 2)

f x, xÝ,t; 
0 2
(4. 3)
In terms of t, Eq. (4.1) becomes
Ý x 
xÝ
Thus, one can always assume that the unperturbed frequency is equal to unity, and forget about
the auxiliary variable, .
Eq. (4.3) can be transformed into a pair of first order coupled equations in two unknowns:
xÝ y
yÝ  x   f x, xÝ,t; 
(4. 4)
or

d x  0 1x  0
  
   

dt y  1 0y  f x,y,t; 
(4. 5)
where 0 has been replaced by 1. One can also use polar coordinates
x  cos
y  sin
(4. 6)
to obtain
rÝ  sin f
Ý 1   cos  f

r
(4. 7)
 1
(4. 8)
Now, defining a new angular variable,
Nonlinear Dynamics
-2-
we obtain
 gr,,t; 
d r  sint    f


    
   
dt  
cost   r f  hr,,t; 
(4. 9)
4.2 Error estimates [9]
With either Eqs. (4.5) or (4.9), the problem has been reduced from a slightly perturbed harmonic
oscillator (second order differential equation in one variable) to two first order coupled equations
in two variables with a small nonlinear perturbation. We shall, therefore, address a somewhat
generalized problem of the form
dx
  f x,t; 
dt
x t  0  x 0
(4.10)
x, f  D  R
n
where f satisfies the Lipschitz condition in the domain D. A unique solution x(t) of Eq. (4.10)
exists for some time interval 0≤t≤t1, due to the uniqueness and existence theorem. Assume that
a function y(t)Rn is considered as an approximation for the exact (but unknown) solution of Eq.
(4.10). We do not expect y(t) to obey Eq. (4.10) but, rather,
dy
  f y,t;   t 
dt
yt  0  y0
x 0  y 0   
(4.11)
For y to be a good approximation for x, (t;) and () must be small in some sense. In fact,
they should satisfy (at least)
t; ,  
 
00
(4.12)
Our goal now is to estimate the quality of approximation y(t) constitutes for x(t) as well as the
time span over which that quality is retained. To this end we consider the equation satisfied by
xy.
d
  f yt   t ,t;   f yt ,t;  t; 
dt
t  0   
(4.13)
The formal solution of Eq. (4.13), at least for 0≤t≤t1, is
t
t
0
0
t       s; ds    f ys  s,s;   f ys,s; ds
(4.14)
One is tempted to make a Taylor expansion in the third term in Eq. (4.14), but one can do better
using the following trick:
-3-
Y. Zarmi
1
1
0
0

2

d2
Z  1  
f x  y  i  j d  1   2 f x  y  d


x i x j
d
(Summation over dummy indices is assumed). Integration by parts gives
1
1

df   df
df
Z  1      d     0  f   1  f   0
d 0 d
d

0


f yi  f y     f y 
yi
Eq. (4.14) becomes
t
t        s; ds 
0
t
1


 hi ys,s; i s   1  gij ys  s,s;  i s j sdds


0
4.15)
0
hi 
f
xi
 2f
x i x j
gij 
For an appropriately "well behaved" f, and a "sufficiently good" approximation y(t), assume that
h M
g N
  0
(4.16)
x D
t T 
with M, N, 0 constants. With these bounds, Eq. (4.15) yields an inequality
t
t
t       0 t   M  s ds  12  N  s ds
0
2
0
t
(4.17)
t
     0 T     M  s ds  12  N  s ds
2
0
0
S t 
In a manner similar to the proof of the Gronwall lemma, Eq. (4.17) yields
dS
  M   12  N 
dt
2
dS
  dt
S M  12 N S
  S M  12 N S 

St  0  L
S M  12 N L
 exp  M t 
L M  12 N S


Nonlinear Dynamics
-4-
M
exp  Mt 
M  12 N L
S
L
1
NL
1 2 1
exp  M t 
M 2NL
(4.18)
Consequently,
t       0 T  C
C
M exp M T
,
M  N L exp M T   1
1
2
t  T  (4.19)
Thus, while all we know about the solution for Eq. (4.10) is that a unique solution exists for x(t)
for some interval in time, the properties of f(x,t;) and y(t) [Eq. (4.16)] guarantee that Eq. (4.13)
for (t) has a unique and bounded solution for 0≤t≤T/. But, given the approximation y(t), this
implies that x(t) has a solution for 0≤t≤T/, satisfying
x t   yt   t 
xt   yt   t      0 T C
(4.20)
0 t  T 
The error estimate theorem just proven is a basic tool. What is the significance of the result?
The deviation of the approximation, y(t), from the exact solution, x(t), over an extended time
interval (of O(1/)) is bounded by the combined effect of the error in the initial condition, ||()||,
and of the largest possible deviation of the integral over the error in the differential equation
(4.11), 0(T/). Thus, if the maximal deviation in the equation satisfies
0 

 
 00

then =xy constitutes a small deviation, so that y(t) is a good approximation for x(t) for
t≤O(1/). For instance, assume that y(t) is given by
N
yt     k y k
(4.21)
k 1
where yk are bounded and that 0, the maximum error in the equation, as well as (), the error
in the initial condition, satisfy
0  O N  1
   O N  1
(4.22)
(naturally, since y is given up to O(N)). Despite the fact that the error in the initial condition is
O(eN+1), and that y(t) includes terms through O(N), the overall error incurred in the
approximation is O(N) (~0/). This is a typical example for the weakness of proofs that rely on
the Gronwall lemma. Unless additional information is provided, the bounds obtained are
somewhat weak. In fact, the following is easy to see.
Theorem
-5-
Y. Zarmi
In an expansion through Nth order of an approximation y(‡) to the solution of Eq. (4.10), 0 and
||()|| are both O(N+1) (thus, the error estimate is O(N)). In addition, the N+1st order term,
yN+1, is bounded relative to yN for all t≤O(1/). Then the error estimate can be improved:
N
xt     k  O N  1 
k 0
t  O1  
(4.23)
Nonlinear Dynamics
-6-
Proof
N
xt     k  xt  
k0
N 1

k

N 1
k0
yN 1 
(4.24)
xt  
N 1

k
k 0


O  N  1 for t O 1  
(error estimate theorem)
  N 1 yN 1

 O N  1 
t  O1  

O  N 1 for t O 1  
(boundednessof y N  1 )
Thus, additional information about the next order in the expansion (here, the fact that yN+1 is
bounded relative to yN), enables us to improve the error estimate in a given order. This explains
why the elimination of secular terms (terms that grow indefinitely in time in a perturbation
expansion), a topic discussed extensively in the following chapters, is so important - their
occurrence increases the estimated errors. The issue of error estimates is studied in [9-12] for
harmonic motion with a small nonlinear perturbation (i.e., second order equations in one
unknown).
4.3 Example-Harmonic oscillator with modified frequency
Consider the equation
Ý 1  2  x  0  
xÝ
 x  cos 1  2  t 
 
xÝ  1  2 sin  1  2 t 
x 0  1, xÝ0   0 
 
(4.25)
Now, go over to two first order coupled equations
xÝ y
yÝ  x  2 x
(4.26)
Transforming to polar coordinates, x = cos , y =  sin , we obtain
Ý sin2

(4.27)
Ý 1   1  cos2 

(4.28)
In terms of the slow angular variable , where  = t + , we find
Ý sin2 t  

(4.27a)
Ý  1  cos2 t   

(4.28a)
Let
˜  t 
˜ 1, 

x˜  cos1   t, y˜  sin1   t
(4.29)
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Y. Zarmi
be an approximation to the exact solution (see Eq. (4.25)). The equation satisfied by the
approximation is
  
 t; 



 
˜
˜ sin 2 t  
˜  0  
d 

     
˜     1  cos 2 t  
dt 
˜

 
 
sin 2 t  ˜ 

t;   
 cos 2 t  ˜ 


0
    
0

t;   2 
(4.30)
   0

Consequently, based on the error estimate theorem,
 ˜
t           0 T    2T  O1
˜ 
 


(4.31)
Thus, for t=O(1/), the error estimate based on the theorem is O(1). This, as we shall see
immediately, is a poor estimate, which demonstrates the weakness of the error estimate theorem:
The bound it provides is based on minimal information. In the present example we have the best
possible information, namely, the exact solution. Therefore, we can check how good the
approximation (4.29) is directly, by expanding the solution of Eq. (4.24)

x 2  y 2  1   1  cos2 1  2 t 
(4.32)
which yields
  1  O 
for all t  0
Similarly,
1 
y 
  tan    tan 1  1  2 tan 1  2 t  
 x 
t  t  12 sin 2t  12  2 t 2cos 2t  1  12  2 1  12 cos 2 t  
t  t  O ,

(4.33)
t  O1  
Thus,
   t  O,
t  O1  
(4.34)
In summary, while the error estimate theorem yields an O(1) estimate for t ≤ O(1/), the
additional information (in this case, knowledge of the actual solution) yields a far better error
estimate.
Nonlinear Dynamics
-8-
4.4 Example-Precession of Mercury around the sun [9]
Kepler's laws constitute an excellent approximation for planetary motion. However, they are
slightly modified by general relativistic effects (the potential deviates slightly from the 1/r law).
The equations of motion for the radius, r(t), and the angle, (t), yield an elliptic orbit in the
plane. They can be reduced into a single equation relating r and : Measuring the radius in units
of the average radius,r (for Mercury it is roughly equal to 5.83109 m), one defines
r
(4.35)
u 
rt 
and obtains for the unperturbed motion
GM r 

a 


l 2 
d 2u
u a
d 2
(4.36)
Thus, the variable u obeys an oscillator's equation, with  playing the role of "time". Here a is
the dimensionless short radius, G is the gravitational constant, M is the solar mass and l is the
angular momentum per unit mass of the planet. Eq. (4.36) is solved by
u  a  b  a cos
(4.37)
where b is the dimensionless long radius.
The perturbation due to general relativistic effects modifies Eq. (4.36) into
d 2u
2
2  u  a  u
d
 3GM
 7 
 
 (4.38)
2  10

cr

The formal solution of this equation is given by the equivalent integral equation:

u  a  b  a cos     sin   u  d
2
(4.39)
0
In naive perturbation theory one substitutes the zero-order approximation for u(t) inside the
integral in Eq. (4.39). This leads to an expansion procedure that starts as follows:



u  a  b  a cos     sin    a 2  b  a cos 2   2 a b  acos  d  O 2 
2
0
(4.40)
 a  b  acos 

 a2 
1
2

b  a 2 1  cos   16 b  a  2 cos   cos 2  a b  asin
 O 2 
 v  O 2 
Where v denotes the combined contribution of the zero and first orders.
-9-
Y. Zarmi
Here we encounter an example of a "secular term". The term
 a b  a  sin
in Eq. (4.40) becomes unbounded in time. Owing to its appearance, the O() character of the first
order correction is retained only for t≤O(0). The origin of the concept is the fact that as is very
small, it takes this problematic term centuries to become sizable (siècle=century in French).
In terms of the error estimate theorem this can be looked upon as follows. If we use v() as an
approximation for u(), then the equation satisfied by v will be
d 2v
 v  a   v 2  t; 
d 2
(4.41)
0
0
t;      O  O  


where the O(0) terms in brackets are combinations (the exact structure of which is not important
for the present analysis) of constants and trigonometric functions. Thus, based on the error
estimate theorem, v() is an O(0) approximation for u() only for short times. Over times of
O(1/) it becomes a bad, O(1/), approximation - again, an indication of the weakness of the
theorem. Direct inspection of v() indicates that it may be a better approximation than the one
concluded by the error estimate theorem by one order of  (O() and O(0), respectively). This
can be made even more outstanding by choosing a better approximation. Write
w0  b  a
uaw
(4.42)
which, when substituted in Eq. (4.38) yields
d 2w
2
2  w   a  w
d
(4.43)
Now go over to polar coordinates, to obtain
w  r cos
dw
 rsin 
d
dr
  sin  a  rcos 
d
d
cos
1  
a  r cos
d
r
r  0  b  a
(4.44)
  0   0
Separating the slow from the fast  dependence in the angular variable, we find


dr
2
  sin   a  r cos   
d
d
cos   
2
 
a  r cos   

d
r
(4.45)
Nonlinear Dynamics
-10-
, the slow part of the phase, and the radius, r, vary very little as  varies over a whole 2 cycle.
Thus, we can obtain an approximate solution by considering the averages of the right hand sides
of the equations in Eq. (4.45) over a period of 2 in . Averaging is carried out with r and 
frozen. (This, essentially, is the method of averaging that will be discussed in detail later on).
dr
0
d
d
  a
d
r  r0  b  a
   0   a 
(4.46)
yielding
(4.47)
w  w0  b  a cos1   a 
We now want to insert r0 and 0 into Eqs. (4.45). Since they will not satisfy them exactly, the
error generating term (t;) will arise:
2
dr0
 0   sin   0 a  r0 cos   0    r t; 
d
cos   0 
d 0
2
  a  
a  r0 cos   0    t; 

d
r0
(4.48)
2

sin  0 a  r0 cos   0 
r  


t;      
2

  
cos   0  r0 a  r0 cos   0   a
a 2  14 r0 2 sin   0   ar0 sin 2    0   14 r0 2 sin 3   0 


  1 2 3 2

1
 a  4 r0 cos   0   a cos 2    0   4 r0 cos 3    0 
r0

(4.49)
With no error in the initial conditions, the error estimate theorem yields
rt   r0 t  

  
  0 T  
 t   0 t 
t T 
(4.50)
where
0   Const.
Thus, based on the theorem, the error is O(0) for t≤O(1/), again, weaker than what can be
shown directly. Indeed, let us define
r  r0 
R     
  0 
This definition yields
-11-
Y. Zarmi
dR d r  r0 

    
d d   0 
 sin  a  r cos   2



  cos   a  r cos    2
 



 a


r
(4.51)
a 2  14 r 2 sin     a r sin2      14 r 2 sin3   


 1
 a 2  34 r 2 cos    acos 2      14 r cos 3  

r

The residual on the r.h.s. of Eq. (4.51) is proportional to . It depends on r and on trigonometric
functions of + that average to zero over a 2 cycle in . The constant contribution, a, in the
Nonlinear Dynamics
-12-
lower component has been eliminated. The solution for R is obtained by integrating Eq. (4.51):

a 2  14 r 2 sin     a r sin2     14 r 2 sin 3  


R   1 2
d

 a  34 r 2 cos    acos 2     14 r cos 3    
r

0
(4.52)

a 2  14 r 2 sin   a r sin2   14 r 2 sin3

 d
  1
2
2
3
1
1  d
 a  4 r cos   acos 2   4 r cos 3

r
d
0
Through the error estimate theorem, r0 is shown to be an O(0) approximation for r for t≤O(1/).
Thus, using Eq. (4.45) for d/d, the integral in Eq. (4.52) becomes a product of  times a
combination of trigonometric functions. As each of the latter is bounded by 1, R will be O() (at
least for times O(1/) over which r0 and 0 do not vary appreciably). This better estimate is
achieved via the method of averaging, which singles out the problematic a term in Eq. (4.45)
and eliminates it from Eq. (4.51). Otherwise, this term would have generated a secular
contribution, a, in the approximate solution.
Exercises
4.1 Given
t

Zt;    A    B Zs;   Zs; 
2
ds
t  0, 0  «1
0
A0
B0
prove that
Zt;  
 A Be B
B   Ae B  1
0  t 1 
4.2 Consider the Duffing equation
x0 1, xÝ0  0
Ý
xÝ x   x 3
Calculate the error for t≤(1/) that accumulates in the approximations:
a.
x  cos t
b.
x  cos1  38 t
Hint: Convert the equation into first order equations for two unknowns, x and y, by defining
xÝ y
yÝ xÝ
Ý  x   x 3
-13-
Y. Zarmi
4.3 Consider the equation
Ý  cost xÝ2  x  0
xÝ
x0 1, xÝ0  0
a) Transform it into a set of two first order equations in the variables x and y ( xÝ).
b) Change the resulting equations into polar coordinates r, .
c) Change into the slow variables r,  (=t+).
d) The solution of the equations obtained in step c) is approximated by
r˜ 
1
1  38  t
˜ 0

(1) Calculate the approximations for x and y;
(2) Calculate ||()||, the norm of the deviation of the approximate solution at t=0 from the
initial condition;
(3) Calculate ||()||, the norm of the deviation between the exact equation and the equation
for rfi and fi.
(4) Based on the error estimate theorem, what is the error incurred between the approximate
solution and the exact one and for what time range?
(5) What is the error estimate and for what time range if we are given that the approximation
is a first term in a series expansion in , obeying the conditions of the theorem of Eq. (4.23)?