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

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 Langrangian along an extremal, for the nonlinear quartic oscillator evaulated on extremals.


INTRODUCTION
The action where L qo = m 2 ( dy dt ) 2 − k 4 y 4 4 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 (t a , y a ), (t b , y b ).
We begin in a well-known way by adding and subtracting 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. where E is the energy on the extremal (See e.g. Goldstein [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 S qo is obtained. (Some authors 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 S qo depends only on the end-point data in space-time.
In Part II Alternative derivation of the Quartic Oscillator Solution, we present an approach in which we arrive at the linearization map in [2]. This maps the solutions to Newtons 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 θ (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 quadrature involving a known function of θ , 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 III Integration of the momentum integral, the results in Part II lead to an integration of (1.2). This is a new result and an extension of the results in [3].
In Part IV Derivation of S qo , using the results in Part II and Part III, we derive a classical action S qo evaluated on an extremal in terms of space-time end-point data and show that Hamilton's equations are satisfied.
In Part V Equivalent Actions, we present two equivalent actions as variations on the result in Part IV. By equivalent we mean they are equal in value on extremals and they produce the same Hamilton's equations.
In Part VI Conclusion, we indicate briefly how the approach in Parts III-IV can be directly extended to all members of a hierarchy with potential energies V 2n (y 2n ) = 1 2n k 2n y 2n 2n (t)| n≥1 .

Alternative Derivation of the Quartic Oscillator Solution
To begin with, we must establish the sign conventions implied by (1.2) for the quartic oscillator Taking advantage of the periodicity of any extremal for the quartic oscillator qo, we execute a change of variable to the angular variable θ by setting 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 unnecessarily heavy notation. Making the signs explicit, and for the quartic oscillator. It involves a parametization 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 θ . Now differentiating (2.4) yields Or from Newtons equation of motion for the quartic oscillator Thus, it follows from (2.3) that we obtain the equation that yields t involving θ Or, its integrated form which yields t (in quadrature) involving a known function of θ The inverse of (2.9a) is given by and its integrated form is given by where the integration is along an extremal.
The equivalence to the linearization map given in [2] is specified by setting spring constant of the harmonic oscillator ho andt equals the time of the ho corresponding to t of the 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 where (θ a − θ a ) is given by (2.10b).
Finally, in this paragraph, given the end-point data how does one determine all other quantities.
One is given (y a ,t a ) and (y b ,t b ) on an qo extremal. The linearization map yields x a and x b on the corresponding ho extremal as well as E ho = E qo = E. This implies from (2.12) the ho time differences (t a −t b ) and (t b −t o ), whereˆrefers to ho times, are known. Now we can set t o =t o .
Effecting the integration by parts, where d Finally, from (2.9b), we have where θ − θ 0 is given by (2.10b).

Part IV
Determination of an S qo .
The developments in Part II and Part III lead directly to the following determination of S qo .
It follows from (3.3) that (1.2) is given by Therefore, using (2.10b), we obtain This is expressed in the endpoint variables as required. This implies After using (2.11) this checks with m times (2.4) for p qo b and ∂ /∂t b obviously checks.
The a-differentiations parallel the b-differentiations and yield Part V

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.
Similarly for the a endpoint, thus we obtain the result reported in [4].
The results given in [4] were obtained before the integration result reported here in Part IV was obtained.
This then parallels the development in Part IV.
The linearization map for these cases is given in [2].