## Abstract

The alternative Hamiltonians for systems with one degree of freedom are solved directly from the Hamilton’s equations. These new Hamiltonians produce the same equation of motion with the standard one (called the Newtonian Hamiltonian). Furthermore, new Hamiltonians come with an extra-parameter, which can be used to recover the standard Hamiltonian.

### Keywords

- Hamiltonian
- Lagrangian
- nonuniqueness
- variational principle
- inverse problem of calculus of variations

## 1. Introduction

It was well known that the Lagrangian possesses the nonuniqueness property. It means that the constant can be added or multiplied into the Lagrangian:

Furthermore, the total derivative term can also be added to the Lagrangian without alternating the equation of motion:

where

Obviously, the last two terms contribute only at the boundary. Then the variation

The standard Lagrangian takes the form

where

Recently, it has been found that actually there is an alternative form of the Lagrangian called the multiplicative form [1, 2, 3]:

where

where

These new Lagrangians

The problem studied in [1, 2, 3] that is actually related to the inverse problem of calculus of variations in the one-dimensional case. The well-known result can be dated back to the work of Sonin [4] and Douglas [5].

Theorem (Sonin): For every function

What we did in [1, 2, 3] is that we went further to show that actually Eq. (8) admits infinite solutions.

In the present chapter, we will construct the Hamiltonian hierarchy for the system with one degree of freedom. In Section 2, the multiplicative Hamiltonian will be solved directly from Hamilton’s equations. In Section 3, the physical meaning of the parameter

## 2. The multiplicative Hamiltonian

To obtain the Hamiltonian, we may use the Legendre transformation:

where

which is nothing but the total energy of the system. The action is then

With the variations

which are known as a set of Hamilton’s equations.

We now introduce a new Hamiltonian, called the multiplicative Hamiltonian, in a form

where

Replacing

Now we define

where

where

where

where

Inserting Eq. (20) into Eq. (14), we find that

which is the equation of motion of the system. Then this new Hamiltonian Eq. (20) gives us the same equation of motion as Eq. (10).

For the case

gives back the standard Hamiltonian. The constant

We find that the multiplicative Hamiltonian Eq. (20) can also be directly obtained from the Legendre transformation:

where

Inserting Eqs. (24) and (5) into Eq. (23), we obtain

which is identical to Eq. (20).

Furthermore, we can rewrite the multiplicative Hamiltonian Eq. (20) in terms of the series:

where

From the structure of Eqs. (26) and (6), it must be a hierarchy of the Legendre transformation. To establish such hierarchy, we start to rewrite the momentum Eq. (24) in the form

where

Then the Legendre transformation Eq. (23) becomes

Eq. (29) holds if

which are the Legendre transformations for each pair of the Hamiltonian

Next, we consider the total derivative

Eq. (31) holds if

Eq. (32) can be considered as the modified Hamilton’s equations for each

From the structure of the multiplicative Hamiltonian Eq. (20), it seems to suggest that the exponential of the function, defined on phase space, is always a solution of the Eq. (14). Then we now introduce an ansatz form of the Hamiltonian as

where

We find that if we take

or

We immediately see that actually Eq. (36) is a consequence of the conservation of the energy of the system:

Then what we have here is another equation that can be used to determine for the Hamiltonian subject to the equation of motion Eq. (21). To see this, we may start with the standard form of the Hamiltonian

Using Eqs. (21) and (38), it can be rewritten in the form

Since

where

Next, we put

or

We see that both sides of Eq. (42) are independent to each other. Then Eq. (42) holds if both sides equal to a constant

where

where

where

We see that with Eq. (36) the Hamiltonian can be easily determined. Here we come with the conclusion that in every function

The existence of solutions of Eq. (46) implies that actually we can do an inverse problem of the Hamiltonian for the systems with one degree of freedom.

Remark: The perspective on nonuniqueness of Hamiltonian, as well as Lagrangian, here in the present work is quite different from those in Aubry-Mather theory [6, 7] (see also [8]). What they had been investigating is the modification of the Tonelli Lagrangian

## 3. Harmonic oscillator

In this section, we give an explicit example, e.g., the harmonic example, and also give the physical interpretation of the parameter

Then the multiplicative Hamiltonian for the harmonic oscillator is

Now we introduce

where

Inserting Eq. (48) into Eq. (49), we obtain

where

where

This means that we can choose any Hamiltonian in the hierarchy to work with. The physics of the system remains the same but with a different time scale. Then we may say that the parameter

Next we consider the standard Lagrangian of the harmonic oscillator

and the multiplicative Lagrangian is

where

where

The action of the system is given by

where

The variation

where

which produce the equation of motions

associated with different time variables. Again in this case, we have the same structure of equation of motion for each Lagrangian in hierarchy but with a different time scale. From Eq. (58), we see that as

## 4. Redundancy

From previous sections, we see that there are many forms of the Hamiltonian that you can work with. One may start with the assumption that any new Hamiltonian is written as a function of the standard Hamiltonian

where

## 5. Summary

We show that actually there exist infinite Hamiltonian functions for the systems with one degree of freedom. We may conclude that there exists the reverse engineering of the calculus of variation on phase space (see Eq. (44)). Furthermore, the solution of Eq. (44) exists not only as one but infinite. Interesting fact here is that these new Hamiltonians come with the extra-parameter called

In the case of many degrees of freedom, the problem turns out to be very difficult. Even in the case of two degrees of freedom, the problem is already hard to solve from scratch. We may start with an anzast form of the Lagrangian:

Furthermore, promoting the Hamiltonian Eq. (20) to be a quantum operator in the context of Schrodinger’s equation is also an interesting problem. This seems to suggest that an alternative form of the wave function for a considering system is possibly obtained. This can be seen as a result from that fact that with new Hamiltonian operator, we need to solve a different eigenvalue equation, and of course a new appropriate eigenstate is needed. From the Lagrangian point of view, extension to the quantum realm in the context of Feynman path integrals is quite natural to address. However, this problem is not easy to deal with since the Lagrangian multiplication is not in the quadratic form. Then a common procedure for deriving the propagator is no longer applicable. Further study is on our program of investigation.

## Acknowledgments

The author would like to thank Kittikun Sarawuttinack, Saksilpa Srisukson, and Kittapat Ratanaphupha for their interests and involving themselves in the investigation on this topic.

## Appendix

In this section, we will demonstrate how to solve the multiplicative Lagrangian Eq. (5). We introduce here again the Lagrangian

The least action principle states that the system will follow the path which

Eq. (64) can be rewritten in the form

Using equation of motion, we observe that the coefficient of the second term depends only

We find that it is not difficult to see that the function

where

and the solution

where

where

the standard Lagrangian at the limit

which can be considered as the one-parameter extended class of the standard Lagrangian.