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Fractional embedding of differential operators and Lagrangian systems

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Abstract

This paper is a contribution to the general program of embedding theories of dynamical systems. Following our previous work on the Stochastic embedding theory developed with S. Darses, we define the fractional embedding of differential operators and ordinary differential equations. We construct an operator combining in a symmetric way the left and right (Riemann-Liouville) fractional derivatives. For Lagrangian systems, our method provide a fractional Euler-Lagrange equation. We prove, developing the corresponding fractional calculus of variations, that such equation can be derived via a fractional least-action principle. We then obtain naturally a fractional Noether theorem and a fractional Hamiltonian formulation of fractional Lagrangian systems. All these constructions are coherents, i.e. that the embedding procedure is compatible with the fractional calculus of variations. We then extend our results to cover the Ostrogradski formalism. Using the fractional embedding and following a previous work of F. Riewe, we obtain a fractional Ostrogradski formalism which allows us to derive non-conservative dynamical systems via a fractional generalized least-action principle. We also discuss the Whittaker equation and obtain a fractional Lagrangian formulation. Last, we discuss the fractional embedding of continuous Lagrangian systems. In particular, we obtain a fractional Lagrangian formulation of the classical fractional wave equation introduced by Schneider and Wyss as well as the fractional diffusion equation.
arXiv:math/0605752v1 [math.DS] 30 May 2006
FRACTIONAL EMBEDDING OF DIFFERENTIAL
OPERATORS AND LAGRANGIA N SYS TEMS
by
Jacky CRESSON
Abstract. This paper is a contribution to the general progr am of embedding theories of
dynamical systems. Following our previous work on the Stochastic embedding theory devel-
oped with S. Darses, we define the fractional emb edding of differential operators and ordinary
differential equations. We construct an o perator co mbining in a symmetric way the left and
right (Riemann-Liouville) fractional derivatives. For Lagrang ian systems, our method pro-
vide a fractiona l Euler-Lagrange equation. We prove, developing the corresponding fractional
calculus of variations, that such eq uation can be derived via a fractional least-action prin-
ciple. We then obtain naturally a fractional Noether theorem a nd a fractional Hamiltonian
formulation of fractional Lagrangian systems. All these constructions are coherents, i.e. that
the embedding procedure is compatible with the fractional calculus of variations. We then
extend our results to cover the Ostrogradski formalism. Using the fractional embedding and
following a previous work of F. Riewe, we obtain a fractional Ostrogradski formalism which
allows us to derive non-conservative dynamical systems via a fractiona l generalized least-
action principle. We also discuss the Whittaker equation and obtain a fractional Lagr angian
formulation. Last, we discuss the fractional embedding of continuous Lagrangian systems.
In particular, we obtain a fractional Lagrangian formulation of the classical fractional wave
equation introduced by Schneider and Wyss as well as the fractional diffusion equation.
MSC: 34L99 - 49S05 - 49N99 - 26A33 - 26 A2 4 - 70S05
Contents
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .. . . .. . . . . . . . . . . . . . . . 3
1. Embedding theories. . . .. . . . .. . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5
2. Emergence of fractional derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 6
3. Deformation theories and the fractional framework. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 8
4. Plan of the paper. . . . . . .. . . .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Part I. Fractional operators.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1. Fractional differential operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1. Left and Ri ght Riemann-Liouville derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2. Left and r ight fractional derivatives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 13
2. The extension problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3. The fractional operator of order (α, β), α > 0, β > 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4. Product rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .. . . . . . .. . . . . . . . . . . . . . . . . . . 17
2 JACKY CRESSON
Part II. Fractional embedding of differential operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1. Fractional embedding of differential operators. . . . . . .. . . . . . . . .. . . . . . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . .. 19
2. Fractional embedding of differential equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 20
3. About time-reversible dynamics. . . . . . .. . . . . . . . .. . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . 21
Part III. Fractional embedding of Lagrangian systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1. Reminder about Lagrangian systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2. Fractional Euler-Lagrange equation. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . . . . 24
3. The coherence problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Part IV. Fractional calculus of variati ons. . . .. . . . . . . . . . . . . . . . . . .. . . .. . . . . . . . . . . . . . . . . . .. . . .. . . . . . . . . . . . 27
1. Fractional functional. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . . .. . . . .. . . . .. . . . . . . . . . . . . .. . . . . . . . . 27
2. Space of variations and extremals. . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . 27
3. The fractional Euler-Lagrange equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4. Coherence. . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .. . . .. . . . . . . . . . . . . . . . . . . . .. . . .. . . . . . . . . . . . . . . . 29
Part V. Symmetries and the Fract ional Noether theorem. . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1. Invariance of fractional functionals. . . . . .. . . . .. . . . . . . . . . . . . . . . .. . . . .. . . . . . . . .. . . . .. . . . . . . . . . . . . .. . . . .. . . . . . . 33
2. The fractional Noether theorem. . . . . . . .. . . . . . . . .. . . . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . .. . . . . . . 34
3. Toward fractional integrability and conservation laws. . .. . . . . . . . .. . . . . . . . .. . . . . . . . . . . . .. . . . . . . . .. . . . . . . . . . 35
Part VI. Fractional Hamiltonian systems. . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . 37
1. Hamil tonian systems and Legendre proper ty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .. . . . . . . . . . . . . . . . . . . . . . . 37
2. The fractional momentum and Hamiltonian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .. . . .. . . .. . . .. . . . . . . . . . . . . 38
3. Fractional Hamilton least-action principle. . . .. . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . .. . . . . . . . 39
Part VII. Fractional Ostrogradski formalism and Nonconservative dynamical systems. . . . . . . 41
1. Ostrogradski formalism and fractional embedding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2. Fractional Ostrogradski formalism... . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . 44
2.1. Space of variations and extremals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.2. The fractional generalized Euler-Lagrange equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.3. Coherence. . . . . .. . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . . . . .. 45
3. Nonconservative dynamical systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1. Linear friction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . 46
3.2. The Whittaker equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Part VIII. Frational embedding of continuous Lagrangian systems. . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. 49
1. Continuous Lagrangian systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .. . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . 49
2. Fractional continuous lagrangian systems. . . . . .. . . . . . . . .. . . . .. . . . . . . . .. . . . .. . . . . . . . . . . . . .. . . . . . . . . . . . . .. . .. 51
3. The fractional wave equation. . . . . . . . .. . . . . . . . .. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4. A r emark on fractional/classical equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5. The fractional diffusion equation. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 55
Conclusion and perspectives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 57
Notations.. . . . . .. . . . .. . . . . . . . . . . . . .. . . . .. . . . . . . . .. . . . .. . . . . . . .. . . . . . . . . . . . . .. . . . .. . . . . . . . .. . . . .. . . . . . . . . . . . . .. 59
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
FRACTIONAL EMBEDDING 3
”Il y a de l’apparence qu’on tirera un jour
des consequences bien utiles de ces paradoxes
(1)
,
car il n’y a gueres de paradoxes sans utilit´e.”
(2)
Leibniz, Letter to L‘Hospi tal, September 30, 1695
Comme la construction du monde est la plus parfaite possible
et qu’elle est due `a un Cr´eateur infiniment sage,
il n’arrive rien dans le monde qui ne pr´esente quelques propret´e de maximum ou minimum.
Aussi ne peut-on douter qu’il soit possible de eterminer tous les effets de l’Univers
par leurs causes finales, `a l’aide de la ethode des maxima et minima
avec tout autant de succ`es que par leurs causes efficientes.
Methodus inveniendi lineas curveas maximi minimive proprietate gaudentes
in Eulerii Opera omnia, S´erie I, 24,
Berlin-Basel-Boston-Stuttgart, Lipsiae-Birkhauser Verlag, 1911
INTRODUCTION
The aim of this paper is to introduce a general procedure called the fractional embedding
procedure, which roughly speaking allows us to associate a fractional analogue of a given
ordinary differential equation in a more or less canonical way. The fractional embedding
procedure is part of a global point of view on dynamical systems called embedding theories
of dynamical systems [15]. Before describing the general strategy underlying all embedding
theories and the particular fractional embedding procedure, we provide a set of problems
which lead us to our point of view:
- Turbulence: Fluid dynamics is modelled by par tial differential equations. Solutions
of these equations must be sufficiently smooth. However, there exits turbulent behavior
which correspond to very irregular trajectories. If the underlying equation has a physical
meaning, then one must give a sense to this equation on irregular functions. This remark
is the starting point of Jean L eray’s work on fluid mechanics [29]. He introduces what
he calls quasi-derivation and the notion of weak-solutions for PDE. This first work has a
long history and descendence going trough t he definition of Laurent Schwartz’s distribu-
tion and Sobolev s paces. We refer to [2] f or an overview of Jean Leray’s work in this domain.
(1)
Derivatives of non-integer order
(2)
It will lead to a paradox, from which one day useful consequences will be drawn.
4 JACKY CRESSON
- Deformation Quantization problems: The problem here is to go from classical mechan-
ics to quantum mechanics trough a deformation involving the Planck constant. Roughly
speaking, we have a one parameter h family of spaces and operators such that they
reduce to usual spaces and operators when h goes to zero. For example, we can look for
a deformation of the classical derivative using its algebraic characterization trough the
Leibniz rule. Another way is following L. Notta le [35] to assume that the space-time at
the atomic scale is a non-differentiable manifold. In that case, we obtain a one parameter
smooth deformation of space-time by smoothing at different scales. The main problem is
then to look for the deformation of the classical derivative during this process. We refer
to [16] for more details.
- Long term behavior of the Solar-system: The dynamics of the Solar system is usually
modelled by a n-body problem. However, the study of the long-term behavior must
include several perturbations terms, like tidal effects, perturbations due to the oblatness
of the sun, general relativity effects etc. We do not know the whole set of perturbations
which can be o f importance for the long term dynamics. In particular, it is not clear that
the remaining perturbations can be modelled using ordinary differential equations. Most
of stability results uses in the Solar systems dynamics make this a ssumption implicitly
[30]. An idea is to try to look for the dynamics of the initial equation on more general
objects like stochastic processes, by extending the ordinary derivative. Then one can look
for the stochastic perturbation of t he underlying stochastic equation which contains the
original one. As a consequence, we can provide a set of dynamical behaviors which have
a strong significance being stable under very general perturbation terms. This strategy is
developed in ([18] [20]) and applied in [21].
These problems although completely distinct have a common core: we need to extend
the classical derivative to a more general functional space. This extension being given, we
have a natural, but not always canonical , associated equation.
However, we are lead to two completely distinct theories depending on the nature of the
extension we make for the classical derivative.
For the Solar-system problem, we need to extend the classical derivat ive to stochastic
processes by imposing that t he new operator reduces to the classical derivative on differen-
tiable deterministic processes. In that case, the initial equation is present in the extended
FRACTIONAL EMBEDDING 5
one and we use the terminology of embedding theory. We describe the strategy with more
details in the next section.
For quantization problems, in particular first quantization of classical mechanics, we
have an extra parameter h (the Planck constant) and the extended operator denoted by
D
h
reduces to the classical derivative when h = 0. The initial equation is not contain in
the extended one, but we have a continuous deformation o f this equation dep ending on h.
We then use the t erminology of de f ormation theory. We give more details on this type of
theories in the following.
1. Embedding theories
The general scheme underlying embedding theories of ordinary or partial differential
equations is the following:
Fix a functional space F and a mapping ι : C
0
F.
Extend the ordinary derivative on F by impo sing a gluing rule.
Extend differential operators.
Extend ordinary or partial differential equations.
Let us denote by D the extended derivative o n F. The gluing rule impose that the extended
derivative reduces to the ordinary derivative on ι(C
1
), i.e. that we have Dι(x) = ι( ˙x), for
all x C
1
, where ˙x = dx/dt. As a consequence, the o riginal equation can be recovered
via the embedded equation by restricting the underlying f unctional space to ι(C
k
), k
depending on the order o f the original equation.
Most of the time, we have in mind applications to physics where Lagrangian systems
play a fundamental role. The importance of these systems is related to the fact that they
can be derived via a first principle, the least-action principle. Moreover, in some cases
of importance we can find a symmetric representation of these systems as Hamiltonian
systems. Hamiltonian systems are f undamental by many aspects. Of particular interest
is the fact that they can be quantized in order to obtain quantum analogues of classical
dynamical systems.
6 JACKY CRESSON
An embedding procedure can always be applied to Lagr angian systems. We obtain to
embed objects which are the embed action function al and the embed Euler-Lagrange equa-
tion. At this point, an embedding procedure can be considered as a particular quantization
procedure related to the underlying functional space. However, we have a more elabora t e
picture here. As we have an embed action functional, we can develop the associated cal-
culus o f variations, that we call the embed calculus of variations in what follows. We then
obtain an embed least-action principle with an associated Euler-Lagrange equation. As a
consequence, we have two kinds of embed Euler-Lagrange equations and it is not clear that
the one obtain directly from the embedding of the classical Euler-Lagrange equation and
the one obtain via the embed least-action principle coincide. This problem is recurrent in
all embedding theories of Lagrangian systems and is called the coherence prob l em. This
problem is far from being trivial in most of the already existing embedding theories, like
the stochastic one ([18] [20]) or the quantum o ne [17].
2. Emergence of fractional derivatives
In this paper, we develop an embedding theory of ordinary differential equations and
Lagrangian systems using fractional derivatives. Precise definitions will be given in section
I. We only give a n heuristic introduction for t hese operators and some basic problems
where they arise naturally.
A fractional derivative is an operator which gives a sense to a real power of the classical
differential operator d/dt, i.e. that we want to consider a n expression like
(.1)
d
α
dt
α
, α 0.
The previous problem appears for the first time in a letter from Leibniz to L’Hˆopital in
1695: Can the meaning of derivatives with integer order be generalized to derivatives
with non-integer orders ?”. Many mathematicians have contributed to this topic including
Leibniz, Liouville, Riemann , etc. We refer to [36] or [32] for a historical survey. A
number of definitions have emerged over the years including Riemann-Liouville fractional
derivative, Grunwald-Letnikov fractional derivative, Caputo fractional derivative, etc.
In this article we restrict our attention to the Riemann-Liouville fractional derivative,
although the embedding theory can be developed for an arbitrary given fractional calculus
with different technical difficulties.
FRACTIONAL EMBEDDING 7
The main difficulties when dealing with fractional derivatives are related to the following
properties:
(i) fractional differential operators are not local operators
(ii) the adjoint of a fractional differential operator is not the negative of itself
Property (i) is widely use in applications and explain part of the interest for these
operators to model phenomenon with long mem ory (see for example [12]).
Other problems arise during computations. Developing the fractional calculus of va r i-
ation and the associated results (the fr actional Noether theorem) we have encountered
difficulties linked with the following facts:
(i) the classical Le i b niz rule (f g)
= f
g + fg
is more complicated (see [40])
(ii) there exists no simple formula for the fractional analogue of the chain rule in
classical differential calculus (see [40])
This last difficulty is of special importance in the derivation of the f ractional Noether
theorem.
The fractional framework has been used in a wide variety of problems. We note in
particular applications in turbulence [10], chaotic dynamics [46] and quantization [34].
In this paper, we will frequently quote the work of F. Riewe ([38],[39]) which proposes
a fractional approach to nonconservative dynamical systems. The main property o f these
systems is that they induce an arrow of time due to irreversible dissipative effects. The
relation between fractional derivatives, nonconservative systems and irreversibility have
been discussed for example in the book [31].
Irreversibility implies that we look for the past P
t
and the future F
t
of a given given
dynamical process x(s), s R at time t, i.e. on the information P
t
= {x(s), a s t}
and F
t
= {x(s), t s b} where a and b can be chosen and depends on the amount of
information we are keeping from the past and the future. This induce the fact that we
look for two quantities, not yet defined that we denote by d
x(t) and d
+
x(t) from the
8 JACKY CRESSON
point of view o f derivatives.
The past and f uture information can be weighted, i.e. that we look not for x(s)
but to w(s, t)x(s) where w(s, t) give the importance of the information at time s with
respect to time t. This can be achieved using a weight
1
|ts|
α+1
and regularizing the
corresponding function. We then are lead to two quantities d
α
x(t) and d
α
+
x(t), which
represent a weighted information on the past and future behavior of the dynamical process.
The previous idea is well formalized by the left and right (Riemann-Liouville) derivatives
[40]. We refer to part I for precise definitions.
3. Deformation theories and the fractional framework
The fr actional framework follows the general strategy outline in the previous section.
However, a new ingredient comes into play which makes the fractional embedding different
from the existing stochastic or quantum embedding theories. We have used in this paper
the left and right Riemann-Liouville derivatives with different indices for the left and right
differentiation, i.e. we consider
a
D
α
t
and
t
D
β
b
. The extended operator depends naturally
on t hese two operators and is denoted D
α,β
. However, this operato r does not reduce to
the ordinary derivative on the set of differentiable functions. We recover the ordinary
derivative only when α = β = 1. As a consequence, we can associate to a given o rdinary
differential equation a two parameters family of fractional differe ntial equations. The
original equation being recovered for a special choice of these parameters, i.e. α = β = 1.
In that case, we propose to the use the terminology of fractional deformation and to keep
the terminology of fractional embedding for the procedure which associated a fractional
analogue of an ordinary differential equation.
Deformation theory can be formalized as follows:
A deformation theory is the data of:
A finite set P = {(p
1
, . . . , p
ν
), p
i
A} of parameters where A is a given interval of
R.
A ν-parameter family o f functional spaces F = {F
P
}
PA
ν
.
Operators D
P
defined on F
P
such that there exists P
0
A
ν
satisfying C
1
F
P
0
and
D
P
0
(x) = ˙x for x C
1
.
FRACTIONAL EMBEDDING 9
The condition on A is only here to be sure that we have a continuous dependance of the
whole construction o n the parameters.
The main difference between deformation and embedding lies in the fact t hat it is no
usually easy to obtain information on the initial equation from the deformed one. We must
use asymptotic methods, looking for the behavior of the deformed equation when p p
0
.
This is not the case for a tr ue embedding theory as the initial equation is already present
in the embedded one.
4. Plan of the paper
Our paper has the same architecture as our previous monograph [20] with ebastien
Darses about the stochastic embedding of dynamical systems. As a consequence, the
comparison between the two embedding procedure will be easier.
In part I we recall the definitions of the left and right fractional derivatives. We also
define left and right fractional derivatives which have satisfy a semi-group property and
the adequate functional spaces on which they are defined following a previous work of
Erwin and Roop [23]. We a lso recall a product rule formula.
In part II we define the fractional embedding of differential operators and ordinary
differential equations.
In part III we study the fractional embedding of La grangian systems. We obtain a
fractional analogue of the Euler-Lagrange equations.
In part IV we develop a fractional calculus of variations associated to the fractional
embedding of classical functionals. generalizing a previous work of O.P Agrawal [1] We
prove two versions of the least a ction principle depending on the underlying authorized
space of variations. We prove in particular a coherence theorem, which roughly speaking
state that the fractional embedding of the Euler-Lagrange equation coincide with the
fractional Euler-Lagrange equation obtained via the fractional least-action principle.
Part V study the behavior of symmetries under the fractional embedding procedure.
In particular, we prove a fractional Noether theorem which generalizes a recent result of
10 JACKY CRESSON
Frederico and Torres [24].
In part VI we derive the analo gue of the Hamilton formalism for our fractional La-
grangian systems.
In part VII we extend results of parts III and IV to cover the Ostrogradski formalism
for Lagrangian systems. In this unified framework, we recover classical results of F. Riewe
([38] [39]). Precisely, we obta in a fractional Lagrangian derivation of Nonconservative
systems.
In part VIII we study the fractional embedding of continuous Lagr angian systems.
In particular, we prove that the classical fractional wave equation introduced by W.R.
Schneider and W. Wyss [41] under ad-hoc assumptions, is the fractional embedding of the
classical wave equation which respects t he underlying continuous Lagrangian structure of
the equation. An analogous result is o bta ined for the fractional diffusion equation.
We then conclude with some open problems and perspectives.
FRACTIONAL EMBEDDING 11
PART I
FRACTIONAL OPE RATORS
In [1] Agrawal has studied Fractional varia t io na l problems using the Riemann-Liouville
derivatives. He notes that even if the initial functional problems only deals with the
left Riemann-Liouville derivative, the right Riemann-Liouville derivative appears naturally
during the computations. In this section, we construct an operator combining the left and
right Riemann-Liouville (RL) derivative. We remind some results concerning functional
spaces associated to the left and right RL derivative. In particular, we discuss the possibility
to obtain a law of exponents.
1. Fractional differential operators
1.1. Left and Right Riemann-Liouville derivatives. We define the left and right
Riemann-Liouville derivatives following ([36] [40] [37] [32]).
Definition I.1 (Left Riemann-Liouville Fractional integral)
Let x be a function defined on (a, b), and α > 0. T hen the left Riemann-Liouville
fra ctional integral of order α is defined to be
(I.1)
a
D
α
t
x(t) :=
1
Γ(α)
Z
t
a
(t s)
α1
x(s)ds.
Definition I.2 (Right Riemann-Liouville Fractional integral)
Let x be a function defined o n (a, b), and α > 0. Then the right Riemann-Liouville
fra ctional integral of order α is defined to be
(I.2)
t
D
α
b
x(t) :=
1
Γ(α)
Z
b
t
(s t)
α1
x(s)ds.
Left and right (RL) integrals satisfy some impor tant properties like the semi-group
property. We refer to [40] fo r more details.
Definition I.3 (Left and Right Riemann-Liouville fractional derivative)
Let α > 0, the le f t and right Riemann-Liouville deriv ative of o rder α, denoted by
a
D
α
t
and
t
D
α
b
res pectively, are defined by
(I.3)
a
D
α
t
x(t) =
1
Γ(n α)
d
dt
n
Z
t
a
(t s)
nα1
x(s)ds,
12 JACKY CRESSON
and
(I.4)
t
D
α
b
x(t) =
1
Γ(n α)
d
dt
n
Z
b
t
(t s)
nα1
x(s)ds,
where n is such that n 1 α < n.
If α = m, m N
, and x C
m
(]a, b[) we have
(I.5)
a
D
m
t
x =
d
m
x
dt
m
,
t
D
m
b
=
d
m
x
dt
m
.
This last relation which ensures the gluing of the left and right Riemann-Liouville (R L )
derivative to the classical derivative will be of fundamental impo rt ance in what follows.
If x(t) C
0
with left and right-derivatives at point t denoted by
d
+
x
dt
and
d
x
dt
respec-
tively then
(I.6)
a
D
m
t
x =
d
+
x
dt
,
t
D
m
b
=
d
x
dt
.
In what follows, we denote by
α
a
E, E
β
b
and
α
a
E
β
b
the functional spaces defined by
(I.7)
α
a
E = {x C([a, b]),
a
D
α
t
x exists}, E
β
b
= { x C([a, b]),
t
D
β
b
x exists},
and
(I.8)
α
a
E
β
b
=
α
a
E E
β
b
.
Remark I.1. O f course the set
α
a
E
β
b
is non-empty. Following ([40] Lemma 2.2 p.35)
we have AC([a, b])
α
a
E
β
b
, where AC([a, b]) is the set of absolutely continuous functions
on the interval [a, b] (see [40] Definition 1.2).
The operators of o rdinary differentiation of integer order satisfy a commutativity prop-
erty and the law of exponents (the semi-group property) i.e.
(I.9)
d
n
dt
n
d
m
dt
m
=
d
m
dt
m
d
n
dt
n
=
d
n+m
dt
n+m
.
These two properties in general fail to be satisfied by the left and right fractional RL
derivatives. We r efer to ([32] §.IV.6) and ([26] p.233) for more details and examples. These
bad properties are responsible for several difficulties in the study of fractional differential
equations. We refer to [37] for more details.
FRACTIONAL EMBEDDING 13
1.2. Left and right fractional derivatives. In some cases, we need that our frac-
tional operators satisfy additional properties like the semi-group property. Following [23]
we introduce the lef t and right fractional derivatives as well as convenient functional spaces
on which we have the semi-group property.
Definition I.4 (Left fractional derivative). Let x be a function defined on R, α >
0, n be the smallest integer greater than α (n 1 α < n), and σ = n α. Then the left
fra ctional derivative of order α is defined to be
(I.10) D
α
x(t) :=
D
α
t
x(t) =
d
n
dt
n
D
α
t
x(t) =
1
Γ(σ)
d
n
dt
n
Z
t
−∞
(t s)
σ1
x(s)ds.
Definition I.5 (Right fractional derivative). Let x be a function defined on R,
α > 0, n be the smallest integer greater than α (n 1 α < n), and σ = n α. Then the
right fractional derivative of order α is defined to be
(I.11) D
α
x(t) :=
t
D
t
x(t) = (1)
n
d
n
dt
n
t
D
α
x(t) =
(1)
n
Γ(σ)
d
n
dt
n
Z
t
(s t)
σ1
x(s)ds.
If
Supp(x) (a, b) we have D
α
x =
a
D
α
t
x and D
α
x =
t
D
α
b
x.
In [23] several useful functional spaces are introduced. Let I R be an open interval
(which may be unbounded). We denote by C
0
(I) the set of all functions x C
(I) that
vanish outside a compact subset K of I.
Definition I.6 (Left fractional derivative space). Let α > 0. Define the semi-
norm
(I.12) | x |
J
α
L
(R)
:=k D
α
x k
L
2
(R)
,
and norm
(I.13) k x k
J
α
L
(R)
:=
k x k
2
L
2
(R)
+ | x |
2
J
α
L
(R)
1/2
.
and let J
α
L
(R) denote the closure of C
0
(R) with respect to k · k
J
α
L
(R)
.
Similarly, we can defined the right f ractional deriva t ive space.
Definition I.7 (Right fractional derivative space). Let α > 0. Define the semi-
norm
(I.14) | x |
J
α
R
(R)
:=k D
α
x k
L
2
(R)
,
14 JACKY CRESSON
and norm
(I.15) k x k
J
α
R
(R)
:=
k x k
2
L
2
(R)
+ | x |
2
J
α
R
(R)
1/2
.
and let J
α
R
(R) denote the closure of C
0
(R) with respect to k · k
J
α
R
(R)
.
We now assume that I is a bounded open subinterval of R. We restrict the fractional
derivative spaces to I.
Definition I.8. Defi ne the spaces J
α
L,0
(I), J
α
R,0
(I) as the closure of C
0
(I) unde r their
res pective norms.
These spaces have very interesting properties with respect to D and D
. In particular,
we have the following semi-group property:
Lemma I.1. For x J
β
L,0
(I), 0 < α < β we have
(I.16) D
β
x = D
α
D
βα
x
and similarly for x J
β
R,0
(I),
(I.17) D
β
x = D
α
D
βα
x.
We refer to ([23] Lemma 2.9) for a proof.
The fractional derivative spaces J
α
L,0
(I) and J
α
R,0
(I) have been characterized when α > 0.
We denote by H
α
0
(I) the fractional Sobolev space.
Theorem I.1. Let α > 0. Then the J
α
L,0
(I), J
α
R,0
(I) and H
α
0
(I) spaces are equal.
We refer to ([23] Theorem 2.13) for a proof. In fact, when α 6= n 1/2, n N we have
a stronger result as the J
α
L,0
(I), J
α
R,0
(I) a nd H
α
0
(I) spaces have equivalent semi-norms and
norms.
2. The extension problem
As we want to deal with dynamical systems exhibiting the arrow of time, we need to
consider the operator
a
D
α
t
and
t
D
β
b
, in order t o keep tra ck of the past and future of the
dynamics. The fact that we consider α 6= β is only here for convenience. This can be used
to take into account a different quantity of inform a tion from the past and the future.
Let
a
D
α
t
and
t
D
β
b
be given. We look for an operator D
α,β
of the form
(I.18) D
α,β
= M(
a
D
α
t
,
t
D
β
b
),
FRACTIONAL EMBEDDING 15
where M : R
2
C is a mapping which does not dep ends on (α, β), satisfying the following
general principles:
i) Gluing property: If x(t) C
1
then when α = β = m, m N
, D
m,m
x(t) =
d
m
x
dt
m
.
ii) M is a R-linear mapping.
iii) Reconstruction: The mapping M is invertible.
Condition i) is fundamental in the embedding framework. It follows that all or con-
structions can be seen as a continuous two-parameters deformation of the corresponding
classical one
(3)
. This can be of importa nce dealing with the fractional quantization problem
in classical mechanics.
Condition ii) does not have a particular meaning. This is only the simp l est dependence
of the o perator D with r espect to
a
D
α
t
and
t
D
β
b
.
Condition iii) is important. It means that the data of D
α,β
on a given function x at point
t allows us to recover the left and right RL derivatives of x at t, so information about x in
a neighborhood of x(t).
Lemma I.2. O perators satisfying conditions i), ii) and iii) are of the form
(I.19) D
α,β
= [p
a
D
α
t
+ (p 1)
t
D
β
b
] + iq [
a
D
α
t
+
t
D
β
b
],
where p, q R an d q 6= 0.
Proof. By ii), we denote M(x, y) = px + qy + i(rx + sy), with p, q, r, s R. By i), we
must have with y = x corresponding to the o perator a choice of operato r s d/dt, d/dt
(I.20) p q = 1, r s = 0.
We then already have an operato r of the form (I.19). The reconstruction assumption only
impose that q 6= 0 in (I.19).
A more rigid form for these operators is obtained imposing a symmetry condition.
(3)
Condition i) is not the usua l condition underlying the stochastic or quantum embedding theories. In
general, we have an injective mapping ι from the set of differentiable functions C
1
in a bigger functional
space E such that the operator D that we define on E reduces to the classical deriva tive on ι(C
1
) meaning
that for x C
1
, D(ι(x)) = ι(x
(t)) wher e x
(t) = dx/dt. As a cons e quence, we have a true embedding
in this case, meaning that the embed theory already contain the cla ssical one via the mapping ι. Here,
the classical theo ry is not contained in the embedded theo ry but can be recovered by a continuous two-
parameters deformation.
16 JACKY CRESSON
iv) Let x C
0
be a real valued function possessing left and right classical derivatives
at point t, denoted by
d
+
x
dt
and
d
x
dt
respectively. If
d
+
x
dt
=
d
x
dt
, t hen we impo se
that
(I.21) D
1,1
x(t) = i
d
+
x
dt
.
Condition iv) must be seen as the non-differentiable pendant of condition i). Indeed,
condition i) can be rephrased as follows: if x C
0
is such that d
+
x and d
x exist and
satisfy d
+
x = d
x then D
1,1
x = d
+
x. Condition iv) is then equivalent to the commutativity
of the f ollowing diagram, where R
2
is seen as the R-vector space associate to C:
(I.22)
C
τ
C
x ix 7− x + ix
M M
C
τ
C,
a 7− ia
where τ : C C is defined by τ(z) = iz, z C and we have used the fact that M(x, x) =
0 following condition i).
Lemma I.3. T h e unique operator satisfying condition i), ii) , iii) and iv) is given by
(I.23) D
α,β
=
1
2
[
a
D
α
t
t
D
β
b
] + i
1
2
[
a
D
α
t
+
t
D
β
b
].
Proof. By iv), we must have p + (p 1) = 0 and 2q = 1, so that p = q = 1/ 2.
3. The fractional operator of order (α, β), α > 0, β > 0
Lemma I.3 leads us to the following definition of a fractional operator of order (α, β):
Definition I.9. For all a, b R, a < b, the fractional operator of order (α, β), α > 0,
β > 0, denoted by D
α,β
µ
, is defined by
(I.24) D
α,β
µ
=
1
2
h
a
D
α
t
t
D
β
b
i
+
1
2
h
a
D
α
t
+
t
D
β
b
i
,
where µ C.
When α = β = 1, we obtain D
α,β
µ
= d/dt.
The free parameter µ can be used to reduce the operator D
α,β
µ
to some special cases of
importance. Let us denoted by x(t) a given real valued function.
FRACTIONAL EMBEDDING 17
- For µ = i, we have D
α,β
µ
=
a
D
α
t
then dealing with an operator using the future state
denoted by F
t
(x) of the underlying function, i.e. F
t
(x) = {x(s), s [a, t[}.
- For µ = i, we obtain D
α,β
µ
=
t
D
β
b
then dealing with en operator using the past state
denoted by P
t
(x) of the underlying function, i.e. P
t
(x) = {x(s), s ]t, b]}.
As a consequence, our operato r can be used to deal with problems using
a
D
α
t
,
t
D
β
b
, o r
both operators in a unified way, only particularizing the value of µ at the end to recover
the desired f ramework.
When a = −∞ a nd b = , we denote the associated operator D
α,β
µ
by D
α,β
µ
, i.e.
(I.25) D
α,β
µ
=
1
2
D
α
D
β
+
1
2
D
α
+ D
β
,
where µ C.
4. Product rules
The classical product rule for Riemann-Liouville derivatives is for all α > 0
(I.26)
Z
b
a
a
D
α
t
f(t)g(t)dt =
Z
b
a
f(t)
t
D
α
b
g(t)dt,
as long a s f(a) = f(b) = 0 or g(a) = g(b) = 0.
This formula gives a strong connection between
a
D
α
t
and
t
D
α
b
via a generalized integra-
tion by part. This r elation is responsible for the emergence of
t
D
α
b
in problems of fractional
calculus of variations only dealing with
a
D
α
t
. See section 3 for more details. This result
also justifies our approach to the construction of a fractional operator which put on the
same level the left and right RL derivatives.
As a consequence, we obtain the following formula for our fractional operator:
Lemma I.4. For all f, g
α
a
E
β
b
, we ha ve
(I.27)
Z
b
a
D
α,β
µ
f(t)g(t)dt =
Z
b
a
f(t)D
β
µ
g(t)dt,
provide that f(a) = f(b) = 0 or g(a) = g(b) = 0.
18 JACKY CRESSON
Proof. We have
(I.28)
Z
b
a
D
α,β
µ
f(t)g(t)dt =
Z
b
a
f(f)
h
(
t
D
α
b
a
D
β
t
) + (
t
D
α
b
+
a
D
β
t
)
i
(g(t)) dt.
Exchanging the role of α and β in (
t
D
α
b
a
D
β
t
) + (
t
D
α
b
+
a
D
β
t
), we obtain the op erator
(
t
D
β
b
a
D
α
t
) + (
t
D
β
b
+
a
D
α
t
) which can be written as
(I.29)
h
(
a
D
α
t
t
D
β
b
) (
t
D
β
b
+
a
D
α
t
)
i
= −D
α,β
µ
.
This concludes the proof.
Here again, we see that it is convenient to keep the parameter µ free.
FRACTIONAL EMBEDDING 19
PART II
FRACTIONAL EMBEDDING OF DIFFERENTIAL OPERATORS
1. Fractional embedding of differential operators
Let d N be a fixed integer and a, b R, a < b be given. We denote by C([a, b]) the
set of continuous functions x : [a, b] R
d
. Let n N, we denote by C
n
([a, b]) the set of
functions in C([a, b]) which are differentiable up to order n.
Let f : R × C
d
C be a function, r eal valued on real arguments. We denote by F the
corresponding operator acting on functions x and defined by
(II.1) F :
C([a, b]) C([a, b])
x 7− f(, x()),
where f(, x()) is the function defined by
(II.2) f(, x()) :
[a, b] C,
t 7− f(t, x(t)).
Let f = {f
i
}
i=0,...,n
be a finite family of functions, f
i
: R × C
d
C, and F
i
, i = 1, . . . , n
the corresponding family of operators. We denote by O
f
the differential operator defined
by
(II.3) O
g
f
=
n
X
i=0
F
i
·
d
i
dt
i
G
i
,
where · is the standard product of operators, i. e . if A and B are two operators, we denote
by A · B the operator defined by (A · B)(x) = A(x)B(x) and the usual composition,
i.e. (A B)(x) = A(B(x)), with the convention that
d
dt
0
= Id, where Id denotes the
identity mapping on C.
Definition II.1 (Fractional embedding of operators). Let f = {f
i
}
i=0,...,n
and
g = {g
i
}
i=0,...,n
be finite families of functions, f
i
: R × C
d
C and g
i
: R × C
d
C
res pectively, and F
i
, G
i
, i = 1, . . . , n the corresponding families of operators, and O
g
f
the
associated diff e rential operator.
20 JACKY CRESSON
The (α, β)-fractional embedd i ng of O
g
f
written as (II.3), denoted by
α
a
Emb
β
b
(µ)(O
g
f
) is
defined by
(II.4)
α
a
Emb
β
b
(µ)(O
g
f
) =
n
X
i=0
F
i
·
D
α,β
µ
i
G
i
.
Note that the embedding procedure acts on operators of a given form and not on
operators like abstract data, i.e. this is not a mapping on the set of operators.
We can solve this indetermin acy using a formal representation of an operator.
Let f = {f
i
}
i=0,...,n
and g = {g
i
}
i=0,...,n
be finite families of functions, f
i
: R × C
d
C
and g
i
: R × C
d
C respectively, and F
i
, G
i
, i = 1, . . . , n the corresponding families of
operators. We denote by
O
g
f
the operator acting on E C
n
E defined by
(II.5)
O
g
f
=
n
X
i=0
F
i
d
i
dt
i
G
i
,
where is the standard tensor product.
We denote by O
the set of operators of the form (II.5) and O the set of differential
operators of the form (II.3). We define a mapping π from O
to O by
(II.6) π(
O
g
f
) = µ(
n
X
i=0
F
i
(D
α,β
µ
)
i
G
i
) =
n
X
i=0
F
i
· (D
α,β
µ
)
i
G
i
,
where µ is the projection µ : E E E E, µ(x y z) = x · (y z ).
A differential operator being given, its fractional emb edding depends on its writing as
an element of O
.
2. Fractional embedding of differential equations
Let k N be a fixed integer. Let f = {f
i
}
i=0,...,n
and g = {g
i
}
i=0,...,n
be finite families of
functions, f
i
: R × C
kd
C and g
i
: R × C
kd
C respectively, and F
i
, G
i
, i = 1, . . . , n the
corresponding families of operators. We denote by O
g
f
the operator acting on (C
n
[a, b])
k
defined by
(II.7) O
g
f
=
n
X
i=0
F
i
·
d
i
dt
i
G
i
,
FRACTIONAL EMBEDDING 21
The ordinary differential equation associated to O
g
f
is defined by
(II.8) O
g
f
(x,
dx
dt
, . . . ,
d
k
x
dt
k
) = 0, x C
n+k
([a, b]).
We then define the fractional embedding of equation (II.8) as follow:
Definition II.2. The fractional embedding of equation (II.8) of order (α, β), α, β > 0
is defin ed by
(II.9)
α
a
Emb
β
b
(µ) (O
g
f
) (x, D
α,β
µ
x, . . . ,
D
α,β
µ
k
x) = 0, x
α
a
E
β
b
(n + k).
Note that as long as the form of the operator is fixed the fractional embedding procedure
associates a unique fractional differential equation.
3. About time-reversible dynamics
The fractional embedding procedure associate a natural fractional counterpart to a given
ordinary differential equation. In some case, the underlying equation possesses specific
properties which have a physical meaning. One of this property is the time-reversib l e
character of the dynamics:
A dynamics on a space U is time-reversible if there exists an invertible map i of U such
that i
2
= Id, i.e. i is an involution, and if we denote by φ
t
the flow describing the dynamics
we have
(II.10) i φ
t
= φ
t
i,
meaning that if x(t) is a solution then i(x(t)) is also a solution of the underlying equation.
Time-Reversibility is closely r elated to a specific property of the classical derivative under
time-reversal:
(II.11)
d
dt
(x(t)) =
dx
dt
(t),
We define a notion of reversibility directly on operators:
Definition II.3. We denote by Rev the C- linear operator defined by Rev(
a
D
α
t
) =
t
D
α
b
, Rev(
t
D
α
b
) =
a
D
α
t
.
The action o f Rev on D
α,β
µ
is non-trivial:
Lemma II.1. We have Rev(D
α,β
µ
) = −D
β
µ
.
22 JACKY CRESSON
We then have the following analogue of reversibility in the fractional setting:
Definition II.4. Let O
g
f
be a differential opera tor of the form (II.7) such that the
dynamics of the associated differential equation (II.8) is time-revers i b l e, i.e.
(II.12) Rev(O
g
f
) = O
g
f
.
The fractional embedding
α
a
Emb
β
b
(µ) is called reversible if
(II.13) Rev(
α
a
Emb
β
b
(µ) (O
g
f
)) =
α
a
Emb
β
b
(µ) (O
g
f
) .
The main consequence of lemma II.1 is that there exists a unique way to do a fractional
embedding conserving the reversibility symmetry.
Theorem II.1. The reversibility symmetry is preserved by a fractional embedding if
and only i f α = β and µ = 0.
Proof. The reversibility symmetry is preserved if and o nly if we always have
Rev(D
α,β
µ
) = −D
α,β
µ
. By lemma II.1 this is only possible when µ = 0 and α = β.
In what follows we denote by R ev
α
the fractional embedding
α
a
Emb
α
b
(0).
FRACTIONAL EMBEDDING 23
PART III
FRACTIONAL EMBEDDING OF LAGRANGIAN SYSTEMS
In this section, we derive the fractional embedding of a particular class of ordinary differ-
ential equations called Euler-Lagrange equations which governs the dynamics of Lagrangian
systems.
1. Reminder about Lagrangian systems
Lagrangian systems play a central role in dynamical systems and physics, in particular
for classical mechanics. We refer to [3] for more details.
Definition III.1. An admissible Lagrangian function L is a function L : R×R
d
×C
d
7→
C such that L(t, x, v) is h olomorphic with respect to v, differentiable with respect to x and
real when v R.
A Lagrangian function defines a functional on C
1
(a, b), denoted by
(III.1) L
a,b
: C
1
(a, b) R, x C
1
(a, b) 7−
Z
b
a
L(s, x(s),
dx
dt
(s))ds, a, b R.
The classical calc ulus of va riations analyzes the behavior of L under small perturbations
of the initial function x. The main ingredient is a notion of differentiable functional and
extremals.
Definition III.2 (Space of variations). We d e note by Var(a, b) the set of func-
tions in C
1
(a, b) such that h(a) = h(b) = 0.
A functional L is differentiab l e at point x C
1
(a, b) if and only if
(III.2) L(x + ǫh) L(x) = ǫdL(x, h) + o(ǫ),
for ǫ > 0 and all h Var(a, b).
Using the notion of differentiability for functionals one is lead to consider extremum of
a given Lagrangian functional.
Definition III.3. An extremal for the functional L is a function x C
1
(a, b) such
that dL(x, h) = 0 for all h Var(a, b).
24 JACKY CRESSON
Extremals of the functional L can be characterized by an ordinary differential equation
of order 2, called the Euler-Lagrange equation.
Theorem III.1. The ex tremals of L coincide with the solutions of the Euler-Lagrange
equation denoted by (EL) and defin ed by
(III.3)
d
dt
L
v
t, x(t),
dx
dt
(t)

=
L
x
t, x(t),
dx
dt
(t)
.
This equation can be seen as the action of the differential operator
(III.4) O
(EL)
=
d
dt
L
v
L
x
on the couple (x(t),
dx
dt
(t)).
2. Fractional Euler-Lagrange equation
The fractional embedding procedure allows us to define a natural extension o f the clas-
sical Euler-Lagrange equation in the fractional context. The main result of this section
is:
Theorem III.2. Let L be an admissibl e Lagrangian function. The
α
a
Emb
β
b
(µ)-
fra ctional Euler-La g range equation associated to L is give n by
D
α,β
µ
L
v
t, x(t), D
α,β
µ
x(t)
=
L
x
t, x(t), D
α,β
µ
x(t)
. (FEL
µ
α,β
)
In what follows, we will simply speak about the f r actional Euler-Lagrange equation
when there is no confusion on the underlying embedding procedure.
The proof is based on the following lemma:
Lemma III.1. Let L be an admissible La grangi an function. The f ractional embedding
of the Euler-Lagra nge differential operator O
(EL)
is given by
(III.5)
α
a
Emb
β
b
(µ)(O
(EL)
) = D
α,β
µ
L
v
L
x
.
Proof. The operator (III.4) is first considered as acting on (C
1
([a, b])
2
, i.e. for all
(x(t), y(t)) C
1
[a, b] × C
1
[a, b] we have
(III.6) O
(EL)
(x(t), y(t)) =
d
dt
L
v
(t, x(t), y(t))
L
x
(t, x(t), y(t)).
FRACTIONAL EMBEDDING 25
This operator is of t he form O
g
f
with
(III.7) f =
1,
L
x
,
and
(III.8) g =
L
v
, 1
,
where 1 : R × C
2
C is the constant function 1(t, x, y) = 1. As a consequence, O
(EL)
is
given by
(III.9) O
(EL)
= 1 ·
d
dt
L
v
L
x
· Id 1,
with the convention that
d
dt
0
= Id. We then obtain equation (III.5) using definition
II.1.
We can now conclude the proof of theorem III.2 using definition II.2. The fractional
embedding of equation (II I.3) is given by
(III.10)
α
a
Emb
β
b
(µ)
O
(EL)
x, D
α,β
µ
x
= 0 ,
which reduces to equation (FEL
µ
α,β
) thanks to lemma III.1.
3. The c oherence problem
The fractional embedding procedure allows us to define a nat ural fractional analogue of
the Euler-Lagrange equation. This result is satisfying because the procedure is fixed. How-
ever, Lagrangian systems possess very special features. In particular, the classical Euler-
Lagrange equation can be obtained using a variational p rinciple, called the least-action
principle and denoted LAP. The least action principle asserts that the Euler-Lagrange
equation characterizes the extremals of a given functional associated to the Lagrangian.
We then are lead to the following problem:
i) Develop a calculus of variation on fractional functionals.
ii) State the corresponding fractional least-action principle, in particular explicit the
associated fractional Euler-Lagrange equation denoted by FEL
flap
.
iii) Compare the result with the embedded Euler-Lagrange equation (FEL
µ
α,β
)
26 JACKY CRESSON
An embedding procedure is called coherent when the two Euler-Lagrange equations are the
same, i.e. if
(III.11) FEL
flap
=
α
a
Emb
β
b
(µ)(EL),
assuming that FEL
flap
is obtained from the embedding of the classical functional using
the same embedding procedure.
As we will see, an embedding procedure is not always coherent. Although we obtain in
general equations of the same form, we usually have some torsion between the embedding
of the f unctional and the Euler-Lagrange equation which cancel o nly in particular cases.
The fractional calculus of variations is developed in §.IV as well as the corresponding
fractional least-action principle. The coherence of fractional embedding procedures is dis-
cussed in §.4.
FRACTIONAL EMBEDDING 27
PART IV
FRACTIONAL CALCULUS OF VARIATIONS
This section is devoted to the fractional calculus of variations using our fractional op-
erator. The functional is obta ined under the f r actional emb edding procedure. We refer to
the work of O.P. Agrawal [1] for related results.
1. Fractional functional
Let L be an admissible Lagrangian function on R × R
d
× C
d
, d 1, and L the associated
functional. Using the Fractional embedding procedure
α
a
Emb
β
b
(µ), we define a natural
Fractional functional associated to L.
We denote by
α
a
E
β
b
the set of functions x such that
a
D
α
t
x and
t
D
β
b
x are defined.
Definition IV.1. The Fractional functional associated to L is defined by
(IV.1) L
α,β
a,b
:
α
a
E
β
b
R, x
α
a
E
β
b
7−
Z
b
a
L(s, x(s), D
α,β
µ
x(s))ds, a, b R;
The extension property implies that L
α,β
a,b
reduce to the classical functional L
a,b
when
α = β = 1.
2. Space of variations and extremals
Let us denote by
α
a
E
β
b
(0, 0) the set of curves h
α
a
E
β
b
satisfying h(a) = h(b) = 0 and
a
E
α
b
:=
α
a
E
α
b
. We denote by Var
α
(0, 0) the set defined to be
(IV.2) Var
α
(a, b) = {h
a
E
α
b
, h(a) = h(b) = 0 and
a
D
α
t
h =
t
D
α
b
h }.
We denote by P the set
α
a
E
β
b
(0, 0) or Var
α
(a, b).
Definition IV.2. Let x be a given curve. A P-variation of x is a one- parameter ǫ R
family of curves of the form
(IV.3) y
ǫ
= x + ǫh, h P.
A notion of differentiability can now be defined for fractional functionals. In the follow-
ing, we write L
a,b
indifferently for L
α,β
a,b
when P =
α
a
E
β
b
(0, 0) and L
α,α
a,b
when P = Var
α
(a, b).
28 JACKY CRESSON
Definition IV.3. Let L be an admissible Lagrangian function and L
a,b
the associated
fra ctional functional. The functional L
a,b
is called P-differentiable at x if
(IV.4) L
a,b
(x + ǫh) L
a,b
(x) = ǫdL
a,b
(x, h) + o(ǫ),
for all h P, ǫ > 0, where dL
a,b
(x, h) is a linear functional of h.
The linear functional dL
a,b
(x, h) is called the P-differential of the fractional functional
L
a,b
at point x.
An extre mal for L
a,b
is then defined by:
Definition IV.4. A P-extremal for the f unc tion al L
a,b
is a function x such that
dL
a,b
(x, h) = 0 for all h P.
The following lemma gives the explicit form of the differential of a fractional functional:
Theorem IV.1. Let L be an admissible Lagrangian function and L
α,β
a,b
the a ssoc i ated
fra ctional functional. T he f unc tion al L
α,β
a,b
is differen tiab l e at any x
α
a
E
β
b
(x
a
, x
b
) and for
all h
α
a
E
β
b
(0, 0) the diffe rential is given by
(IV.5) dL
α,β
a,b
(x, h) =
Z
b
a
−D
β
µ
L
v
t, x(t), D
α,β
µ
x(t)
+
L
x
(t, x(t), D
α,β
µ
x(t))
h(t)dt.
Proof. As the left and right RL derivatives are linear operato r s we have
(IV.6) D
α,β
µ
(x + ǫh) = Dx + ǫ D
α,β
µ
h.
As a consequence, we obtain
(IV.7) L
α,β
a,b
(x + ǫh) =
Z
b
a
L(s, x(s) + ǫh(s), D
α,β
µ
x(s) + ǫ D
α,β
µ
h(s))ds
which implies, doing a Taylor expansion of L(s, x(s) + ǫh(s), D
α,β
µ
x(s) + ǫ D
α,β
µ
h(s)) in ǫ
around 0
(IV.8)
L
α,β
a,b
(x + ǫh) = ǫ
Z
b
a
L
x
(s, x(s), D
α,β
µ
x(s))h(s) +
L
v
(s, x(s), D
α,β
µ
x(s))D
α,β
µ
h(s)
ds
+L
α,β
a,b
(x) + o (ǫ).
Using the product rule (I.27) we obtain
(IV.9)
Z
b
a
L
v
(s, x(s), D
α,β
µ
x(s))D
α,β
µ
h(s)ds =
Z
b
a
D
β
µ
L
v
(s, x(s), D
α,β
µ
x(s))h(s)ds.
Replacing this expression in (IV.8), we deduce formula (IV.5).
FRACTIONAL EMBEDDING 29
3. The fractional Euler-Lagrange equation
We obtain the fo llowing analogue o f the least-action p ri nciple in classical Lagrangian
mechanics:
Theorem IV.2 (Fractional least-action principle). Let L[x] be a functional of the
form
(IV.10) L[x] =
Z
b
a
L(s, x(s), D
α,β
µ
x(s))ds
defined on
α
a
E
β
b
(x
a
, x
b
).
A necessary and sufficient condition for a given function x
α
a
E
β
b
to be a
α
a
E
β
b
-extremal
for L[x] with fixed end po i nts x(a) = x
a
, x(b) = x
b
, is that it satisfies the fractional
Euler-Lagrange equation (FEL):
(IV.11) D
β
µ
L
v
t, x(t), D
α,β
µ
x(t)
=
L
x
(t, x(t), D
α,β
µ
x(t)).
Note that this equation is different from the one obtained via the fra ctional embedding
procedure.
Proof. Using the classical D u Bois Reymond lemma ([4],p.108) and theorem IV.1 we
obtain (IV.11).
The weak analogue using the space of variation Var
α
(a, b) is given by:
Theorem IV.3 (Weak fractional least-action principle). Let L[x] be a func-
tional of the form
(IV.12) L[x] =
Z
b
a
L(s, x(s), D
α
µ
x(s))ds
defined on
a
E
α
b
(x
a
, x
b
).
A necessary and sufficient condition f or a given function x
a
E
α
b
to be a Var
α
(a, b)-
extremal for L[x] with fixed end points x(a) = x
a
, x(b) = x
b
, i s that it satisfies the fractional
Euler-Lagrange equation FEL
α,α
µ
We denote FEL
α
µ
for FEL
α,α
µ
in the following.
4. Coherence
The coherence problem can now b e studied in details. We have the following theorem,
which is only a rewriting of theorem IV.2 and definition III.2:
30 JACKY CRESSON
Theorem IV.4. Let L be an a dmissible Lagrangian function, a, b R, a < b, α, β > 0,
then the following dia gram commutes:
(IV.13)
L(t, x(t), dx/dt)
α
a
Emb
β
b
(µ)
L(t, x(t), D
α,β
µ
x)
LAP FLAP
EL
β
a
Emb
α
b
(µ)
FEL
µ
β
.
Theorem IV.4 is not a coherence result in the spirit of ([