perturbation analysis

7/30/2019 Perturbation Analysis

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Lecture 2: Time-dependent pertubation theoryand secular terms

Learning outcomes: By the end of this lecture, the students should be able to

recognise secular terms as a cause for nonuniform perturbative expansions,

dene a uniform asymptotic expansion,

improve perturbative expansions by means of renormalisation or the method of

strained parameters,

derive approximate solutions to differential equations with oscillating solutions using

variation of parameters and temporal averaging.

I. EXAMPLE: WEAKLY ANHARMONIC OSCILLATOR

Ordinary differential equations arise in virtually all areas of physics. In most cases,

these equations cannot be solved exactly, but perturbation theory can be used to obtain an

approximate solution. As an example, consider the motion of a classical mass m at position

x moves under the inuence of a weakly nonlinear spring with anharmonic potential

V (x) = 12 kx2 1 +

2x2a20

where k the Hookean a spring constant, a0 is a characteristic length scale of the of the

anharmonic potential and is a small dimensionless parameter describing the strength of

the latter. The equation of motion for this system is

mx2 = V (x) = kx 1 + x2

a20.

We choose the particle to be initially at some position x(0)= x0 and at rest x(0)=0. This

simple nonlinear ordinary differential equation is known as the Duffing equation. To solve

it, we rst bring it into a simpler form by introducing the dimensionless variable u = x/a 0

and dening 0 = k/m :u + 20 u +

20 u

3 = 0 with u(0) = a, u(0) = 0 .

where a = x0 /a 0 .

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We try to nd a perturbative solution as an asymptotic series in powers of the small

parameter :

u(t) =

n =0

un (t)n .

Substituting this Poincare-type expansion into the Duffing equation and equating powers of

, we nd the leading-order equation

u0 + 20 u0 = 0 with u0 (0) = a, u0 (0) = 0

and its rst-order correction

u1 + 20 u1 = 20 u

30 with u1 (0) = 0 , u1 (0) = 0 .

The zero-order equation is that of an undamped, free harmonic oscillator with solution

u0 (t) = a cos(0 t).

We substitute it into the rst-order equation and make use of the identity cos 3 = 34 cos +14 cos(3), which follows from taking the real part of e

3i =(cos +isin )3 . The result is an

equation for a driven harmonic oscillator,

u1 (t) + 20 u1 (t) = 20 a34

[3cos(0 t) + cos(30 t)] with u0 (0) = 0 , u0 (0) = 0 .

The solution to this inhomogeneous second-order differential equation can be obtained via

the method of variation of parameters. It reads

u1 (t) = a3

32[cos(0 t) cos(30 t)]

3a3

80 t sin(0 t) .

Up to rst-order perturbation theory, the solution to the Duffing equation is hence

u(t) = a cos(0 t) a3 132 [cos(0 t) cos(30 t)] 38 0 t sin(0 t) + O(

2 ) .

The weak nonlinearity of the spring seems to have to effects on the oscillator: It leads

to the generation of higher harmonics cos(3 0 t) and it give rise to a linearly growing term,

which is due to the resonant driving term 34 20 a3 cos(0 t) on the right-hand side of the

rst-order equation. In particular, the solution is unbounded in the limit t . Is this

a genuine feature or an indication that our simple perturbative approach has failed? The

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latter is the case, since energy conservation prohibits unbounded solutions. Multiplying the

Duffing equation by x and integrating over t, we nd that the total energy

E = 12 mx2 + V (x)

is a conserved quantity, i.e.,

u2 + 20 u2 +

220 u

4 = 20 a2 +

220 a

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remains constant at all times, showing that u(t) must be bounded. An unbounded, unphys-

ical term arising in time-dependent perturbation theory is called secular term . Before

solving the problem posed by secular terms, let us quantify it by introducing some general

notions.

II. DEFINITION: UNIFORM ASYMPTOTIC EXPANSION

The perturbative solution for the Duffing equation obtained in the previous part is an

example for a non-uniform asymptotic expansion. It can be dened by introducing the big-O

and little-o notations for functions of two variables

Denition. The notation f (x,) = O(g(x,)) for 0 means that there are nite real

constants M and 0 such that | f (x,)| M |g(x,)| 0 there is a

constant 0 such that |f (x)| |g(x)|


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Remarks: Our perturbative expansion of the Duffing equation is nonuniform due to the

unbounded domain for t. Alternatively, perturbative solutions to differential equations can

become nonuniform if the perturbative parameter appears as a factor in front of the highest

derivative or if the differential equation exhibits a singularity.

III. RENORMALISATION

Returning to our example, we combine the secular term in our solution with the zero-

order contribution to write a[cos(0 t) 38 a20 t sin(0 t)]. Can this term be simplied in any

way by using the addition theorem cos( x + y)=cos x cos y sin x sin y? Indeed it can: If we

write 38 a20 tsin 38 a

20 t and 1cos 38 a

20 t , we nd

a cos(0 t) 38 a20 t sin(0 t) = a cos 38 a

20 t cos(0 t) sin 38 a

20 t sin(0 t) + O(2 )

= a cos(t) + O(2 )

with = 0 1 + 38 a2 . This modication, which is correct up to the considered order

O(2 ), will lead to a uniform perturbative expansion. To achieve this, we have introduced a

parameter into the solution, which is given by an expansion in terms of the perturbative

parameter . This is the essence of the renormalisation technique for removing secular terms:

Method (Renormalisation) . Suppose that we are trying to solve a differential equation

u + 20 u2 = f (u, u).

with u = u(t) for some small parameter where a Poincare-type perturbative expansion

u(t)

n =0

un (t)n

is nonuniform. A uniform expansion can then be obtained by introducing a new variable

s = t with

n =0

nn ,

expanding

u(s/ )

n =0

un (s/ )n

in terms of and determining n such that secular terms are cancelled.

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Example: Returning to our example of a weakly anharmonic oscillator, we introduce a new

variable s = t with = 0 + 1+ O(2 ) such that

0 t =0

s = 1 10 s + O(2 ) .

Substituting this into our Poincare-type perturbative solution and expanding in terms of ,

we nd

u(s) = a cos(s) + a10ssin s a3 132 [coss cos(3s)]

38 s sin(s) + O(

2 ) .

The secular term is cancelled if 1 = 38 a2 0 . Resubstituting s = t, we then have the uniform

asymptotic expansion

u(t) = a cos(t) a3 132 [cos(t) cos(3t)]+ O(2 ) .

with = 0 1+ 38 a2 .

Remarks: The renormalisation technique can be used quite generally whenever the solution

from a direct perturbative expansion contains a parameter that might be changed by the

perturbation. Examples could be frequency, wave number, wave speed, energy level etc.

Renormalisation is commonly used in quantum eld theories.

IV. METHOD OF STRAINED PARAMETERS

Following the renormalisation technique, we have rst obtained a perturbative solution of

the simple Poincare type and then removed secular terms by applying a change of variables to

this solution. Alternatively, one can perform a change of variables from the very beginning.

This is the central idea of the LindstedtPoincare method:

Method (LindstedtPoincare method) . Suppose that we are trying to solve a differential

equation

u + 20 u2 = f (u, u).

with u = u(t) for some small parameter where a Poincare-type perturbative expansion

u(t)

n =0

un (t)n

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is nonuniform. A uniform expansion can then be obtained by introducing a new variable

s = t with

n =0

nn

and solving the transformed equation

2 u (s) + 20 u2 (s) = f [u(s), u(s)]

by means of a Poincare-type expansion:

u(s)

n =0

un (s)n .

Here, both and u(s) are expanded simultaneously and the n have to be determined such

that secular terms are avoided.

Example: In terms of the transformed variable s = t, the Duffing equation assumes the

form

2 u (s) + 20 u2 (s)u + 20 u

3 (s) = 0 with u(0) = a, u (0) = 0 .

Inserting the expansions u(s)= u0 (s)+ u1 (s)+ O(2 ) and = 0 + 1+ O(2 ), and collecting

equal powers of , we nd the leading-order equation

u 0 (s) + u0 (s) = 0 with u0 (0) = a, u

0 (0) = 0

and the rst-order correction

u 1 (s) + u1 (s) = u30 (s) 2

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u 0 (s) with u1 (0) = 0 , u

1 (0) = 0 .

With the solution

u0 (s) = a cos(s)

to the zero-order equation, the rst-order equation reads

u 1 (s) + u1 (s) = a3

4[3 cos(s) + cos(3 s)] + 2a

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cos(s) with u0 (0) = 0 , u 0 (0) = 0 ,

recall that cos 3 = 34 cos +14 cos(3). The resonant driving terms proportional to cos( s) can

be cancelled by choosing 1 = 38 a2 0 . The resulting rst-order equation

u 1 (s) + u1 (s) = a3

4cos(3s) with u0 (0) = 0 , u0 (0) = 0 ,

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is easily solved by an ansatz u1 (s)= c cos(3s), leading to

u1 (s) =a3

32cos(3s) .

Combining the zero- and rst-order solutions and resubstituting s = t, we again nd theuniform expansion

u(t) = a cos(t) a3 132 [cos(t) cos(3t)]+ O(2 ) .

Remarks: At rst glance, the renormalisation technique seems to be the simpler option,

because involves a variable substitution applied to the end result of a simple Poincare-type

expansion. The LindstedtPoicare method requires a variable transformation at a much

earlier stage. This seeming complication has the advantage that we remove resonant drivingterms as a cause for secular terms, tackling the problem at an earlier stage with the benet

of a much simplied differential equation.

Note that standard perturbation theory of quantum mechanics is an example of the

LindstedtPoicare method where both the solutions to a differential equation (the eigen-

states) and a parameter (the eigenenergy) are perturbed. As the parameter perturbations

are not independent, but must be suitable chosen to obtain a uniform expansion, the method

is also known as the method of strained parameters . As a more powerful alternative,one can use a more general coordinate transformation t = n =0 n (s)

n where the straining

functions n (s) are determined such that a uniform expansion is obtained. This is called the

Lighthill technique or method of strained coordinates.

V. VARIATION OF PARAMETERS AND TEMPORAL AVERAGING

So far, our approximative methods have been improvements of the straightforwardPoincare-type perturbative expansions which have led to systematic asymptotic series in

terms of . We now turn to an approximate method that aims at preserving the physical

behaviour of solutions and which is particularly useful when these solutions are expected to

contain both fast oscillations and slowly evolving components.

We again consider a differential equation of the type

u + 20 u2 = f (u, u).

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This time, we start from the general solution to the unperturbed equation ( = 0), which

reads

u0 (t) = a cos(0 t + 0 )

where the parameters amplitude a and initial phase 0 are determined by the initial con-ditions. We now assume that an approximate solution to the perturbed equation has the

same form, but with slowly varying functions a = a(t) and 0 = 0 (t):

u(t) = a(t)cos[0 t + 0 (t)] .

This approach is know as variation of parameters and is commonly used to solve inhomoge-

neous linear differential equations. As we have introduced two functions a(t) and 0 (t) to

parametrise one solution, we are free to impose a constraint. We demand that u(t) has thesame value as in the unperturbed case,

u(t) = 0 a(t) sin[0 t + 0 (t)] .

Evaluating this derivative explicitly,

u(t) = a(t)cos[0 t + 0 (t)] 0 a(t) sin[0 t + 0 (t)] 0 (t)a(t) sin[0 t + 0 (t)] ,

our constraint takes the explicit form

a cos 0 a sin = 0

with (t)= 0 t + 0 (t). Calculating

u(t) = 0 a(t) sin[0 t + 0 (t)] 20 a(t)cos[0 t + 0 (t)] 0 0 (t)a(t)cos[0 t + 0 (t)] ,

the original differential equation reads

0 a sin + 0 0 a cos = f (a cos , 0 a sin ) .

Solving this equation, together with the constraint for a and 0 , we nd

a = 1

0sin f (a cos , 0 a sin ) ,

0 = 1

a0cos f (a cos , 0 a sin ) .

To leading order in , we may use our zero-order solution on the right-hand side of these

equations, 0 t. This means that a(t) and 0 (t) do not vary much during one free

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oscillation period T =2 / 0 and we may replace the right-hand sides by their time average

over one period:

sin f (a cos , 0 a sin ) 1T

T

0dt sin(0 t)f [a cos(0 t), 0 a sin(0 t)]

= 12

2

0d sin f (a cos , 0 a sin ) ,

cos f (a cos , 0 a sin ) 1

2 2

0d cos f (a cos , 0 a sin ) .

with this approximation, we obtain:

Method (Variation of parameters and temporal averaging) . An approximate solution to the

differential equation

u + 2

0 u2

= f (u, u)with small is given by

u(t) = 0 a(t) sin[0 t + 0 (t)] + O()

where the slowly varying parameters a(t) and 0 (t) are solutions to

a = 1

20 2

0d sin f (a cos , 0 a sin ) ,

0 =

12a 0

2

0 d cos f (a cos , 0 a sin ) .This method is also known as the Kyrilov-Bogoliubov technique .

Example: For the weakly anharmonic oscillator, the equations for amplitude and phase

read

a =a3 02

2

0d sin cos3 = 0 ,

0 =a2 0

2

2

0

d cos4 = 38 a2 0 .

These equations are solved by a(t)= a0 and (t)= 38 a2 0t, so that

u(t) = a0 cos 0 1 + 38 a2 t + O() .

We note that the amplitude and phase behaviour are faithfully represented by our solution,

while the higher-harmonic has been lost due to our temporal averaging procedure.

Remarks: The WKB-approximation used for solving quantum scattering problems is based

on very similar ideas, but applied to a spatial rather then temporal coordinate.

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FURTHER READING

Perturbation Methods , A. H. Nayfeh, Chaps. 3 and 5, pp. 56-109, 159-227 (Wiley-VCH,

Weinheim, 2004).

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perturbation analysis

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