The action

where equals the Lagrangian for the quartic oscillator in 1 + 1 dimensions, is integrated along an extremal and expressed in terms of the spacetime end-point data.

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 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 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 depends only on the end-point data in space-time.

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 solutions 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 quadrature 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, using the results in Part 2 and Part 3, we derive a classical action evaluated on an extremal in terms of space-time endpoint data and show that Hamilton’s equations are satisfied.

In Part 5 Equivalent Actions, we present two equivalent 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 approach in Parts 3 and 4 can be directly extended to all members of a hierarchy with potential energies

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

where

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

where

and

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 unnecessarily heavy notation. Making the signs explicit, (2.1)-(2.2) yield

and

Note, for future use (2.3) implies

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 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. Now differentiating (2.4) yields

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

we obtain

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

Or, it s integrated form which yields (in quadrature) involving a known function of

The inverse of (2.9a) is given by

and it’s 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, where constant of the harmonic oscillator and equals the time of the corresponding to of the.

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

Equation (2.2) plus equation (2.4) imply

where is given by (2.10b).

Finally, in this paragraph, given the end-point data how does one determine all other quantities.

One is given and on an extremal. The linearization map yields and on the corresponding extremal as well as. This implies from (2.12) the time differences

and, where refers to times, are known. Now we can set.

From [4], as a result of mapping extremals for the onto extremals to the, we have from [4],

and

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

where and yields.

Everything else follows from the development in Part 3.

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

Effecting the integration by parts, where and yields

Finally, from (2.9b), we have

where is given by (2.10b).

The developments in Part 2 and Part 3 lead directly to the following determination of.

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 times (2.4) for and obviously checks.

The -differentiations parallel the -differentiations and yield

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

which implies that (4.2) transforms to the expression

Second Variation:

Equation (5.2) is equivalent to

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

The identity

follows from

Similarly for the 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.

One can parallel the development in Parts 3 and 4 for an hierarchy with potential energies

Starting with setting

one can parallel Part 3.

Then integration by parts in these cases is effected by

and.

This then parallels the development in Part 4.

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