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Research Article Helmholtz Theorem for Nondifferentiable Hamiltonian Systems in the Framework of Cresson’S Quantum Calculus

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Research Article Helmholtz Theorem for Nondifferentiable Hamiltonian Systems in the Framework of Cresson’S Quantum Calculus Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2016, Article ID 8073023, 8 pages http://dx.doi.org/10.1155/2016/8073023 Research Article Helmholtz Theorem for Nondifferentiable Hamiltonian Systems in the Framework of Cresson’s Quantum Calculus Frédéric Pierret1 and Delfim F. M. Torres2 1 Institut de Mecanique´ Celeste´ et de Calcul des Eph´ em´ erides,´ Observatoire de Paris, 75014 Paris, France 2Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal Correspondence should be addressed to Delfim F. M. Torres; [email protected] Received 23 January 2016; Accepted 28 April 2016 Academic Editor: Taher S. Hassan Copyright © 2016 F. Pierret and D. 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. 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 the study of PDEs. Euler-Lagrange equations for variational functionals with Lagrangians containing multiple quantum Several types of quantum calculus are available in the lit- derivatives, depending on a parameter or containing higher- erature, including Jackson’s quantum calculus [1, 2], Hahn’s order quantum derivatives, are studied in [20]. Variational quantum calculus [3–5], the time-scale -calculus [6, 7], the problems with constraints, with one and more than one power quantum calculus [8], and the symmetric quantum independent variable, of first and higher-order type are inves- calculus [9–11]. Cresson introduced in 2005 his quantum cal- tigated in [21]. Recently, problems of the calculus of variations culus on a set of Holder¨ functions [12]. This calculus attracted and optimal control with time delay were considered [22]. attention due to its applications in physics and the calculus In [18], a Noether type theorem is proved but only with of variations and has been further developed by several dif- the momentum term. This result is further extended in [23] ferent authors (see [13–16] and references therein). Cresson’s by considering invariance transformations that also change calculus of 2005 [12] presents, however, some difficulties, and the time variable, thus obtaining not only the generalized in 2011 Cresson and Greff improved it [17, 18]. Indeed, the momentum term of [18] but also a new energy term. In [13], quantum calculus of [12] let a free parameter, which is present nondifferentiable variational problems with a free terminal in all the computations. Such parameter is certainly difficult point, with or without constraints, of first and higher-order to interpret. The new calculus of [17, 18] bypasses the problem are investigated. Here, we continue to develop Cresson’s by considering a quantity that is free of extra parameters and quantum calculus in obtaining a result for Hamiltonian reduces to the classical derivative for differentiable functions. systems and by considering the so-called inverse problem of It is this new version of 2011 that we consider here, with a the calculus of variations. brief review of it being given in Section 2. Along the text, Aclassicalprobleminanalysisisthewell-knownHelm- by Cresson’s calculus we mean this quantum version of 2011 holtz’sinverseproblemofthecalculusofvariations:finda [17, 18]. For the state of the art on the quantum calculus necessary and sufficient condition under which a (system of) of variations we refer the reader to the recent book [19]. differential equation(s) can be written as an Euler-Lagrange With respect to Cresson’s approach, the quantum calculus or a Hamiltonian equation and, in the affirmative case, find of variations is still in its infancy: see [13, 17, 18, 20–22]. In all possible Lagrangian or Hamiltonian formulations. This [17] nondifferentiable Euler-Lagrange equations are used in condition is usually called the Helmholtz condition.The 2 Discrete Dynamics in Nature and Society − Lagrangian Helmholtz problem has been studied and solved quantum derivatives of , denoted, respectively, by and + by Douglas [24], Mayer [25], and Hirsch [26, 27]. The Hamil- , are defined by tonian 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 frame- (1) workhasbeendonein[29,30],inthediscreteLagrangian (+) −() + () = . 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 Remark 3. The -left and -right quantum derivatives of a continuous function correspond to the classical derivative equation to be an Euler-Lagrange equation on time scales. For the Hamiltonian case it has been done for the discrete of the -mean function defined by calculus of variations in [32] using the framework of [33] + = ∫ , =±. and in [34] using a discrete embedding procedure derived ( ) ( ) (2) in [35]. In the case of time-scale calculus it has been done in [36]; for the Stratonovich stochastic calculus see [37]. Here The next operator generalizes the classical derivative. we give the Helmholtz theorem for Hamiltonian systems 0 in the case of nondifferentiable Hamiltonian systems in the Definition 4 (the -scale derivative [17]). Let ∈C (, R ). >0 ◻ /◻ framework of Cresson’s quantum calculus. By definition, the For all ,the -scale derivative of , denoted by , nondifferentiable calculus extends the differentiable calculus. is defined by Such as in the discrete, time-scale, and stochastic cases, we ◻ 1 + − + − = [( + ) + ( − )] , (3) recover the same conditions of existence of a Hamiltonian ◻ 2 structure. The paper is organized as follows. In Section 2, we give where is the imaginary unit and ∈ {−1, 1, 0, −,. } some generalities and notions about the nondifferentiable Remark 5. If is differentiable, then one can take the limit of calculus introduced in [17], the so-called Cresson’s quantum calculus. In Section 3, we remind definitions and results the scale derivative when goestozero.Wethenobtainthe classical derivative / of . about classical and nondifferentiable Hamiltonian systems. In Section 4, we give a brief survey of the classical Helmholtz We also need to extend the scale derivative to complex Hamiltonian problem and then we prove the main result valued functions. of this paper—the nondifferentiable Hamiltonian Helmholtz 0 theorem. Finally, we give two applications of our results in Definition 6 (see [17]). Let ∈C (, C ) be a continuous Section5,andweendinSection6withconclusionsand complex valued function. For all >0,the-scale derivative future work. of , denoted by ◻/◻, is defined by ◻ ◻Re () ◻ Im () = + , (4) 2. Cresson’s Quantum Calculus ◻ ◻ ◻ We briefly review the necessary concepts and results of the where Re() and Im() denote the real and imaginary part quantum calculus [17]. of ,respectively. 2.1. Definitions. Let X denote the set R or C , ∈N,and In Definition 4, the -scale derivative depends on , let be an open set in R with [, ], ⊂ <. We denote by which is a free parameter related to the smoothing order 0 F(, X ) the set of functions :→X and by C (, X ) of the function. This brings many difficulties in applications to physics, when one is interested in particular equations the subset of functions of F(, X ) which are continuous. that do not depend on an extra parameter. To solve these ∈C0(, R) problems, the authors of [17] introduced a procedure to Definition 1 (Holderian¨ functions [17]). Let . Let ∈.Function is said to be -Holderian,¨ 0<<1,at extract information independent of but related with the mean behavior of the function. point if there exist positive constants >0and >0such that | − |⩽implies ‖() − ( )‖ ⩽ | − | for all ∈, C0 ( × ]0, 1], R)⊆C0( × ]0, Definition 7 (see [17]). Let conv where ‖⋅‖is a norm on R . 0 1], R ) be such that for any function ∈Cconv( × ]0, 1], R ) (, ) ∈ the lim→0 exists for any .Wedenoteby acom- The set of Holderian¨ functions of Holder¨ exponent ,for C0 ( × ]0, 1], R) C0( × ]0, 1], R) (, R) plementary space of conv in . some , is denoted by . The quantum derivative is We define the projection map by defined as follows. 0 0 :Cconv (×]0, 1] , R )⊕→Cconv (×]0, 1] , R ) Definition 2 (the -left and -right quantum derivatives [17]). 0 (5) Let ∈C (, R ).Forall>0,the-left and -right conv + → conv Discrete Dynamics in Nature and Society 3 and the operator ⟨⋅⟩ by Extremals of the functional L can be characterized by the well-known Euler-Lagrange equation (see, e.g., [38]). ⟨⋅⟩ : C0 (×]0, 1] , R)→C0 (, R) 1 (6) Theorem 14. The extremals ∈C (, R ) of L coincide with → ⟨ ⟩ : → ()(,) . lim→0 the solutions of the Euler-Lagrange equation The quantum derivative of without the dependence of [ (, () , ̇ ())] = (, () , ̇ ()). is introduced in [17]. V (12) Definition 8 (see [17]). The quantum derivative of in the The nondifferentiable embedding procedure allows us to 0 space C (, R ) is given by define a natural extension of the classical Euler-Lagrange equation in the nondifferentiable context. ◻ ◻ =⟨ ⟩. (7) ◻ ◻ Definition 15
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