Extending Bauer's corollary to fractional derivatives

David W Dreisigmeyer¹ and Peter M Young²

¹Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA
²Department of Electrical and Computer Engineering, Colorado State University, Fort Collins, CO 80523, USA

Email: dreisigm@math.colostate.edu and pmy@engr.colostate.edu

Received 12 December 2003
Published 2 March 2004

PACS numbers: 42.20. - d, 02.30. - f

1. Introduction

In 1931 Bauer proved the following corollary [3]: `The equations of motion of a dissipative linear dynamical system with constant coefficients are not given by a variational principle'. There are a few methods that may allow us to get around Bauer's corollary. For example, we could allow additional equations of motion to result. Bateman used this technique in [2]. If we use the Lagrangian

where C is a constant, we would have the following equations of motion:

Bateman's method uses the loophole that Bauer's proof assumed that no additional equations arise.

Riewe pointed out that Bauer's proof also implicitly assumes that all the derivatives are integer ordered [6]. This has led to attempts to use fractional derivatives in the actions to model nonconservative systems [4-6]. Here we will close this second loophole by extending Bauer's corollary to include fractional derivatives.

Our paper is organized as follows. In section 2 we review the background material needed for our result. The extension of Bauer's corollary is proved in section 3. A brief discussion follows in section 4.

2. Background material

Here we develop the relevant mathematics for our proof. A fuller discussion of this material can be found in [4]. Fractional derivatives can be defined using the theory of distributions. First, define the generalized functions

and

where Γ(α) is the gamma function. The left fractional derivatives (LFDs) of a function q(t) is given by

where we set q(t) ≡ 0 for t < a. When α  =  n, n an integer, (6) becomes

where is the generalized derivative. Right fractional derivatives (RFDs) are defined similarly

where now q(t) ≡ 0 for t > b. Instead of (7), we have

In [4] the actions were treated as Volterra series. The Volterra series are a generalization to functionals of the power series of a function. For a functional , define the symmetric kernels

Now introduce the notation

Then can be written as

where we set K^(s)₀ ≡ 0.

The Φ^±_α(t) are now treated as kernels in a Volterra series. We can then take the functional derivative of the series to derive our equations of motion. An example should make this clearer. We will restrict our attention to the action

where K₂(t, τ) in (13) is an arbitrary kernel, i.e., not necessarily symmetric as in (10). (Equation (13) would be sufficiently general to handle the nonconservative harmonic oscillator.) Now let K₂(t, τ) be given by

where 0 < γ < 2 and C is a constant. So (13) becomes

The functional derivative of (15) is

If we require the advanced and retarded parts of (16) to vanish separately, we have

where in (18) we used the fact that . This is the method presented in [4] for deriving the equations of motion.

3. The result

In section 2 we reviewed the procedure Dreisigmeyer and Young proposed in [4] for deriving a system's equations of motion. From (17) and (18) we see that two equations are actually derived: an advanced one and a retarded one. So this is, effectively, a generalization of Bateman's method (see (1)-(3)). That is, extra equations of motion are allowed to result from the action's variation.

We desire to have a single, retarded equation of motion to result from a variational principle. From (14) we see that the derivative operators are always contained in the K₂(τ₁, τ₂) kernel. Perhaps it is possible to use some kernel other than the Φ^±_α(τ₁  -  τ₂) to have a fractional derivative arise from an action's variation? The following theorem shows that this is not possible within the formalism presented in [4].

Theorem 3.1. There does not exist a , such that the variation of the quantity

will result in for α ≠ 2n, n an integer.

Proof. The variation of is given by

We will assume that

and arrive at a contradiction. We require that (21) holds for every q(t). Then we must have

Interchanging ρ and t in (22) gives us

Hence, unless Φ^± _- α(ρ  -  t) is symmetric in ρ and t, (22) and (23) cannot both hold. That is, unless α  =  2n, n an integer, there does not exist a , such that (21) holds.        

Theorem 3.1 shows that, in general, the fractional mechanics formalisms presented in [4-6] cannot derive a single, retarded equation of motion. In order to overcome this difficulty, Riewe suggested approximating RFDs with LFDs [5, 6]. Dreisigmeyer and Young showed in [4] that this is not a sound idea and, instead, allowed for an extra, advanced equation of motion. The latter technique is not, itself, entirely satisfactory.

4. Discussion

Theorem 3.1 shows that some revision of our concept of an action may be in order if we desire a variational principle to work for nonconservative systems. How could we derive a single, retarded equation of motion for systems? Our result holds even if K₂(τ₁, τ₂) is allowed to be complex. We would also require that q(τ₁)  =  q(τ₂) for τ₁  =  τ₂ in (19). That is, we do not want to employ Bateman's method, as was done in [4].

One possible method proposed by Tonti [7] (see also [1]) is to use the convolution product in our Lagrangians. This leads to actions of the form

This method does allow the derivation of a single retarded equation of motion for, e.g., the driven nonconservative harmonic oscillator. Unfortunately, it does not seem possible to naturally generalize Tonti's method to higher ordered potentials. That is, using quantities such as

in the action will not lead to the correct form for the potential energy terms when n > 2. This situation should be contrasted with that in [4]. There terms such as

were able to treat the potential energy terms correctly. However, as theorem 3.1 demonstrates, the formalism in [4] is unable to deal correctly with the fractional derivative terms.

An interesting feature of (24) versus (19) is the presence of t in the action along with τ₁ and τ₂. Our stated goal is to derive purely retarded equations of motion using a variational principle. However, to achieve this we need to find the correct kernels for our Volterra series action. Theorem 3.1 states that kernels of the form K₂(τ₁, τ₂) are not sufficient for our purposes. Equation (24) suggests that we look instead at the kernels K(t, τ₁,  ..., τ_n) for our actions.

It is desirable to be able to use the same type of kernel for the fractional derivatives as well as for the higher ordered potential terms. This assumption is necessary so that the same perturbation of q(τ) can be used in the kinetic and potential energy terms of the action. It allows us to reject using kernels of the form K(t  -  τ₁  -  ···  -  τ_n), as in (25). Also, it prevents us from using terms such as (24) for the kinetic energy and terms such as (26) for the potential energy, within the Lagrangian formalism. Finding the correct kernels K(t, τ₁,  ..., τ_n) for our Volterra series actions is a line of research that we are actively pursuing at this time.

Acknowledgments

The authors would like to thank the NSF for grant no 9732986. Our deepest appreciations to the referee for bringing references [1, 7] to our attention.

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