Non-differentiable variational principles

NON DIFFERENTIABLE VARIATIONAL PRINCIPLES by Jacky CRESSON

Abstract. — We developp a calculus of variations for functionals which are defined on a set of non differentiable curves. We first extend the classical differential calculus in a quantum calculus, which allows us to define a complex operator, called the scale derivative, which is the non differentiable analogue of the classical derivative. We then define the notion of extremals for our functionals and obtain a characterization in term of a generalized Euler-Lagrange equation. We finally prove that solutions of the Schr¨ odinger equation can be obtained as extremals of a non differentiable variational principle, leading to an extended Hamilton’s principle of least action for quantum mechanics. We compare this approach with the scale relativity theory of Nottale, which assumes a fractal structure of space-time. R´ esum´ e (Principes variationnels non diff´ erentiables). — Nous d´eveloppons un calcul des variations pour des fonctionnelles d´efinies sur un ensemble de courbes non diff´erentiables. Pour cela, nous ´etendons le calcul diff´erentiel classique, en calcul appel´e calcul quantique, qui nous permet de d´efinir un op´erateur ` a valeur complexes, appel´e d´eriv´ee d’´echelle, qui est l’analogue non diff´erentiable de la d´eriv´ee usuelle. On d´efinit alors la notion d’extremale pour ces fonctionnelles pour lesquelles nous obtenons une caract´erisation via une ´equation d’Euler-Lagrange g´en´eralis´ee. On prouve enfin que les solutions de l’´equation de Schr¨ odinger peuvent s’obtenir comme solution d’un probl`eme variationnel non diff´erentiable, ´etendant ainsi le principe de moindre action de Hamilton au cadre de la m´ecanique quantique. On discute enfin la connexion entre ce travail et la th´eorie de la relativit´e d’´echelle d´evelopp´ee par Nottale, et qui suppose une structure fractale de l’espace-temps.

Key words and phrases. — Non differentiable functions - variational principle - leastaction principle - Schr¨ odinger’s equation.

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JACKY CRESSON

Contents 1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

2. Quantum calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

3. Non differentiable calculus of variations. . . . . . . . . . . . . . . . . . .

7

4. Proof of lemma 3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5. Application :

least action principle and non-linear

Schr¨odinger equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1. Introduction Lagrangian mechanics describes motion of mechanical systems using differentiable manifolds. Motions of Lagrangian systems are extremals of a variational principle called “Hamilton’s principle of least action” (see [1],p.55). However, some important physical systems can’t be put in such a framework.

For example, generic trajectories of quantum mechanics are not

differentiable curves [12], such that a classical Lagrangian formalism is not possible (see however [11]). In this article we extend the calculus of variations in order to cover sets of non differentiable curves. We first define a quantum calculus allowing us to analyze non differentiable functions by means of a complex operator, which generalizes the classical derivative. We then introduce functionals on H¨olderian curves and study the analogue of extremals for these objects. We prove that extremals curves of our functionals are solutions of a generalized Euler-Lagrange equation, which looks like the one obtain by Nottale [17] in the context of the scale relativity theory. We then prove that the Schr¨ odinger equation can be obtain as extremals of a non differentiable variational problem.

NON DIFFERENTIABLE VARIATIONAL PRINCIPLES

3

The non differentiable calculus of variations gives a rigorous basis to the scale relativity principle developped by Nottale [17] in order to recover quantum mechanics by keeping out the differentiability assumption of the spacetime. 2. Quantum calculus In this section we define the quantum calculus, which extends the classical differential calculus to non differentiable functions. We refer to [4] and [9] for analogous ideas and the underlying physical framework leading to this extension. 2.1. Basic definitions. — We denote by C 0 the set of continuous real valued functions defined on R. Definition 2.1. — Let f ∈ C 0 . For all ² > 0, we call ² left and right quantum derivatives the quantities f (t + σ²) − f (t) , σ = ±. ² The ² left and right quantum derivatives of a continuous function correspond

(1)

∆σ² f (t) = σ

to the classical derivatives of the left and right ²-mean function defined by Z σ t+σ² σ (2) f² (t) = f (s)ds, σ = ±. ² t Using ² left and right derivatives, we can define an operator which generalize the classical derivative. Definition 2.2. — Let f ∈ C 0 . For all ² > 0, the ² scale derivative of f at point t is the quantity denoted by 2² f /2t, and defined by 2² f − + − (3) (t) = (∆+ ² f (t) + ∆² f (t)) − i(∆² f (t) − ∆² f (t)). 2t If f is differentiable, we can take the limit of the scale derivative when ² goes to zero. We then obtain the classical derivative of f , f 0 . In the following, we will frequently denote 2² x for 2² x/2t.

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JACKY CRESSON

We also need to extend the scale derivative to complex valued functions. Definition 2.3. — Let f be a continuous complex valued function. For all ² > 0, the ² scale derivative of f , denoted by 2² f /2t is defined by 2² f 2² Re(f ) 2² Im(f ) (t) = +i , 2t 2t 2t

(4)

where Re(f ) and Im(f ) denote the real and imaginary part of f . This extension of the scale derivative in order to cover complex valued functions is far from being trivial. Indeed, it mixes complex terms in a complex operator. 2.2. Basic formulas. — For all ² > 0 the scale derivative is not a derivation(1) on the set of continuous functions(2) Indeed, we have : Theorem 2.1. — Let f and g be two functions of C 0 . For all ² > 0 we have (5) 2² (f g) = 2² f.g + f.2² g + ²i [2² f ¯² g − ¯² f 2² g − 2² f 2² g − ¯² f ¯² g] , where ¯f is the complex conjugate of 2f . Of course, when we restrict our attention to differntiable functions, taking the limit of (5) when ² goes to zero, we obtain the classical Leibniz rule (f g)0 = f 0 .g + f.g 0 .

Proof. — Formula (5) follows from easy calculations. In particular, we use the fact that (6)

∆²σ (f g) = ∆²σ f.g + f.∆²σ g + σ²∆²σ f.∆²σ g, σ = ±,

which is a standard result of the calculus of finite differences (see [14]).

(1)

We recall that a derivation on an abtract algebra A is a linear application D : A → A such that D(xy) = D(x).y + x.D(y) for all x, y ∈ A. (2) A classical result says that there exists no derivations on the set of continuous functions except the trivial one, define by D(f ) = 0 for all f ∈ C 0 .

NON DIFFERENTIABLE VARIATIONAL PRINCIPLES

5

As a consequence, we have (7)

£ ¤ − − − + − − − 2² (f g) = 2² f.g+f.2² g+² (∆+ ² f ∆² g − ∆² f ∆² g) − i(∆² f ∆² g + ∆² f ∆² g) . Moreover, we have the following formula : ¤ 1£ + − + + + − − (8) 2² f 2² g = (∆² f ∆² g + ∆− ² f ∆² g) − i(∆² f ∆² g − ∆² f ∆² g) , 2 and ¤ 1£ + + − + − − + (9) 2² f ¯² g = (∆² f ∆² g + ∆− ² f ∆² g) − i(∆² f ∆² g − ∆² f ∆² g) . 2 We then obtain + − − ∆+ = 2² f ¯² g + ¯² f 2² g, ² f ∆² g + ∆² f ∆² g (10) + + − −i(∆² f ∆² g − ∆² f ∆− g) = 2² f 2² g − ¯² f ¯² g. ² We deduce then the following equality (11)

− − − + − − − (∆+ ² f ∆² g − ∆² f ∆² g) − i(∆² f ∆² g + ∆² f ∆² g) = i(2² f 2² g − ¯² f ¯² g) − i(2² f ¯² g + ¯² f 2² g).

This concludes the proof. We have the following integral formula : ¯ Z b ¢ ¡ + ¢¤¯b 1 £¡ + − − f² (t) + f² (t) − i f² (t) − f² (t) ¯¯ . (12) 2² f (t)dt = 2 a a When ² goes to zero, we deduce Z b 2² f (t)dt = f (t)|ba . (13) lim ²→0 a

2.3. H¨ olderian functions. — In the following, we consider a particular class of non differentiable functions called H¨ olderian functions [22]. Definition 2.4. — A continuous real valued function f is H¨ olderian of H¨ older exponent α, 0 < alpha < 1, if for all ² > 0, and all t, t0 ∈ R such that | t − t0 |≤ ², there exists a constant c such that (14)

| f (t) − f (t0 ) |≤ c²α .

In the following, we denote by H α the set of continuous functions which are H¨olderian of H¨older exponent α. Moreover, we say that a complex valued function y(t) belongs to H α if its real and imaginary part belong to H α .

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JACKY CRESSON

We then have the following lemma : Lemma 2.1. — If x ∈ H α then 2² x ∈ H α for all ² > 0. This follows from the definition of 2² x(t) and simple calculations. 2.4. A technical result. — We derive a technical result about the scale derivative, which will be used in the last section. Theorem 2.2. — Let f (x, t) be a C n+1 function and x(t) ∈ H 1/n , n ≥ 1. For all ² > 0 sufficiently small, we have n

2² f (x(t), t) ∂f X 1 ∂ j f = + (x(t), t)²j−1 a²,j (t) + o(²1/n ), 2t ∂t j! ∂xj

(15)

j=1

where (16)

a²,j (t) =

¢ ¡ ¢¤ 1 £¡ ² j (∆+ x) − (−1)j (∆²− x)j − i (∆²+ x)j + (−1)j (∆²− x)j . 2

The proof follows easily from the following lemma : Lemma 2.2. — Let f (x, t) be a real valued function of class C n+1 , n ≥ 1, and x(t) ∈ H 1/n . For all ² > 0 sufficiently small, the right and left quantum derivatives of f (x(t), t) are given by n

(17)

∆²σ f (x(t), t)

X 1 ∂if ∂f (x(t), t)+σ = (x(t), t)²−1 (σ²∆²σ x(t))i +o(²1/n ), ∂t i! ∂xi i=1

for σ = ±. Proof. — This follows from easy computations. First, we remark that, as x(t) ∈ H 1/n , we have | ²∆²σ X(t) |= o(²1/n ). Moreover, f (x(t + ²), t + ²) = f (x(t) + ²∆²+ x(t), t + ²). By the previous remark, and the fact that f is of order C n+1 , we can make a Taylor expansion up to order n with a controled remainder. n X 1 X ∂kf f (x(t + ²), t + ²) = f (x(t), t) + (²∆²+ x(t))i ²j i j (x(t), t) k! ∂ x∂ t k=1

i+j=k

+o((²∆²+ x(t))n+1 ).

NON DIFFERENTIABLE VARIATIONAL PRINCIPLES

7

As a consequence, we have n X ∂kf 1 X (²∆²+ x(t))i ²j i j (x(t), t) + o((²∆²+ x(t))n+1 ). ²∆²+ f (x(t), t) = k! ∂ x∂ t k=1

i+j=k

By selecting terms of order less or equal to one in ² in the right of this equation, we obtain

"

n

X 1 ∂if ∂f ²∆²+ f (x(t), t) = ² (x(t), t) + (x(t), t)²−1 (²∆²+ x(t))i ∂t i! ∂xi i=1 +o(²2 ∆²+ x(t)).

#

Dividing by ², we obtain the lemma. 3. Non differentiable calculus of variations 3.1. Functionals. — The classical calculus of variations is concerned with the extremals of functions whose domain is an infinite-dimensional space : the space of curves, which is usually the set of differentiable curves. We look for an analogous theory on the set of non differentiable curves. In all the text, α is a real number satisfying 0 < α < 1, and ² is a parameter, which is assumed to be sufficiently small, i.e. 0 < ² 0, a functional Φ² : C α (a, b) → C is defined by Z (19)

Φ² (γ) =

b

a

L(x(t), 2² x(t), t)dt,

for all γ ∈ C α (a, b). Of course, when we consider differentiable curves, we can take the limit of (19) when ² goes to zero, and we obtain the classical functional (see [1],p.56) : Z b (20) Φ(γ) = L(x(t), x(t), ˙ t)dt, a

where x˙ = dx/dt. 3.2. Variations. — We first define variations of curves. Definition 3.2. — Let γ ∈ C α (a, b). A variation γ 0 of γ is a curve (21) γ 0 = {(t, x(t)+h(t)), x ∈ H α , h ∈ H β , β ≥ α1[1/2,1] +(1−α)1]0,1/2[ , h(a) = h(b) = 0}. We denote this curve by γ 0 = γ + h. As in the usual case, we look for paths of a given regularity class with prescribed end points. The condition β ≥ α1[1/2,1] + (1 − α)1]0,1/2[ for the variation is a technical assumption, which will be used in the derivation of the non differentiable analogue of the Euler-Lagrange equation (see §.3.3). The minimal condition on β for which the problem of variations makes sense is β ≥ α, in order to ensure that γ + h is again in C α (a, b). In the following, we always consider variations of a given curve γ of the form γµ = γ + µh, where µ is a real parameter.

NON DIFFERENTIABLE VARIATIONAL PRINCIPLES

9

Definition 3.3. — A functional Φ is called differentiable on C α (a, b) if for all variations h ∈ C β (a, b), we have (22)

Φ(γ + h) − Φ(γ) = F (γ, h) + R(γ, h),

where F depends linearly on h, i.e. F (h1 + h2 ) = F (h1 ) + F (h2 ) and F (ch) = cF (h), and R(γ, h) = O(h2 ), i.e. for | h |< µ and | 2² h |< µ, we have | R |< Cµ2 . The functional F is called the differential of Φ. In the case of functionals of the form (19), we have : Theorem 3.1. — For all ² > 0, the functional Φ² (γ) defined by (19) is differentiable, and its derivative is given by the formula (23)

Z b·

µ ¶¸ ∂L 2² x 2² ∂L 2² x = (x(t), , t) − (x(t), , t) h(t)dt 2t ¶ 2t ∂2² x 2t aZ ∂x µ b 2² ∂L + h(t) dt + iR²γ (h), 2t ∂2 x ² a

F²γ (h)

with (24) R²γ (h) = ²

Z a

b

[2² f² (t)2² h(t) − ¯² f² (t)2² h(t) − 2² f² (t) ¯² h(t) − ¯² f² (t) ¯² h(t)] dt,

where (25)

f² (t) =

∂L (x(t), 2² x(t), t). ∂2² x

Proof. — We have Φ (γ + h) − Φ(γ) = Z ²b [L(x(t) + h(t), 2² x(t) + 2² h(t), t) − L(x(t), 2² x(t), t)] dt, aZ · ¸ b ∂L ∂L = (x(t), 2² x(t), t)h(t) + (x(t), 2² x(t), t)2² h(t) dt + O(h2 ), ∂x ∂2 x ² a = F²γ (h) + R(h), where F²γ (h)

Z b· = a

¸ ∂L ∂L (x(t), 2² x(t), t)h(t) + (x(t), 2² x(t), t)2² h(t) dt, ∂x ∂2² x

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JACKY CRESSON

and R(h) = O(h2 ). Using (5), we deduce : (26) F²γ (h) =

Z b·

∂L 2² x 2² (x(t), , t) − ∂x 2t 2t aZ µ ¶ b 2² ∂L + h(t) dt ∂2² x aZ 2t

µ

¶¸ ∂L 2² x (x(t), , t) h(t)dt ∂2² x 2t

b

+i² a

with f² (t) =

[2² f² (t)2² h(t) − ¯² f² (t)2² h(t) − 2² f² (t) ¯² h(t) − ¯² f² (t) ¯² h(t)] dt,

∂L (x(t), 2² x(t), t). This concludes the proof. ∂2² x

3.3. Extremal curves and Euler-Lagrange equation. — The functional derivative of Φ² mix terms which are either divergent when ² goes to zero, or tending toward 0 with ². In order to simplify our problem and to take into account only dominant terms in ², we introduce the following operator : Definition 3.4. — Let ap (²) be a real or complex valued function, with parameters p. We denote by [.]² the linear operator defined by : i. ap (²) − [ap (²)]² −→²→0 0, ii. [ap (²)]² = 0 if lim²→0 ap (²) = 0. The quantity [ap (²)]² is called the ²-dominant part of ap (²). For example, if a(²) = ²−1/2 + 2² + 2, then [a(²)]² = ²−1/2 + 2. We deduce the following properties : Lemma 3.1. — The ²-dominant part is unique. Proof. — This comes from the relation [[.]² ]² = [.]² . Indeed, by definition we have ap (²) = [ap (²)]² + r(²) with lim²→0 r(²) = 0. Applying [.]² directly on this expression, we obtain [ap (²)]² = [[ap (²)]² ]² using ii.

NON DIFFERENTIABLE VARIATIONAL PRINCIPLES

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Remark 3.3. — Unicity comes from condition ii. Indeed, if we cancel this condition, we can obtain many different quantities satisfying i. For example, if a(²) = α²−1/2 + ² + 2, then without ii), we have the choice between [a(²)]² = ²−1/2 + 2 + ² and [a(²)]² = ²−1/2 + 2. ii. This operator can be used in the definition of left and right quantum operators by considering δ²σ x(t) = [∆σ² x(t)]² , σ = ±. However, using such kind of operators lead to many difficulties from the algebraic point of view, in particular with the derivation of the analogue of the Leibniz rule. We now introduce the non differentiable analogue of the notion of extremals curves in the classical case (see [1],p.57 ). Definition 3.5. — Let 0 < α ≤ 1. An extremal curve of the functional (19) on the space of curves of class C β (a, b), β ≥ α1[1/2,1] + (1 − α)1]0,1/2[ , is a curve γ ∈ C α (a, b) satisfying [F²γ (h)]² = 0,

(27)

for all ² > 0 and all h ∈ C β (a, b). The following theorem gives the analogue of the Euler-Lagrange equations for extremals of our functionals. Theorem 3.2. — We assume that the function L defining the functional (19) satisfies (28)

k D(∂L/∂v) k≤ C,

where C is a constant, D denotes the differential, and k . k is the classical norm on matrices. The curve γ : x = x(t) is an extremal curve of the functional (19) on the space of curves of class C β (a, d), (29)

β ≥ α1[1/2,1] + (1 − α)1]0,1/2[ ,

if and only if it satisfies the following generalized Euler-Lagrange equation µ ¶¸ · 2² x 2² ∂L 2² x ∂L (x(t), , t) − (x(t), , t) = 0, (30) ∂x 2t 2t ∂2² x 2t ² for ² > 0.

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JACKY CRESSON

Remark 3.4. — Our Euler-Lagrange equation (30) looks like the one obtain by Nottale [17] in the context of the scale relativity theory (see §.5.2). Proof. — The proof follow the classical derivation of Euler-Lagrange equation (see for example [21],p.432-434). By theorem 3.1, we have µ ¶¸ Z b· ∂L ∂L 2² x 2² 2² x γ F² (h) = (x(t), , t) − (x(t), , t) h(t)dt 2t ¶ 2t ∂2² x 2t aZ ∂x µ b ∂L 2² + h(t) dt ∂2² x aZ 2t b +i² [2² f² (t)2² h(t) − ¯² f² (t)2² h(t) − 2² f² (t) ¯² h(t) − ¯² f² (t) ¯² h(t)] dt, a

with f² (t) =

∂L (x(t), 2² x(t), t). ∂2² x

In order to conclude, we need the following lemma : Lemma 3.2. — Let 0 < ², a, b ∈ R, h ∈ H β , β ≥ α1[1/2,1] + (1 − α)1]0,1/2[ , such that h(a) = h(b) = 0, and f² : R → C such that (31)

sup s∈{t,t+σ²}

| f² (s) |≤ C²α−1 ,

for all t ∈ [a, b]. Then, we have (32) Z a

b

2² (f² (t)h(t))dt = O(²α+β−1 ), and ² 2t

where Op² and

Op0²

Z a

b

Op² (f² )Op0² (h)dt = O(²α+β ).

are either 2² or ¯² .

The proof is given in the next section. Using condition (28), we obtain sup s∈{t,t+σ²}

| ∂L/∂2² x |≤ C 0 ²α−1 ,

as sups∈{t,t+σ²} [max(| x(s) |, | 2² x(s) |, | s |)] ≤ C”²α−1 .

NON DIFFERENTIABLE VARIATIONAL PRINCIPLES

13

Using lemma 3.2 with f² (s) = (∂L/∂2² t)(x(s), 2² (s), s), and condition (29), we deduce that µ µ ¶ ¶ Z b Z b 2² ∂L ∂L lim h(t) dt = 0 and lim ² Op² Op0² (h(t))dt = 0, ²→0 a 2t ²→0 ∂2² x ∂2² x a for Op² and Op0² which are either 2² or ¯² . Hence, applying the operator [.]² , we obtain ·Z b · µ ¶¸ ¸ ∂L 2² x 2² ∂L 2² x γ [F² (h)]² = (x(t), , t) − (x(t), , t) h(t)dt , ∂x 2t 2tµ ∂2² x 2t ¶¸ ² Z ba · ∂L 2² x 2² ∂L 2² x = (x(t), , t) − (x(t), , t) h(t)dt. ∂x 2t 2t ∂2² x 2t a ² The rest of the proof follows as in the classical case (see [1],p.57-58). Remark 3.5. — The special form of condition (29) comes from the two following constraints : one must have β ≥ α in order to preserve the regularity of perturbed curves γ + h, and β ≥ 1 − α in order to ensure that the first quantity of equation (32) goes to zero when ² goes to zero. Note that α = 1/2 plays a special role for these sets of conditions, as this is the only one for which the regularity of curves and variations are equals.

4. Proof of lemma 3.2 This comes essentially from the integral formula (12). Indeed, Z b 2² (f² (t)h(t))dt 2t a is a combination of the following quantities Z 1 t+σ² f² (s)h(s)ds, σ = ±, 2² t for t = a or t = b. As h(a) = h(b) = 0 and h ∈ C β (a, b), we have for t = a or t = b, sup s∈{t,t+σ²}

| h(s) |=

sup s∈{t,t+σ²}

| h(s) − h(t) |≤ C²β ,

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JACKY CRESSON

for some constant C. Moreover, using condition (31), we easily obtain sup s∈{t,t+σ²}

where

C0

| f² (s)h(s) | C 0 ²α+β−1 .

is a constant. Using this inequality, we deduce Z b 2² | (f² (s)h(s))ds |= O(²α+β−1 ). 2t a

We only prove the second inequality of equation (32) for Op² = 2² and Op0² = 2² . The remaining cases are proved in the same way. As h ∈ C β (a, b), we have sup | 2² h(t) |≤ C²β ,

t∈[a,b]

for some constant C (see lemma 2.1). Moreover, using (31), we obtain sup | ∆σ² (f² )(s) |≤ C σ ²α−1 ,

s∈[a,b]

for some constant

Cσ,

σ = ±. We deduce sup | 2² f² (s) |≤ C 0 ²α−1 .

s∈[a,b]

As a consequence, we obtain the inequality Z b 2² f² 2² h |² dt |≤ C”²α+β , 2t 2t a for some constant C”. This concludes the proof of lemma 3.2.

5. Application : least action principle and non-linear Schr¨ odinger equations 5.1. Least action principle and the Schr¨ odinger equation. — In this section we gives a variational principle whose extremals are solutions of the Schr¨ odinger equation. We consider the following non-linear Schr¨ odinger’s equation (obtained in [4],[9]) :

NON DIFFERENTIABLE VARIATIONAL PRINCIPLES

(33) "

"

1 2iγm − ψ

µ

∂ψ ∂x

15

# # ¶2 µ ¶ a² (t) ∂ψ a² (t) ∂ 2 ψ + iγ + + = (U (x) + α(x))ψ , 2 ∂t 2 ∂x2 ²

where m > 0, γ ∈ R, U : R → R, a² : R → C, α(x) is an arbitrary continuous function. The main result of this section is an analogue of the Hamilton’s principle of least action (see [1],p.59) for (33). Theorem 5.1. — Solutions of the non-linear Schr¨ odinger equation (33) coincide with extremals of the functional associated to (34)

L(x(t), 2² x(t), t) = (1/2)m(2² x(t))2 + U (x),

on the space of C 1/2 curves, where x(t) and ψ² (x, t) are related by (35)

2² x ∂ ln(ψ(x, t)) = −i2γ , 2t ∂x

and if a² (t) is such that ¤ ¤ 1£ + 1£ + 2 2 (36) a² (t) = (∆² x(t))2 − (∆− (∆² x(t))2 + (∆− ² x(t)) − i ² x(t)) . 2 2 Remark 5.1. — i. The nonlinear Schr¨ odinger equation (33) was derived in [4] using an analogue of the Euler-Lagrange equation (30) proposed by Nottale [17] in the context of the Scale relativity theory. This derivation was done in the framework of the local fractional calculus developped in [3] and under an assumption concerning the existence of solutions to a particular fractional differential equation. However, as proved in ([9],part I,§.4.3, [5]) such assumptions can’t be satisfied. ii. In [9], equation (33) was derived using a “scale quantization procedure”, which gives a way to pass from classical mechanics to quantum mechanics, avoiding the problems of [4]. However, the Euler-Lagrange equation used in [9] comes from scale quantization, which is an abstract and formal way to derive the analogue of (30) from the classical Euler-Lagrange equation (see §.5.2).

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Proof. — As ∂L/∂2² x = m2² x, its differential is given by D(∂L/∂2² x) = (0, m, 0), so that condition (28) is satisfied. By theorem 3.2, extremals of our functional satisfy the Euler-Lagrange equation · ¸ 2² 2² x(t) dU (37) m = (x) . 2t dx ² We denote

∂ ln(ψ(x, t)) (x, t). ∂x We apply theorem 2.2 with n = 2, in order to compute 2² f (x(t), t)/2t. We f (x, t) =

have (38) µ ¶ 2² ∂ ln(ψ) (x(t), t) = 2t ∂x

µ ¶ 2² x ∂ ∂ ln(ψ(x, t)) (x(t), t) 2t ∂x ∂x ¶ µ ∂ ∂ ln(ψ(x, t)) + (x(t), t) ∂t ∂x ¶ µ 2 1 ∂ ∂ ln(ψ(x, t)) + a² (t) 2 (x(t), t) + o(²1/2 ). 2 ∂x ∂x

Elementary calculus gives

µ ¶ µ ¶ ∂ ln(ψ(x, t)) 1 ∂ψ ∂ 1 ∂ψ 1 ∂2ψ 1 ∂ψ 2 = , and = − . ∂x ψ ∂x ∂x ψ ∂x ψ ∂ 2 x ψ 2 ∂x Hence, we obtain µ ¶ µ ¶ 2² x ∂ ∂ ln(ψ(x, t)) ∂ ln(ψ) ∂ ∂ ln(ψ) (x(t), t) = −i2γ (x(t), t), 2t ∂x ∂x ∂x ∂x ∂x " # µ ¶ ∂ ∂ ln(ψ) 2 = −iγ (x(t), t), ∂x ∂x " # µ ¶ 1 ∂ψ 2 ∂ (x(t), t). = −iγ ∂x ψ 2 ∂x We then have µ ¶ 2² ∂ ln(ψ(x, t)) (x(t), t) 2t " ∂x " µ ¶ µ ¶ ## ∂ 1 ∂ψ 2 ∂ ln(ψ) 1 1 ∂2ψ 1 ∂ψ 2 = −iγ 2 + + a² (t) − 2 + o(²1/2 ), ∂x ψ ∂x ∂t 2 ψ ∂x2 ψ ∂x " # µ ¶ µ ¶ ∂ 1 ∂ψ 2 a² (t) 1 ∂ψ a² (t) 1 ∂ 2 ψ = − 2 iγ + + + + o(²1/2 ). ∂x ψ ∂x 2 ψ ∂t 2 ψ ∂x2

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As a consequence, equation (38) is equivalent to " " # # µ ¶ µ ¶ ∂ a² (t) 1 ∂ψ 1 ∂ψ 2 a² (t) 1 ∂ 2 ψ ∂U iγ + + . i2γm − 2 + = 2 ∂x ψ ∂x 2 ψ ∂t 2 ψ ∂x ∂x By integrating with respect to x, we obtain " # µ ¶ µ ¶ 1 ∂ψ 2 a² (t) 1 ∂ψ a² (t) 1 ∂ 2 ψ = U (x)+α(x)+o(²1/2 ), i2γm − 2 iγ + + + ψ ∂x 2 ψ ∂t 2 ψ ∂x2 where α(x) is an arbitrary function. This concludes the proof. A great deal of efforts have been made in order to generalize the classical linear Schr¨odinger equation (see for example De Broglie [6],[7] and Lochak [13]). However, these generalizations are in general ad-hoc one, choosing some particular non linear terms in order to solve some specific problems of quantum mechanics (see for example [2],[19], [20]). On the contrary, the non differentiable least action principle impose a fixed non linear term. In order to recover the classical linear Schr¨ odinger equation, we must specialize the functional space on which we work. Precisely, we have : Theorem 5.2. — Solutions of the Schr¨ odinger equation · ¸ ¯ 2 ∂2ψ ∂ψ h ¯ (39) ih + = U (x)ψ , ∂t 2m ∂x2 ² ¯ = h/2π, coincide with extremals of the functional associated to where h (40)

L(x(t), 2² x(t), t) = (1/2)m(2² x(t))2 + U (x),

on the space of C 1/2 curves γ : x = x(t) satisfying, ¤ ¤ 1£ + 1£ + 2 2 ¯ (41) (∆² x(t))2 − (∆− (∆² x(t))2 + (∆− = −ih/m, ² x(t)) − i ² x(t)) 2 2 where x(t) and ψ² (x, t) are related by (42)

¯ ∂ ln(ψ(x, t)) 2² x h = −i , 2t m ∂x

Proof. — This follows easily from the calculations made in the proof of theorem 5.1.

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For different derivations of the Schr¨ odinger equation, we refer to the work of Nelson on stochastic mechanics ([15],[16]) and Feynman [11], where he developp a principle of least action, different from the one presented here. 5.2. About the scale relativity theory. — This final section is informal and discuss the connexion between our non differentiable variational principle and the scale relativity theory. In the following, we don’t give a precise definition to the word fractal. The only property which is assumed is that fractals are scale dependent objects. We refer to [10] for more details. The scale relativity theory developped by Nottale [17], gives up the assumption of the differentiability of space-time by considering what he calls a fractal space-time, and extending the Einstein’s principle of relativity to scales. One of the consequences of such a theory is that there exists an infinity of geodesics(3) and that geodesics are fractal curves. On such curves, one must developp a new differential calculus taking into account the non differentiable character of the curve. The scale derivative introduced by Nottale is the analogue of the scale derivative introduced in this paper. The scale relativity principle can be state as follows : The equations of physics keep the same form under scale transformations (see [17]). As a consequence, the scale relativity principle allows us to pass from classical mechanics to quantum mechanic via a simple procedure : one must change the classical derivative in Newton’s fundamental equation of dynamics by the scale derivative (see [18]). As Newton’s equation is written via an Euler-Lagrange equation of the form · ¸ d ∂L ∂L (43) = , dt ∂v ∂x (3)

This notion is not well defined, and we refer to [17] for more details.

NON DIFFERENTIABLE VARIATIONAL PRINCIPLES

19

this procedure, called scale quantization in [9], gives a quantum analogue of the form (44)

· ¸ 2² ∂L ∂L = , 2t ∂v ∂x

where v is of course a complex quantity defined by 2² x (45) v= . 2t As a consequence, scale quantization gives an Euler-Lagrange equation similar to the one obtained via the non differentiable variational principle introduced in this paper. The non differentiable variational principle can be considered as an attempt to developp the mathematical foundations of the scale relativity principle. References [1] Arnold V.I., Mathematical methods of classical mechanics, 2d edition, Graduate Texts in Mathematics 60, Springer-Verlag, 1989. [2] Bialynicky-Birula I, Mycielsky J, Ann. Phys. 100, 62, 1976. [3] Ben Adda F, Cresson J, About non differentiable functions, Journ. Mathematical Analysis and Applications 263, pp. 721-737, 2001. [4] Ben Adda F., Cresson J., Quantum derivatives and the Schr¨odinger equation, Chaos, solitons and fractals, Vol. 19, no.5, 1323-1334, 2004. [5] Ben Adda F, Cresson J, Fractional differential equations and the Schr¨odinger equation, 27.p, to appear in Applied Mathematics and Computations, 2004. [6] Broglie de L, Non-linear wave mechanics, Elsevier, Amsterdam, 1960. [7] Broglie de L, Nouvelles perspectives en microphysique, Coll. Champs Flammarion, 1992. [8] Cresson J, Scale relativity for one dimensional non differentiable manifolds, Chaos, Solitons and fractals Vol. 14, no.4, pp. 553-562, 2002. [9] Cresson J., Scale calculus and the Schr¨odinger equation, Journal of Mathematical Physics, Vol. 44, No. 11, 4907-4938, 2003. [10] Falconer K., Fractal geometry. Mathematical foundations and applications, John Wiley and Sons, 1990. [11] Feynman R.P., The development of the space-time view of quantum electrondynamics, Nobel lecture, December 11, 1965. [12] Feynman R, Hibbs A, Quantum mechanics and path integrals, MacGraw-Hill, 1965. [13] Lochak G, Annales de la fondation Louis de Broglie 22, no.1, p.1-22, no.2, p.187217, 1997.

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[14] Milne-Thomson L.M., The calculus of finite differences, Chelsea Publ. Comp., 1981. [15] Nelson E., Dynamical theories of Brownian motion, 2d edition, 2001, Princeton University Press, 1967. [16] Nelson E., Derivation of the Schr¨odinger equation from Newtonian mechanics, Physical Review 150 (1966). [17] Nottale L., Fractal space-time and microphysics, World Scientific, 1993. [18] Nottale L., Scale-relativity and quantization of the universe I. Theoritical framework, Astron. Astrophys. 327, 867-899 (1997). [19] Pardy M, To the nonlinear quantum mechanics, preprint 2002, arxiv:quant-ph/ 0111105 [20] Puszkarz W, On the Staruszkiewicz modification of the Schr¨odinger equation, preprint 1999, arxiv:quant-ph/9912006 [21] Spivak M, A comprehensive introduction to differential geometry, Publish or Perish, Berkeley, 1979. [22] Tricot C, Courbes et dimension fractale, 2d Ed., Springer, 1999.

Jacky CRESSON, Universit´e de Franche-Comt´e, Equipe de Math´ematiques de Besan¸con, CNRS-UMR 6623, Th´eorie des nombres et alg`ebre, 16 route de gray, 25030 Besan¸con cedex, France. • E-mail : [email protected]