First of all, let's define fractional calculus, since it is a not-so-well-known term, and it will be the basis of this report. Fractional calculus is the theory of derivatives and integrals of arbitrary non-integer (even complex) order, and started in 1695 when Leibniz presented his analysis of the derivative of order ½. Then it developed mainly as a theoretical aspect of Mathematics, being considered by some of the great names in Mathematics such as Euler, Lagrange, and Fourier, among many others.
Well, what's this about non-integer-order derivatives and integrals?, and more importantly, is it just a theoretical curiosity or does it have practical applications? Recently, fractional calculus has experienced a revival of interest and has been used in very diverse topics such as fractal theory [1], viscoelasticy [2], electrodynamics [3,4], optics [5,6], and thermodynamics [7], to name a few.
The purpose of this report is to review the application of fractional calculus to variational mechanics, more precisely, the ability to account for non-conservative systems with a fractional Lagrangian formulation [8-10]. Amazingly, Lagrangian (or Hamiltonian) mechanics only describe conservative systems, and there have been some approaches at describing non-conservative systems that will also be presented. Riewe [8,9] proposed an elegant (and general) way to incorporate non-conservative systems to the field of variational mechanics using fractional calculus. Since we are not aiming for a description of fractional calculus, we shouldn't dwell on it for too long; however it it useful to have a quick review of the basic concepts behind it, because we will use them later on. If you are already familiar with fractional calculus, you can continue to the next section (although you should make sure to at least look at the notation we will be using throughout the rest of the report).
Since variational mechanics is a very elegant and powerful formalism when it comes to solving a wide variety of problems in classical mechanics, there have been many approaches meant to include non-conservative systems in such formalism. One of the best known methods is that presented by Rayleigh, introducing a new functional R (which is now called Rayleigh's dissipation function [11,12]), and rewrite the Euler-Lagrange equation in the following way
where L is the system's (traditional) Lagrangian, and
.
This provides us with a way to treat dissipative systems, incorporating forces that are functions of the velocity; however, most of the beauty and power of Lagrangian (Hamiltonian) mechanics is that the analyzed system is characterized with only one scalar functional (the Lagrangian L, or the Hamiltonian H). If we would use Rayleigh's dissipation function, then it would take two scalar functionals to completely define the problem. This is the principal objection to this approach.
As pointed out in [8,9,13,14], several non-conservative problems have been solved using the variational formalism. This is done constructing a Lagrangian that leads to differential equations that are equivalent to the ones obtained with Newton's laws. However, such Lagrangians lead to Hamiltonians and momentum equations that are physically misleading. As an example, we could look at the damped linear oscillator described by Ray [15] but also in [12,13,16], and reviewed by Riewe [8,9]. We would like to obtain the following equation of motion
where λ is the damping coefficient. If we use the Lagrangian
or, in general we could use a Lagrangian of the form
where Q = Q(x,t) is arbitrary but continuously differentiable. For this case, we will use Q=0. This leads to the Euler-Lagrange equation
Hamiltonian
and generalized momentum
We can see that the obtained Euler-Lagrange equation is equivalent to the one we wanted, if we ignore the factor
,
but the Hamiltonian and conjugate momentum are not quite what we wanted. In fact, it has been pointed out by Ray [15] that this Lagrangian actually describes an oscillator with varying mass, not a dissipative oscillator. He then proposes two systems (without dissipation) that are described (with physical meaningful momentum and Hamiltonian), by such a Lagrangian:
Yet another method is that of incorporating an auxiliary coordinate that describes the reverse-time system with negative friction. However, the Hamiltonian leads to additional solutions that must be suppressed, and (again) the momenta are not physically meaningful [8,9].
One can also include the microscopic description of the dissipation into the Lagrangian, but this gets really complicated really fast, and is not a viable way to include dissipation into variational mechanics.
So far, the presented approaches are not completely satisfactory when it comes to dealing with non-conservative variational mechanics. Now we describe a method proposed by Riewe [8,9] that, using fractional calculus, offers a powerful way of describing non-conservative systems. Our Lagrangian will be a functional of the generalized coordinates
 |
 |
where s(n) is any non-negative number (just derivatives, not integrals), and s(0)=0, so that qr,s(0) denotes the coordinate xr (we have labeled coordinates as usual in Physics and Mathematics literature, e.g. x1=x, x2=y, x3=z). In this notation, s(n) is a list of the orders of the derivatives used in the problem, and n is the index used for labeling them.
Since summations over the r index will always be over all values, the convention of summing over repeated indices is used; however, summations over n are not always over all valued, thus all summations over n will be indicated explicitly.
The limits a and b come from the fact that we are looking for an extremum for the action
where α is the variational parameter [9]. Next, the generalized momenta are defined,
and the generalized Euler-Lagrange equation
Then, the Hamiltonian
and Hamilton's equations
For integer-order derivatives, it is shown in detail on [8] that
so, if the Lagrangian does not depend explicitly on time, then the Hamiltonian is a constant of the motion. However, when we have fractional derivatives in our Lagrangian, this is not generally true and the Hamiltonian is not a constant of the motion, allowing us to account for non-conservative systems in variational mechanics.
This report is not meant to discuss the (extensive) mathematical detail involved in the derivation of the above equations, but to review a new method for describing non-conservative systems in variational mechanics, and illustrate its power through an example, which is worked out in the following section.
Now we put the described machinery to work on a simple problem, the same one we have been comparing all along: the linear damped oscillator. We begin with the regular Lagrangian
and add the term
in order to get the full fractional Lagrangian
where
 |
 |
 |
and the Euler-Lagrange equation, which is what we wanted!
then get the generalized momenta
so we can write down the Hamiltonian
and Hamilton's equations just follow from this and give us the same Euler-Lagrange equation (as expected), and identities or definitions, which do not bring new insight and are very easy to work out. Other examples are presented by Rabei et al. [17].
There are several definitions of fractional derivatives and integrals, but the most general is the one of Riemann-Liouville. The notation used is not unique, but here we will use the so-called Davis notation
to denote the operator for the α-th order fractional integral operator with respect to x based on the lower limit of integration a.
for α < 0 and x > a. In the case of fractional derivation (α > 0), to avoid the problems of Γ(-α) when α = 1,2,3,... we use
where n is the smallest integer exceeding α, this is (n-1) ≤ α < n. It is easy to verify that this reduces to our previous knowledge on integral calculus for integer α. From now on we will refer to derivatives and integrals as differintegrals.
Probably the first thing that we notice about fractional differintegrals is that they depend on the value of the lower limit of integration, which means that the are nonlocal.
Now, let's look into some of the most fundamental properties of the fractional differintegral operator:
Also, fractional derivatives can be approximated using finite differences, and there are several definitions for this (different expressions that converge to the same limit). We will only consider forward and backward differences. For forward differences,
and for backward differences,
where
is the binomial coefficient.
© 2007 R. J. Noriega-Manez. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
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