Integration of the Classical Action for the Quartic Oscillator in 1 + 1 Dimensions

(1)

Integration of the Classical Action for the Quartic

Oscillator in 1 + 1 Dimensions

Robert L. Anderson

Department of Physics and Astronomy, University of Georgia, Athens, USA Email: [email protected]

Received July 11, 2013; revised August 11, 2013; accepted August 18, 2013

Copyright © 2013 Robert L. Anderson. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

In this paper, we derive an explicit form in terms of end-point data in space-time for the classical action, i.e. integration

of the Lagrangian along an extremal, for the nonlinear quartic oscillator evaluated on extremals.

Keywords: Action; Integral

1. Introduction

The action





 

extremal

d

, ; , , d

d b

a

t qo a a b b qo

t

S t y t y L y t y t t

t

 

 _ _

 



, (1.1)

where

2 4

4

d

2 d 4

qo

k y

m y

L

t

 

 _ _ 

  equals the Lagrangian for

the quartic oscillator in 1 + 1 dimensions, is integrated along an extremal and expressed in terms of the space- time end-point data



ta,ya

 

, t yb, b



.

We begin in a well-known way by adding and sub- tracting the kinetic energy to the Lagrangian. Thus we obtain from (1.1), after changing the variable of integration in the remaining integral, the following equivalent expression.









extremal

extremal d

, ; , d

d

,

b

a

y

qo a b

y

b a

S t y t y m y y

t

E t t



 



_(1.2)

where is the energy on the extremal (See e.g. Gold- stein [1]). Equation (1.2) is the form of the action that we will start from and then derive by integrating the first term in (1.2), which we call the momentum integral, thus the desired expression for qo is obtained. (Some au-

thors call this momentum integral the action.) For our convenience, we refer to the second term in (1.2) as the energy term. The derived action depends only on the end-point data in space-time.

E

S

qo

S

In Part 2 Alternative derivation of the Quartic Oscillator Solution, we present an approach in which we

arrive at the linearization map in [2]. This maps the solu- tions to Newton’s equations of motion for the quartic oscillator 1 − 1 onto those of the harmonic oscillator in a way which lends itself to integrating the momentum integral in (1.2). It involves a parametization of time t in

terms of an angular coordinate  (a cyclic coordinate which takes advantage of the periodic motion of the quartic oscillator and is intrinsic to the harmonic oscillator ho). This results in the time being given by a quadra-

ture involving a known function of , as in [2]. As stated in [2] R. C. Santos, J. Santos and J. A. S. Lima [3], first demonstrated the possibility of linearization of the quartic oscillator to the harmonic oscillator.

In Part 3 Integration of the momentum integral, the

results in Part 2 lead to an integration of (1.2). This is a

new result and an extension of the results in [3].

In Part 4 Derivation of qo, using the results in Part

2 and Part 3, we derive a classical action qo evaluated

on an extremal in terms of space-time endpoint data and show that Hamilton’s equations are satisfied.

S

In Part 5 Equivalent Actions, we present two equi-

valent actions as variations on the result in Part 4. By

equivalent we mean they are equal in value on extremals and they produce the same Hamilton’s equations.

In Part 6 Conclusion, we indicate briefly how the

ap-proach in Parts 3 and 4 can be directly extended to all

members of a hierarchy with potential energies

 

2

 

2 2 2 2 ₁

1 _.

2

n

n n n n _n

V y k y t

n 

(2)

2. Alternative Derivation of the Quartic

Oscillator Solution

To begin with, we must establish the sign conventions implied by (1.2) for the quartic oscillator

 

2

1

2

1

extremal

d _d

d 2

1 d

y y y

qo y

m y t y

t

V y

E

m y

m E

 

  _  _

 



,

(2.1)

where

 

4 4 4

qo

V y k y

E  E

Taking advantage of the periodicity of any extremal for the quartic oscillator qo, we execute a change of

variable to the angular variable  by setting





 

4 4

2

0 ,

sin

4

qo

V y k y

E E

    (2.2) where

 



0



 

0

 

0 0 0

y  t y  y t y 

and

 





 

0

0 0

d d

d d 2

d d t

y y

y t t t E m

t t

   

and E = energy on the extremal. We have opted not to

change the symbol for a function when it depends on a variable through a nested function in order to avoid un- necessarily heavy notation. Making the signs explicit, (2.1)-(2.2) yield













1 4

0 1 4 2

4 ₀

sin

4 _.

sin

E y

k

   



 

  

   (2.3)

and



0

d 2 _cos _.

d

y E

t  m  



(2.4)

Note, for future use (2.3) implies













1 4

0 1 4 2

4 ₀

cos

4 1

d

2 sin

E y

k

 

d .

 

  

  

   (2.5)

Now, we are in position to present an alternative derivation of the solution to Newton’s equations of motion (2.7) below for the quartic oscillator. It involves a pa-

rametization of time in terms of the angular coordinate. As we shall see, this results in the time being given by a quadrature involving a known function of . Now dif- ferentiating (2.4) yields









2

0 2

d 2 _sin d _.

d d

y E

m t

t

  

   (2.6)

Or from Newton’s equation of motion for the quartic oscillator

2

3 4 2 d

, d

y

m k

t   y (2.7)

we obtain





3 4

0

2 _sin _.

d

k y E

m m

d

t   



  (2.8)

Thus, it follows from (2.3) that we obtain the equation that yields involving t 





_{1 4} _{1 2} ₁



₂





1 4

4 0

dt k E  m 2 sin   d. (2.9a) Or, it s integrated form which yields (in quadrature) involving a known function of

t 

 









0

1 2 _{1 4}

1 4 2

0 4 ₂ sin 0 d .

m

t t k E



   _  _

 

_

 (2.9b)

The inverse of (2.9a) is given by

 

1 2 1 2 2 4

2

d k y

m  _{ } _

  d ,t (2.10a)

and it’s integrated form is given by

 





0

1 2

1 2 2 4 0

2

( ) d ,

t

t k

t y t

m

  _{ } _ 

 



t



(2.10b) where the integration is along an extremal.

The equivalence to the linearization map given in [2]

is specified by setting ₀ , where

constant of the harmonic oscillator and equals the time of the corresponding to of the .



0 tˆ ˆt

   

ho

2

2 spring

k m 

ho tˆ

t qo

Then (2.9b) and (2.10b) are equivalent to one half of the linearization map in [2]. The other half of the linearization map is given by

 





 









 





1 2 2

0

1 2 1 2

4 2

ˆ

sin .

4 2

ho

y t y t x t

E k E k

 

   (2.11)

Equation (2.2) plus equation (2.4) imply

 

1 2

 

1 2









4 4 2 2 2

4

4 ₂ _cos _sin

b a b b a a b a b a

E

y y y y y y

k   



_    _ 



(3)

how does one determine all other quantities. One is given



a,a



tˆ_atˆ_b



and



tˆbtˆo



, where refers to times, are

known. Now we can set

ho

ˆ

o o

t t .



y t and



y tb,b



on an ex-

tremal. The linearization map yields a

qo x and xb on the

corresponding extremal as well as ho qo .

This implies from (2.12) the time differences

ho E E From [4], as a result of mapping extremals for the ho

1

1 

onto extremals to the qo, we have from [4],

E  ho

 









 



 

_



 

_



 



1 2 1 2

2 2

1 2

0 4

sin sin

sin 4 ,

sin

b b a a a b

b a

y y t y y t

t E k

   

 

 _    _

  _ _



 

 

(2.13) and

 





 



 

_



 

_



 



1 2 1 2

2 2

1 2

0 4

cos cos

cos (4 ) ,

sin

b b a a a b

b a

y y t y y t

t E k    

 

  

 _ _ _ 

 

 

  

 

 

(2.14) Now (2.13) and (2.14) imply e.g.





 





 





 

1 2 2

0 ₂ 1 2 1 2

sin tan

cos

b b b a

b

b b b a a

y y

y y y y

   

 



 

  2

a

ˆ

a t

where ˆ ˆ



ˆ ˆ



ˆ and yields

b o a o b

t  t t t  t t tˆ

 

0  .

Everything else follows from the development in

Part 3.

3. Integration of

t



extremal

d

b

a

y

qo qo y

y

m

y

The problem of integrating (1.2) is the problem of integrating (2.1). Therefore, using (2.2) , (2.4) ,and (2.5), we obtain

 













4 4 extremal

extremal 1 4

1 4 2

0 0

4

d 2

d 1 d

d 4

2 _cos 4 1 _sin _cos _d

2

b b

a a

b

a

y y

k y E

m y t y m y

t m E

E E

m

m k



0

    

 

    

 

 _{  }

 

  

  _ _{  }  

   





(3.1)

Effecting the integration by parts, where



2





3 4

0

d d d

, sin

d d d

f g

fg g f f  

       and







00



2cos 3sin

g  

   

 yields





















_



_







 













1/ 4 _{1 2}

1 4 4

1 2 1 4

4

extremal 4

1 4 3 4 ₀

2 2

0 0

0

3 4 1 2

3 4 ₀

2 1 1 4

0 4

d 2 4 1 2 2 1

d

d 2 3

cos

sin d sin

sin

cos 2

4 _sin

3 3 s

b

a

b b

a _a

b

a

y qo

qo y

b a

y _E _E _m

m y m k E

t m k m k E

m E

E

t t

k

 

 _



  

    

 

   

  

  

 _{    }_ _ _

 _ _{    } _

 

    

  _ _ _



 _

  

 

 

   





2

in

(3.2)

Finally, from (2.9b), we have





 



















 



_



_



3 4 1 2

3 4 ₀

2 0 1 4

0

extremal 4

1 2

3 2 ₀

2 4

0

d 4 2 cos

d sin

d 3 ₃ sin

cos

4 1 _,

3 3 2 sin

b b

a _a

b

a

y qo

qo b a

y

b a

y E m E

m y t t

t _k

mk E

t t y t



 

    

 

   



   



 

  _{ } _



 



(3.3)

(4)

4. Determination of a

S

_qo

The developments in Part 2 and Part 3 lead directly to

the following determination of Sqo. It follows from (3.3) that (1.2) is given by















 



_



_







 

_



_



 

_



_







extremal extremal

1 2

3 2 ₀

2 4

0

1 2

3 2 ₀ 3 2 ₀

2 2

4

0 0

d

, ; , d

d

cos

4 1

3 3 2 sin

cos cos

1

3 2 sin sin 3

b

a

b

a

y

qo a b b b a

y

b a b a

b a

S t y t y m y y E t t

t

mk E

t t y t E t t

mk E

y y



    

   

  



 

   _ _  



 

   

 

    _  _ 

  _ _



b a

t t

 

(4.1)

Therefore, using (2.10b), we obtain





 



 



 





 





 









0 0

1 2 1 2

2 2

4 4

1 2

3 2 3 2

2 2

4

1 2 1 2

2 2

4 4

; ; ,

2 2

cos d cos d

1

3 2 2 2 3

sin d sin d

b a

qo a b b

t t

b t a t

t t

S t y t y

k k

y t t y t t

m m

mk E

y y

k k

y t t y t t

m m

 _ _ _ _ 

   

     

   

   

 _ _{ }  _

  _   _ _   _ __

   

     

 



b a

t t

  (4.2)

This is expressed in the endpoint variables as required. This implies

 



 



 







 









 



0

1 2 2 4

1 2

1 2 1 2

2 2

4 4

1 2 ₄

2 4

1 2

1 2 4 2

2

cos d

2

4 _cos _d

2 _sin 2 _d 2

2

2 cos d ,

b

b b

b

t

t t

qo

qo b b t

b t

t t

t

k

y t t

m

S mk mk E k

p y y y t

y k k

y t t

m

k

mE y t t

m

  _ _

 

 _ _    _      _ _

 

     

     _ _      

 

 

  _ _

 _ _

 



4 _t

m

(4.3)

 





 

0

1 2 1 2

3 2 1 2

2 2

4 4

1 2

2 4 2

1 2

1 1

.

3 2 2 3

sin b d

qo

b t b

b

t

S mk k

y y

t k m

y t t

m

 

 

 _       _

    

   _   _ __ 

 

   







E E

After using (2.11) this checks with m times (2.4) for pqo_b and  tb obviously checks.

The a-differentiations parallel the b-differentiations and yield

 



 



 







 









 



0

0 0

1 2 2 4

1 2

1 2 1 2

2 2

4 4

1 2 ₄

2 4

1 2

1 2 4 2

2

cos d

2 4

cos d

2 2 2

sin d

2

2 cos d ,

a a

a

t

t t

qo

qo_a a a t

a t

t t

t

k

y t t

m

S mk mk E k₄

p y y y t

y k k

y t t

m

k

mE y t t

m

  _ _

 

 _ _   _ _   _ _

 

   _ _  _ _ _ _ _ _

     _ _      

   

  _ _

  _ _

 



(5)

 





 

0

1 2 1 2

3 2 1 2

2 2

4 4

1 2

2 4 2

1 2

1 1

.

3 2 2 3

sin d

a

qo

a t a

a

t

S mk k

y y

t k m

y t t

m

 

 

 _{ }       _ _{ }

    

   _   _ __ 

 

   







E E

(4.4)

5. Equivalent Actions

Here, we present two examples of equivalent actions as variations on this result. By equivalent we mean they are equal in value on extremals and they both produce the same Hamilton equations.

First Variation:

This variation follows from the indentities























0 max 0 max

sin sin cos ,

cos cos sin ,

      

     

      



(5.1)

which implies that (4.2) transforms to the expression









 



 



 





 





 





max max

extremal extremal

1 2 1 2

2 2

4 4

1 2

3 2 3 2

2 2

4

1 2 1 2

2 2

4 4

d

, ; , d

d

2 2

sin d sin d

1

3 2 2 2

cos d cos d

b

a

b a

y

qo a a b b b a

y

t t

b t a t

t t

S t y t y m y y E t t

t

k k

y t t y t t

m m

mk

y y

k k

y t t y t t

m m

  

 _ _ _ _

   

 _ _ _ _

  

 _ _{ } 

  _   _ _   _ _

   

    









.

3 b a

E

t t

 

   

 

(5.2)

Second Variation:

Equation (5.2) is equivalent to





 



 



 



 





 



 



 



 

max max max

max max

1 2

3 2 1 2 3 2 1 2 1 2

2 2 2 2 2

4 4 4

3 2 1 2 3 2

2 4 2 2 4

, ; ,

1 2 2 2

cos d cos d sin

3 2

2 2

cos d cos

b b b

a a

qo a a b b

t t t

b b

t t t

t t

a a

t t

S t y t y

mk k k

y y t t y y t t y t

m m

k k

y y t t y

m m

 



    _ _   _ _   _ _

    _ _  _ _  _ _

   

  _ _  

 _ _  _

   





 



4 _d

k

t m

 









max

1 2 1 2

2 _d _sin 2 4 2 _d

1

. 3

a

t

b a

k

y t t y t t

m

E t t

  _ _ _

   

     

 

  

 

 







(5.3)

Comment: The signs and the limits of integration have to be carefully watched in these calculations.

The identity

 





 









3 2 1 2

2

max max max

3 2 3 4

2 2

max max max

cos 3

2 cos cos

b b b

b b

y y y

y

 

   

   

 

2

,

y

 (5.4)

follows from

 

2 1 2

 

2 1 2





max maxcos max .

b b b

y y  y y  

Similarly for the endpoint, thus we obtain the result reported in [4].

a

The results given in [4] were obtained before the integration result reported here in Part IV was obtained.

6. Conclusions

One can parallel the development in Parts 3 and 4 for an

hierarchy with potential energies

 

2

2 2 ₁

1 2

n

n n _n

V y k y

n 

 (6.1) Starting with setting





2

 

2 2

2

0 ,

sin

2

n

n n

V y k y

E nE

    (6.2) one can parallel Part 3.

Then integration by parts in these cases is effected by







₂



 1 2

0

d d d _, _sin

d d d

n n

f g

fg g f f  

  



   

and







 

0 0



cos 1 sin

n g

n

     

  .

(6)

The linearization map for these cases is given in [2].

REFERENCES

[1] H. Goldstein, “Classical Mechanics,” Addison-Wesley Publishing Company, Reading, 1980.

[2] R. L. Anderson, “An Invertible Linearization Map for the Quartic Oscillator,” Journal of Mathematical Physics,

Vol. 51, No. 12, 2010, Article ID: 122904.

[3] R. C. Santos, J. Santos and J. A. S. Lima, “Hamilton- Jacobi Approach for Power-Law Potentials,” Brazilian Journal of Physics, Vol. 36, No. 4A, 2006, pp. 1257-

1261.