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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 propri´et´e de maximum ou minimum.

Aussi ne peut-on douter qu’il soit possible de d´eterminer tous les eﬀets de l’Univers

par leurs causes ﬁnales, `a l’aide de la m´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

|t−s|

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

}

P∈A

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 S´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)

α−1

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)

α−1

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)

n−α−1

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)

n−α−1

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)

σ−1

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)

σ−1

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

β−α

x

and similarly for x ∈ J

β

R,0

(I),

(I.17) D

β

∗

x = D

α

∗

D

β−α

∗

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

+ iµ

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

∗

+ iµ

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,α

−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

) + iµ(

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

) + iµ(

t

D

α

b

+

a

D

β

t

), we obtain the op erator

(

t

D

β

b

−

a

D

α

t

) + iµ(

t

D

β

b

+

a

D

α

t

) which can be written as

(I.29) −

h

(

a

D

α

t

−

t

D

β

b

) − iµ(

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,α

−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,α

−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,α

−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,α

−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,α

.

Theorem IV.4 is not a coherence result in the spirit of ([