Helmholtz Theorem for Nondifferentiable Hamiltonian Systems in the Framework of Cresson’s Quantum Calculus

Pierret, Frédéric; Torres, Delfim F. M.

doi:https://doi.org/10.1155/2016/8073023

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Volume 2016 | Article ID 8073023 | https://doi.org/10.1155/2016/8073023

Helmholtz Theorem for Nondifferentiable Hamiltonian Systems in the Framework of Cresson’s Quantum Calculus

Frédéric Pierret¹and Delfim F. M. Torres²

Academic Editor: Taher S. Hassan

Received23 Jan 2016

Accepted28 Apr 2016

Published26 May 2016

Abstract

We derive the Helmholtz theorem for nondifferentiable Hamiltonian systems in the framework of Cresson’s quantum calculus. Precisely, we give a theorem characterizing nondifferentiable equations, admitting a Hamiltonian formulation. Moreover, in the affirmative case, we give the associated Hamiltonian.

1. Introduction

Several types of quantum calculus are available in the literature, including Jackson’s quantum calculus [1, 2], Hahn’s quantum calculus [3–5], the time-scale -calculus [6, 7], the power quantum calculus [8], and the symmetric quantum calculus [9–11]. Cresson introduced in 2005 his quantum calculus on a set of Hölder functions [12]. This calculus attracted attention due to its applications in physics and the calculus of variations and has been further developed by several different authors (see [13–16] and references therein). Cresson’s calculus of 2005 [12] presents, however, some difficulties, and in 2011 Cresson and Greff improved it [17, 18]. Indeed, the quantum calculus of [12] let a free parameter, which is present in all the computations. Such parameter is certainly difficult to interpret. The new calculus of [17, 18] bypasses the problem by considering a quantity that is free of extra parameters and reduces to the classical derivative for differentiable functions. It is this new version of 2011 that we consider here, with a brief review of it being given in Section 2. Along the text, by Cresson’s calculus we mean this quantum version of 2011 [17, 18]. For the state of the art on the quantum calculus of variations we refer the reader to the recent book [19]. With respect to Cresson’s approach, the quantum calculus of variations is still in its infancy: see [13, 17, 18, 20–22]. In [17] nondifferentiable Euler-Lagrange equations are used in the study of PDEs. Euler-Lagrange equations for variational functionals with Lagrangians containing multiple quantum derivatives, depending on a parameter or containing higher-order quantum derivatives, are studied in [20]. Variational problems with constraints, with one and more than one independent variable, of first and higher-order type are investigated in [21]. Recently, problems of the calculus of variations and optimal control with time delay were considered [22]. In [18], a Noether type theorem is proved but only with the momentum term. This result is further extended in [23] by considering invariance transformations that also change the time variable, thus obtaining not only the generalized momentum term of [18] but also a new energy term. In [13], nondifferentiable variational problems with a free terminal point, with or without constraints, of first and higher-order are investigated. Here, we continue to develop Cresson’s quantum calculus in obtaining a result for Hamiltonian systems and by considering the so-called inverse problem of the calculus of variations.

A classical problem in analysis is the well-known Helmholtz’s inverse problem of the calculus of variations: find a necessary and sufficient condition under which a (system of) differential equation(s) can be written as an Euler-Lagrange or a Hamiltonian equation and, in the affirmative case, find all possible Lagrangian or Hamiltonian formulations. This condition is usually called the Helmholtz condition. The Lagrangian Helmholtz problem has been studied and solved by Douglas [24], Mayer [25], and Hirsch [26, 27]. The Hamiltonian Helmholtz problem has been studied and solved, up to our knowledge, by Santilli in his book [28]. Generalization of this problem in the discrete calculus of variations framework has been done in [29, 30], in the discrete Lagrangian case. In the case of time-scale calculus, that is, a mixing between continuous and discrete subintervals of time, see [31] for a necessary condition for a dynamic integrodifferential equation to be an Euler-Lagrange equation on time scales. For the Hamiltonian case it has been done for the discrete calculus of variations in [32] using the framework of [33] and in [34] using a discrete embedding procedure derived in [35]. In the case of time-scale calculus it has been done in [36]; for the Stratonovich stochastic calculus see [37]. Here we give the Helmholtz theorem for Hamiltonian systems in the case of nondifferentiable Hamiltonian systems in the framework of Cresson’s quantum calculus. By definition, the nondifferentiable calculus extends the differentiable calculus. Such as in the discrete, time-scale, and stochastic cases, we recover the same conditions of existence of a Hamiltonian structure.

The paper is organized as follows. In Section 2, we give some generalities and notions about the nondifferentiable calculus introduced in [17], the so-called Cresson’s quantum calculus. In Section 3, we remind definitions and results about classical and nondifferentiable Hamiltonian systems. In Section 4, we give a brief survey of the classical Helmholtz Hamiltonian problem and then we prove the main result of this paper—the nondifferentiable Hamiltonian Helmholtz theorem. Finally, we give two applications of our results in Section 5, and we end in Section 6 with conclusions and future work.

2. Cresson’s Quantum Calculus

We briefly review the necessary concepts and results of the quantum calculus [17].

2.1. Definitions

Let denote the set or , , and let be an open set in with , . We denote by the set of functions and by the subset of functions of which are continuous.

Definition 1 (Hölderian functions [17]). Let . Let . Function is said to be -Hölderian, , at point if there exist positive constants and such that implies for all , where is a norm on .

The set of Hölderian functions of Hölder exponent , for some , is denoted by . The quantum derivative is defined as follows.

Definition 2 (the -left and -right quantum derivatives [17]). Let . For all , the -left and -right quantum derivatives of , denoted, respectively, by and , are defined by

Remark 3. The -left and -right quantum derivatives of a continuous function correspond to the classical derivative of the -mean function defined by

The next operator generalizes the classical derivative.

Definition 4 (the -scale derivative [17]). Let . For all , the -scale derivative of , denoted by , is defined by where is the imaginary unit and .

Remark 5. If is differentiable, then one can take the limit of the scale derivative when goes to zero. We then obtain the classical derivative of .

We also need to extend the scale derivative to complex valued functions.

Definition 6 (see [17]). Let be a continuous complex valued function. For all , the -scale derivative of , denoted by , is defined bywhere and denote the real and imaginary part of , respectively.

In Definition 4, the -scale derivative depends on , which is a free parameter related to the smoothing order of the function. This brings many difficulties in applications to physics, when one is interested in particular equations that do not depend on an extra parameter. To solve these problems, the authors of [17] introduced a procedure to extract information independent of but related with the mean behavior of the function.

Definition 7 (see [17]). Let be such that for any function the exists for any . We denote by a complementary space of in . We define the projection map by and the operator by

The quantum derivative of without the dependence of is introduced in [17].

Definition 8 (see [17]). The quantum derivative of in the space is given byThe quantum derivative (7) has some nice properties. Namely, it satisfies a Leibniz rule and a version of the fundamental theorem of calculus.

Theorem 9 (the quantum Leibniz rule [17]). Let . For and , one has

Remark 10. For and , one obtains from (8) the classical Leibniz rule: .

Definition 11. We denote by the set of continuous functions such that .

Theorem 12 (the quantum version of the fundamental theorem of calculus [17]). Let be such thatThen,

2.2. Nondifferentiable Calculus of Variations

In [17] the calculus of variations with quantum derivatives is introduced and respective Euler-Lagrange equations derived without the dependence of .

Definition 13. An admissible Lagrangian is a continuous function such that is holomorphic with respect to and differentiable with respect to . Moreover, when ; when .

An admissible Lagrangian function defines a functional on , denoted byExtremals of the functional can be characterized by the well-known Euler-Lagrange equation (see, e.g., [38]).

Theorem 14. The extremals of coincide with the solutions of the Euler-Lagrange equation

The nondifferentiable embedding procedure allows us to define a natural extension of the classical Euler-Lagrange equation in the nondifferentiable context.

Definition 15 (see [17]). The nondifferentiable Lagrangian functional associated with is given by

Let and with . A -variation of is a function of the form , where . We denote by the quantity if there exists the so-called Fréchet derivative of at point in direction .

Definition 16 (nondifferentiable extremals). A -extremal curve of the functional is a curve satisfying for any .

Theorem 17 (nondifferentiable Euler-Lagrange equations [17]). Let with . Let be an admissible Lagrangian of class . We assume that , such that . Moreover, we assume that satisfies condition (9) for all . A curve satisfying the nondifferentiable Euler-Lagrange equation is an extremal curve of functional (13).

3. Reminder about Hamiltonian Systems

We now recall the main concepts and results of both classical and Cresson’s nondifferentiable Hamiltonian systems.

3.1. Classical Hamiltonian Systems

Let be an admissible Lagrangian function. If satisfies the so-called Legendre property, then we can associate to a Hamiltonian function denoted by .

Definition 18. Let be an admissible Lagrangian function. The Lagrangian is said to satisfy the Legendre property if the mapping is invertible for any .

If we introduce a new variableand satisfies the Legendre property, then we can find a function such thatUsing this notation, we have the following definition.

Definition 19. Let be an admissible Lagrangian function satisfying the Legendre property. The Hamiltonian function associated with is given by

We have the following theorem (see, e.g., [38]).

Theorem 20 (Hamilton’s least-action principle). The curve is an extremal of the Hamiltonian functional if and only if it satisfies the Hamiltonian system associated with given bycalled the Hamiltonian equations.

A vectorial notation is obtained for the Hamiltonian equations in posing and , where denotes the transposition. The Hamiltonian equations are then written aswheredenotes the symplectic matrix with being the identity matrix on .

3.2. Nondifferentiable Hamiltonian Systems

The nondifferentiable embedding induces a change in the phase space with respect to the classical case. As a consequence, we have to work with variables that belong to and not only to , as usual.

Definition 21 (nondifferentiable embedding of Hamiltonian systems [17]). The nondifferentiable embedded Hamiltonian system (20) is given byand the embedded Hamiltonian functional is defined on by

The nondifferentiable calculus of variations allows us to derive the extremals for .

Theorem 22 (nondifferentiable Hamilton’s least-action principle [17]). Let with . Let be an admissible -Lagrangian. We assume that , such that . Moreover, we assume that satisfies condition (9) for all . Let be the corresponding Hamiltonian defined by (18). A curve solution of the nondifferentiable Hamiltonian system (23) is an extremal of functional (24) over the space of variations .

4. Nondifferentiable Helmholtz Problem

In this section, we solve the inverse problem of the nondifferentiable calculus of variations in the Hamiltonian case. We first recall the usual way to derive the Helmholtz conditions following the presentation made by Santilli [28]. Two main derivations are available:(i)The first is related to the characterization of Hamiltonian systems via the symplectic two-differential form and the fact that by duality the associated one-differential form to a Hamiltonian vector field is closed—the so-called integrability conditions.(ii)The second uses the characterization of Hamiltonian systems via the self-adjointness of the Fréchet derivative of the differential operator associated with the equation—the so-called Helmholtz conditions.

Of course, we have coincidence of the two procedures in the classical case. As there is no analogous of differential form in the framework of Cresson’s quantum calculus, we follow the second way to obtain the nondifferentiable analogue of the Helmholtz conditions. For simplicity, we consider a time-independent Hamiltonian. The time-dependent case can be done in the same way.

4.1. Helmholtz Conditions for Classical Hamiltonian Systems

In this section we work on , , .

4.1.1. Symplectic Scalar Product

The symplectic scalar product is defined byfor all , where denotes the usual scalar product and is the symplectic matrix (22). We also consider the symplectic scalar product induced by defined for by

4.1.2. Adjoint of a Differential Operator

In the following, we consider first-order differential equations of the formwhere the vector fields and are with respect to and . The associated differential operator is written asA natural notion of adjoint for a differential operator is then defined as follows.

Definition 23. Let . We define the adjoint of with respect to by

An operator will be called self-adjoint if with respect to the symplectic scalar product.

4.1.3. Hamiltonian Helmholtz Conditions

The Helmholtz conditions in the Hamiltonian case are given by the following result (see Theorem , p. 176-177 in [28]).

Theorem 24 (Hamiltonian Helmholtz theorem). Let be a vector field defined by . The differential equation (27) is Hamiltonian if and only if the associated differential operator given by (28) has a self-adjoint Fréchet derivative with respect to the symplectic scalar product. In this case the Hamiltonian is given by

The conditions for the self-adjointness of the differential operator can be made explicit. They coincide with the integrability conditions characterizing the exactness of the one-form associated with the vector field by duality (see [28], Theorem , p. 88).

Theorem 25 (integrability conditions). Let be a vector field. The differential operator given by (28) has a self-adjoint Fréchet derivative with respect to the symplectic scalar product if and only if

4.2. Helmholtz Conditions for Nondifferentiable Hamiltonian Systems

The previous scalar products extend naturally to complex valued functions. Let and let , such that and . We consider first-order nondifferential equations of the formThe associated quantum differential operator is written asA natural notion of adjoint for a quantum differential operator is then defined.

Definition 26. Let . We define the adjoint of with respect to by

An operator will be called self-adjoint if with respect to the symplectic scalar product. We can now obtain the adjoint operator associated with .

Proposition 27. Let be such that . Let , such that and . The Fréchet derivative of (33) at along is then given byAssume that and satisfy condition (9) for any . In consequence, the adjoint of with respect to the symplectic scalar product is given by

Proof. The expression for the Fréchet derivative of (33) at along is a simple computation. Let be such that and . By definition, we have As and satisfy condition (9) for any , using the quantum Leibniz rule and the quantum version of the fundamental theorem of calculus, we obtain Then,By definition, we obtain the expression of the adjoint of with respect to the symplectic scalar product.

In consequence, from a direct identification, we obtain the nondifferentiable self-adjointess conditions called Helmholtz’s conditions. As in the classical case, we call these conditions nondifferentiable integrability conditions.

Theorem 28 (nondifferentiable integrability conditions). Let be a vector field. The differential operator given by (33) has a self-adjoint Fréchet derivative with respect to the symplectic scalar product if and only if

Remark 29. One can see that the Helmholtz conditions are the same as in the classical, discrete, time-scale, and stochastic cases. We expected such a result because Cresson’s quantum calculus provides a quantum Leibniz rule and a quantum version of the fundamental theorem of calculus. If such properties of an underlying calculus exist, then the Helmholtz conditions will always be the same up to some conditions on the working space of functions.

We now obtain the main result of this paper, which is the Helmholtz theorem for nondifferentiable Hamiltonian systems.

Theorem 30 (nondifferentiable Hamiltonian Helmholtz theorem). Let be a vector field defined by . The nondifferentiable system of (32) is Hamiltonian if and only if the associated quantum differential operator given by (33) has a self-adjoint Fréchet derivative with respect to the symplectic scalar product. In this case, the Hamiltonian is given by

Proof. If is Hamiltonian, then there exists a function such that is holomorphic with respect to and differentiable with respect to and and . The nondifferentiable integrability conditions are clearly verified using Schwarz’s lemma. Reciprocally, we assume that satisfies the nondifferentiable integrability conditions. We will show that is Hamiltonian with respect to the Hamiltonian that is, we must show thatWe haveUsing the nondifferentiable integrability conditions, we obtainwhich concludes the proof.

5. Applications

We now provide two illustrative examples of our results: one with the formulation of dynamical systems with linear parts and another with Newton’s equation, which is particularly useful to study partial differentiable equations such as the Navier-Stokes equation. Indeed, the Navier-Stokes equation can be recovered from a Lagrangian structure with Cresson’s quantum calculus [17]. For more applications see [34].

Let and let be such that and .

5.1. The Linear Case

Let us consider the discrete nondifferentiable systemwhere , , , and are constants. The Helmholtz condition is clearly satisfied. However, system (45) satisfies the condition if and only if . As a consequence, linear Hamiltonian nondifferentiable equations are of the formUsing formula (40), we compute explicitly the Hamiltonian, which is given by

5.2. Newton’s Equation

Newton’s equation (see [38]) is given bywith and . This equation possesses a natural Hamiltonian structure with the Hamiltonian given byUsing Cresson’s quantum calculus, we obtain a natural nondifferentiable system given byThe Hamiltonian Helmholtz conditions are clearly satisfied.

Remark 31. It must be noted that Hamiltonian (49) associated with (50) is recovered by formula (40).

6. Conclusion

We proved a Helmholtz theorem for nondifferentiable equations, which gives necessary and sufficient conditions for the existence of a Hamiltonian structure. In the affirmative case, the Hamiltonian is given. Our result extends the results of the classical case when restricting attention to differentiable functions. An important complementary result for the nondifferentiable case is to obtain the Helmholtz theorem in the Lagrangian case. This is nontrivial and will be subject of future research.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work was supported by FCT and CIDMA through project UID/MAT/04106/2013. The first author is grateful to CIDMA and DMat-UA for the hospitality and good working conditions during his visit at University of Aveiro. The authors would like to thank an anonymous referee for careful reading of the submitted paper and for useful suggestions.

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Copyright © 2016 Frédéric Pierret and Delfim F. M. Torres. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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