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 𝑞 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 𝑓, denoted, respectively, by 𝑑𝜖 𝑓 and + by Douglas [24], Mayer [25], and Hirsch [26, 27]. The Hamil- 𝑑𝜖 𝑓, are defined by tonian Helmholtz problem has been studied and solved, up 𝑓 (𝑡) −𝑓(𝑡−𝜖) to our knowledge, by Santilli in his book [28]. Generalization 𝑑−𝑓 (𝑡) = , 𝜖 𝜖 of this problem in the discrete calculus of variations frame- (1) workhasbeendonein[29,30],inthediscreteLagrangian 𝑓 (𝑡+𝜖) −𝑓(𝑡) 𝑑+𝑓 (𝑡) = . case. In the case of time-scale calculus, that is, a mixing 𝜖 𝜖 between continuous and discrete subintervals of time, see [31] 𝜖 𝜖 for a necessary condition for a dynamic integrodifferential Remark 3. The -left and -right quantum derivatives of a continuous function 𝑓 correspond to the classical derivative equation to be an Euler-Lagrange equation on time scales. 𝜖 𝑓𝜎 For the Hamiltonian case it has been done for the discrete of the -mean function 𝜖 defined by calculus of variations in [32] using the framework of [33] 𝜎 𝑡+𝜎𝜖 𝑓𝜎 𝑡 = ∫ 𝑓 𝑠 𝑑𝑠, 𝜎 =±. and in [34] using a discrete embedding procedure derived 𝜖 ( ) ( ) (2) 𝜖 𝑡 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 𝑑 in the case of nondifferentiable Hamiltonian systems in the Definition 4 (the 𝜖-scale derivative [17]). Let 𝑓∈C (𝐼, R ). 𝜖>0 𝜖 𝑓 ◻ 𝑓/◻𝑡 framework of Cresson’s quantum calculus. By definition, the For all ,the -scale derivative of , denoted by 𝜖 , nondifferentiable calculus extends the differentiable calculus. is defined by Such as in the discrete, time-scale, and stochastic cases, we ◻ 𝑓 1 + − + − 𝜖 = [(𝑑 𝑓+𝑑 𝑓) + 𝑖𝜇 (𝑑 𝑓−𝑑 𝑓)] , (3) recover the same conditions of existence of a Hamiltonian ◻𝑡 2 𝜖 𝜖 𝜖 𝜖 structure. The paper is organized as follows. In Section 2, we give where 𝑖 is the imaginary unit and 𝜇 ∈ {−1, 1, 0, −𝑖,. 𝑖} some generalities and notions about the nondifferentiable Remark 5. If 𝑓 is differentiable, then one can take the limit of calculus introduced in [17], the so-called Cresson’s quantum 𝜖 calculus. In Section 3, we remind definitions and results the scale derivative when goestozero.Wethenobtainthe classical derivative 𝑑𝑓/𝑑𝑡 of 𝑓. 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 𝑑 theorem. Finally, we give two applications of our results in Definition 6 (see [17]). Let 𝑓∈C (𝐼, C ) be a continuous Section5,andweendinSection6withconclusionsand complex valued function. For all 𝜖>0,the𝜖-scale derivative future work. of 𝑓, denoted by ◻𝜖𝑓/◻𝑡, is defined by

◻ 𝑓 ◻𝜖Re (𝑓) ◻𝜖 Im (𝑓) 𝜖 = +𝑖 , (4) 2. Cresson’s Quantum Calculus ◻𝑡 ◻𝑡 ◻𝑡 We briefly review the necessary concepts and results of the where Re(𝑓) and Im(𝑓) denote the real and imaginary part quantum calculus [17]. of 𝑓,respectively.

𝑑 𝑑 𝑑 2.1. Definitions. Let X denote the set R or C , 𝑑∈N,and In Definition 4, the 𝜖-scale derivative depends on 𝜖, let 𝐼 be an open set in R with [𝑎, 𝑏] ⊂𝐼, 𝑎<𝑏. We denote by which is a free parameter related to the smoothing order 𝑑 𝑑 0 𝑑 F(𝐼, X ) the set of functions 𝑓:𝐼→X and by C (𝐼, X ) of the function. This brings many difficulties in applications 𝑑 to physics, when one is interested in particular equations the subset of functions of F(𝐼, X ) which are continuous. that do not depend on an extra parameter. To solve these 𝑓∈C0(𝐼, R𝑑) problems, the authors of [17] introduced a procedure to Definition 1 (Holderian¨ functions [17]). Let . 𝜖 Let 𝑡∈𝐼.Function𝑓 is said to be 𝛼-Holderian,¨ 0<𝛼<1,at extract information independent of but related with the mean behavior of the function. point 𝑡 if there exist positive constants 𝜖>0and 𝑐>0such 󸀠 󸀠 󸀠 𝛼 󸀠 that |𝑡 − 𝑡 |⩽𝜖implies ‖𝑓(𝑡) − 𝑓(𝑡 )‖ ⩽ 𝑐|𝑡 −𝑡 | for all 𝑡 ∈𝐼, C0 (𝐼 × ]0, 1], R𝑑)⊆C0(𝐼 × ]0, 𝑑 Definition 7 (see [17]). Let conv where ‖⋅‖is a norm on R . 𝑑 0 𝑑 1], R ) be such that for any function 𝑓∈Cconv(𝐼 × ]0, 1], R ) 𝑓(𝑡, 𝜖) 𝑡∈𝐼 𝐸 𝛼 the lim𝜖→0 exists for any .Wedenoteby acom- The set of Holderian¨ functions of Holder¨ exponent ,for C0 (𝐼 × ]0, 1], R𝑑) C0(𝐼 × ]0, 1], R𝑑) 𝛼 𝐻𝛼(𝐼, R𝑑) plementary space of conv in . some , is denoted by . The quantum derivative is We define the projection map 𝜋 by defined as follows. 0 𝑑 0 𝑑 𝜋:Cconv (𝐼×]0, 1] , R )⊕𝐸󳨀→Cconv (𝐼×]0, 1] , R ) Definition 2 (the 𝜖-left and 𝜖-right quantum derivatives [17]). 0 𝑑 (5) Let 𝑓∈C (𝐼, R ).Forall𝜖>0,the𝜖-left and 𝜖-right 𝑓conv +𝑓𝐸 󳨃󳨀→ 𝑓conv 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 (𝐼×]0, 1] , R𝑑)󳨀→C0 (𝐼, R𝑑) 1 𝑑 (6) Theorem 14. The extremals 𝑞∈C (𝐼, R ) of L coincide with 𝑓 󳨃󳨀→ ⟨ 𝑓 ⟩ :𝑡 󳨃󳨀→ 𝜋(𝑓)(𝑡,) 𝜖 . lim𝜖→0 the solutions of the Euler-Lagrange equation 𝑓 𝑑 𝜕𝐿 𝜕𝐿 The quantum derivative of without the dependence of [ (𝑡, 𝑞 (𝑡) , 𝑞̇ (𝑡))] = (𝑡, 𝑞 (𝑡) , 𝑞̇ (𝑡)). 𝜖 is introduced in [17]. 𝑑𝑡 𝜕V 𝜕𝑥 (12)

Definition 8 (see [17]). The quantum derivative of 𝑓 in the The nondifferentiable embedding procedure allows us to 0 𝑑 space C (𝐼, R ) is given by define a natural extension of the classical Euler-Lagrange equation in the nondifferentiable context. ◻𝑓 ◻𝜖𝑓 =⟨ ⟩. (7) ◻𝑡 ◻𝑡 Definition 15 (see [17]). The nondifferentiable Lagrangian functional L◻ associated with L is given by The quantum derivative (7) has some nice properties. Namely, it satisfies a Leibniz rule and a version of the L : C1 (𝐼, R𝑑)󳨀→R fundamental theorem of calculus. ◻ ◻ 𝑏 ◻𝑞 (𝑠) (13) Theorem 9 (the quantum Leibniz rule [17]). Let 𝛼+𝛽>1. 𝑞 󳨃󳨀→ ∫ 𝐿 (𝑠, 𝑞 (𝑠) , )𝑑𝑠. 𝛼 𝑑 𝛽 𝑑 ◻𝑡 For 𝑓∈𝐻 (𝐼, R ) and 𝑔∈𝐻 (𝐼, R ),onehas 𝑎 𝛽 𝛽 𝑑 ◻ ◻𝑓 (𝑡) ◻𝑔 (𝑡) Let 𝐻 ﬂ {ℎ ∈ 𝐻 (𝐼, R ), ℎ(𝑎) = ℎ(𝑏) = 0} and 𝑞∈ (𝑓 ⋅ 𝑔) (𝑡) = ⋅𝑔(𝑡) +𝑓(𝑡) ⋅ . (8) 0 ◻𝑡 ◻𝑡 ◻𝑡 𝛼 𝑑 𝛽 𝐻 (𝐼, R ) with 𝛼+𝛽>1.A𝐻0 -variation of 𝑞 is a function 𝛽 1 𝑑 1 𝑑 𝑞+ℎ ℎ∈𝐻 𝐷L (𝑞)(ℎ) Remark 10. For 𝑓∈C (𝐼, R ) and 𝑔∈C (𝐼, R ),one of the form ,where 𝑂.Wedenoteby ◻ 󸀠 󸀠 obtains from (8) the classical Leibniz rule: (𝑓 ⋅ 𝑔) =𝑓 ⋅𝑔+ the quantity 󸀠 𝑓⋅𝑔. L◻ (𝑞 + 𝜖ℎ) − L◻ (𝑞) lim (14) 1 𝜖→0 𝜖 Definition 11. We denote by C◻ the set of continuous func- 𝑞∈C0([𝑎, 𝑏], R𝑑) ◻𝑞/◻𝑡 ∈ C0(𝐼, R𝑑) tions such that . if there exists the so-called Frechet´ derivative of L◻ at point 𝑞 in direction ℎ. Theorem 12 (the quantum version of the fundamental theo- 1 𝑑 𝛽 rem of calculus [17]). Let 𝑓∈C◻([𝑎, 𝑏], R ) be such that Definition 16 (nondifferentiable extremals). A 𝐻0 -extremal 𝑏 L 𝑞∈𝐻𝛼(𝐼, R𝑑) ◻𝜖𝑓 curve of the functional ◻ is a curve satisfying lim ∫ ( ) (𝑡) 𝑑𝑡 = 0. (9) 𝐷L (𝑞)(ℎ) = 0 ℎ∈𝐻𝛽 𝜖→0 𝑎 ◻𝑡 𝐸 ◻ for any 0 . Then, Theorem 17 (nondifferentiable Euler-Lagrange equations 𝑏 [17]). Let 0<𝛼,𝛽<1with 𝛼+𝛽>1.Let𝐿 be an admissible ◻𝑓 2 𝛼 𝑑 ∫ (𝑡) 𝑑𝑡 =𝑓 (𝑏) −𝑓(𝑎) . (10) Lagrangian of class C . We assume that 𝛾∈𝐻(𝐼, R ),such 𝑎 ◻𝑡 𝛼 𝑑 that ◻𝛾/◻𝑡 ∈𝐻 (𝐼, R ). Moreover, we assume that 𝐿(𝑡, 𝛾(𝑡), ◻𝛾(𝑡)/◻𝑡)ℎ(𝑡) ℎ∈𝐻𝛽(𝐼, R𝑑) 2.2. Nondifferentiable Calculus of Variations. In [17] the satisfies condition (9) for all 0 . 𝛾 calculus of variations with quantum derivatives is introduced Acurve satisfying the nondifferentiable Euler-Lagrange and respective Euler-Lagrange equations derived without the equation 𝜖 dependence of . ◻ 𝜕𝐿 ◻𝛾 (𝑡) 𝜕𝐿 ◻𝛾 (𝑡) [ (𝑡, 𝛾 (𝑡) , )] = (𝑡, 𝛾 (𝑡) , ) (15) Definition 13. An admissible Lagrangian 𝐿 is a continuous ◻𝑡 𝜕V ◻𝑡 𝜕𝑥 ◻𝑡 𝐿:R × R𝑑 × C𝑑 → C 𝐿(𝑡, 𝑥, V) function such that is holo- is an extremal curve of functional (13). morphic with respect to V and differentiable with respect to 𝑑 𝑥.Moreover,𝐿(𝑡, 𝑥, V)∈R when V ∈ R ; 𝐿(𝑡, 𝑥, V)∈C when 𝑑 V ∈ C . 3. Reminder about Hamiltonian Systems

d 𝑑 An admissible Lagrangian function 𝐿:R × R × C → C We now recall the main concepts and results of both classical 1 𝑑 and Cresson’s nondifferentiable Hamiltonian systems. defines a functional on C (𝐼, R ), denoted by L : C1 (𝐼, R𝑑)󳨀→R 3.1. Classical Hamiltonian Systems. Let 𝐿 be an admissible Lagrangian function. If 𝐿 satisfies the so-called Legendre 𝑏 (11) property,thenwecanassociateto𝐿 a Hamiltonian function 𝑞 󳨃󳨀→ ∫ 𝐿(𝑡,𝑞(𝑡) , 𝑞̇ (𝑡))𝑑𝑡. 𝐻 𝑎 denoted by . 4 Discrete Dynamics in Nature and Society

Definition 18. Let 𝐿 be an admissible Lagrangian function. 3.2. Nondifferentiable Hamiltonian Systems. The nondifferen- The Lagrangian 𝐿 is said to satisfy the Legendre property if the tiable embedding induces a change in the phase space with mapping V 󳨃→ (𝜕𝐿/𝜕V)(𝑡, 𝑥, V) is invertible for any (𝑡, 𝑞, V)∈ respect to the classical case. As a consequence, we have to 𝑑 𝑑 𝑑 𝑑 𝐼×R × C . work with variables (𝑥, 𝑝) that belong to R × C and not 𝑑 𝑑 only to R × R ,asusual. If we introduce a new variable 𝜕𝐿 Definition 21 (nondifferentiable embedding of Hamiltonian 𝑝= (𝑡, 𝑞, V) (16) systems [17]). The nondifferentiable embedded Hamiltonian 𝜕V system (20) is given by 𝐿 and satisfies the Legendre property, then we can find a ◻𝑞 (𝑡) 𝜕𝐻 (𝑡, 𝑞 (𝑡) ,𝑝(𝑡)) function 𝑓 such that = , ◻𝑡 𝜕𝑝 V =𝑓(𝑡,𝑞,𝑝). (23) (17) ◻𝑝 (𝑡) 𝜕𝐻 (𝑡, 𝑞 (𝑡) ,𝑝(𝑡)) =− ◻𝑡 𝜕𝑞 Using this notation, we have the following definition.

𝐿 and the embedded Hamiltonian functional H◻ is defined on Definition 19. Let be an admissible Lagrangian function 𝛼 𝑑 𝛼 𝑑 satisfying the Legendre property. The Hamiltonian function 𝐻 (𝐼, R )×𝐻 (𝐼, C ) by 𝐻 associated with 𝐿 is given by 𝑏 ◻𝑞 (𝑡) H (𝑞, 𝑝) = ∫ (𝑝 (𝑡) −𝐻(𝑡,𝑞(𝑡) ,𝑝(𝑡))) 𝑑𝑡. 𝑑 𝑑 ◻ (24) 𝐻:R × R × C 󳨀→ C 𝑎 ◻𝑡 (18) (𝑡, 𝑞, 𝑝) 󳨃󳨀→ 𝐻 (𝑡, 𝑞, 𝑝) = 𝑝𝑓(𝑡, 𝑞, 𝑝) − 𝐿 (𝑡, 𝑞,𝑓(𝑡,𝑝)). The nondifferentiable calculus of variations allows usto derive the extremals for H◻. We have the following theorem (see, e.g., [38]). Theorem 22 (nondifferentiable Hamilton’s least-action prin- Theorem 20 (Hamilton’s least-action principle). The curve ciple [17]). Let 0<𝛼,𝛽<1with 𝛼+𝛽.Let >1 𝐿 be an 𝑑 𝑑 2 𝛼 𝑑 (𝑞, 𝑝) ∈ C(𝐼, R )×C(𝐼, C ) is an extremal of the Hamiltonian admissible C -Lagrangian. We assume that 𝛾∈𝐻(𝐼, R ), 𝛼 𝑑 functional such that ◻𝛾/◻𝑡 ∈𝐻 (𝐼, R ). Moreover, we assume that 𝐿(𝑡, 𝛾(𝑡), ◻𝛾(𝑡)/◻𝑡)ℎ(𝑡) satisfies condition (9) for all ℎ∈ 𝑏 𝐻𝛽(𝐼, R𝑑) 𝐻 H (𝑞, 𝑝) = ∫ 𝑝 (𝑡) 𝑞̇ (𝑡) −𝐻(𝑡,𝑞(𝑡) ,𝑝(𝑡))𝑑𝑡 (19) 0 .Let be the corresponding Hamiltonian defined 𝑑 𝑑 𝑎 by (18). A curve 𝛾 󳨃→ (𝑡, 𝑞(𝑡), 𝑝(𝑡)) ∈𝐼× R × C solution of the nondifferentiable Hamiltonian system (23) is an extremal if and only if it satisfies the Hamiltonian system associated with 𝑉=𝐻𝛽(𝐼, R𝑑)× 𝐻 given by of functional (24) over the space of variations 0 𝛽 𝑑 𝐻0 (𝐼, C ). 𝜕𝐻 (𝑡, 𝑞 (𝑡) ,𝑝(𝑡)) 𝑞̇ (𝑡) = , 𝜕𝑝 4. Nondifferentiable Helmholtz Problem (20) 𝜕𝐻 (𝑡, 𝑞 (𝑡) ,𝑝(𝑡)) 𝑝̇ (𝑡) =− In this section, we solve the inverse problem of the nondif- 𝜕𝑞 ferentiable calculus of variations in the Hamiltonian case. We first recall the usual way to derive the Helmholtz conditions called the Hamiltonian equations. following the presentation made by Santilli [28]. Two main derivations are available: A vectorial notation is obtained for the Hamiltonian ⊤ ⊤ equations in posing 𝑧 = (𝑞, 𝑝) and ∇𝐻 = (𝜕𝐻/𝜕𝑞, 𝜕𝐻/𝜕𝑝) , (i) The first is related to the characterization of Hamilto- where ⊤ denotes the transposition. The Hamiltonian equa- nian systems via the symplectic two-differential form tions are then written as and the fact that by duality the associated one- differential form to a Hamiltonian vector field is 𝑑𝑧 (𝑡) =𝐽⋅∇𝐻(𝑡, 𝑧 (𝑡)) , (21) closed—the so-called integrability conditions. 𝑑𝑡 (ii) The second uses the characterization of Hamiltonian where systems via the self-adjointness of the Frechet´ derivative of the differential operator associated with the 0𝐼𝑑 equation—the so-called Helmholtz conditions. 𝐽=( ) (22) −𝐼𝑑 0 Of course, we have coincidence of the two procedures in the classical case. As there is no analogous of differential form denotes the symplectic matrix with 𝐼𝑑 being the identity in the framework of Cresson’s quantum calculus, we follow 𝑑 matrix on R . the second way to obtain the nondifferentiable analogue of Discrete Dynamics in Nature and Society 5 the Helmholtz conditions. For simplicity, we consider a time- The conditions for the self-adjointness of the differential independent Hamiltonian. The time-dependent case can be operator can be made explicit. They coincide with the integra- doneinthesameway. bility conditions characterizing the exactness of the one-form associated with the vector field by duality (see [28], Theorem 2.7.3 4.1. Helmholtz Conditions for Classical Hamiltonian Systems. , p. 88). 2𝑑 In this section we work on R , 𝑑≥1, 𝑑∈N. ⊤ Theorem 25 (integrability conditions). Let 𝑋(𝑞, 𝑝) = (𝑋𝑞(𝑞, 𝑝),𝑝 𝑋 (𝑞, 𝑝)) be a vector field. The differential operator 4.1.1. Symplectic Scalar Product. The symplectic scalar product 𝑂𝑋 given by (28) has a self-adjoint Frechet´ derivative with ⟨⋅, ⋅⟩ 2 𝐽 is defined by respect to the 𝐿 symplectic scalar product if and only if ⟨𝑋, 𝑌⟩ = ⟨𝑋,𝐽⋅𝑌⟩ 𝐽 (25) ⊤ 𝜕𝑋𝑞 𝜕𝑋𝑝 𝜕𝑋𝑞 𝜕𝑋𝑝 2𝑑 +( ) =0, , 𝑎𝑟𝑒 𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐. (31) for all 𝑋, 𝑌 ∈ R ,where⟨⋅, ⋅⟩ denotes the usual scalar 𝜕𝑞 𝜕𝑝 𝜕𝑝 𝜕𝑞 product and 𝐽 is the symplectic matrix (22). We also consider 𝐿2 ⟨⋅, ⋅⟩ the symplectic scalar product induced by 𝐽 defined for 4.2. Helmholtz Conditions for Nondifferentiable Hamiltonian 𝑓, 𝑔 ∈ C0([𝑎, 𝑏], R2𝑑) by Systems. The previous scalar products extend naturally to 0<𝛼<1 (𝑞, 𝑝) ∈ 𝑏 complex valued functions. Let and let 𝐻𝛼(𝐼, R𝑑)×𝐻𝛼(𝐼, C𝑑) ◻𝑞/◻𝑡𝛼 ∈𝐻 (𝐼, C𝑑) ⟨𝑓, 𝑔⟩ 2 =∫ ⟨𝑓 (𝑡) ,𝑔(𝑡)⟩ 𝑑𝑡. (26) ,suchthat and 𝐿 ,𝐽 𝐽 𝛼 𝑑 𝑎 ◻𝑝/◻𝑡 ∈𝐻 (𝐼, C ). We consider first-order nondifferential equations of the form 4.1.2. Adjoint of a Differential Operator. In the following, we consider first-order differential equations of the form ◻ 𝑞 𝑋 (𝑞, 𝑝) ( )=( 𝑞 ). 𝑞 𝑋 (𝑞, 𝑝) (32) 𝑑 𝑞 ◻𝑡 𝑝 𝑋𝑝 (𝑞, 𝑝) ( )=( ), (27) 𝑑𝑡 𝑝 𝑋𝑝 (𝑞, 𝑝)

1 The associated quantum differential operator is written as where the vector fields 𝑋𝑞 and 𝑋𝑝 are C with respect to 𝑞 and 𝑝. The associated differential operator is written as ◻𝑞 −𝑋𝑞 (𝑞, 𝑝) 𝑞−𝑋̇ (𝑞, 𝑝) ◻𝑡 𝑞 𝑂◻,𝑋 (𝑞, 𝑝) =( ). (33) 𝑂 (𝑞, 𝑝) =( ). ◻𝑝 𝑋 ̇ (28) −𝑋 (𝑞, 𝑝) 𝑝−𝑋𝑝 (𝑞, 𝑝) ◻𝑡 𝑝

A natural notion of adjoint for a differential operator is then A natural notion of adjoint for a quantum differential opera- defined as follows. tor is then defined. 1 2𝑑 1 2𝑑 Definition 23. Let 𝐴:C ([𝑎, 𝑏], R )→C ([𝑎, 𝑏], R ).We 1 2𝑑 1 2𝑑 ∗ Definition 26. Let 𝐴:C◻([𝑎, 𝑏], C )→C◻([𝑎, 𝑏], C ).We define the adjoint 𝐴𝐽 of 𝐴 with respect to ⟨⋅, ⋅⟩𝐿2,𝐽 by ∗ define the adjoint 𝐴𝐽 of 𝐴 with respect to ⟨⋅, ⋅⟩𝐿2,𝐽 by ⟨𝐴 ⋅ 𝑓,𝑔⟩ =⟨𝐴∗ ⋅𝑔,𝑓⟩ . 𝐿2,𝐽 𝐽 𝐿2,𝐽 (29) ⟨𝐴 ⋅ 𝑓,𝑔⟩ =⟨𝐴∗ ⋅𝑔,𝑓⟩ . 𝐿2,𝐽 𝐽 𝐿2,𝐽 (34) ∗ An operator 𝐴 will be called self-adjoint if 𝐴=𝐴𝐽 with 2 respect to the 𝐿 symplectic scalar product. ∗ An operator 𝐴 will be called self-adjoint if 𝐴=𝐴𝐽 with 2 respect to the 𝐿 symplectic scalar product. We can now 4.1.3. Hamiltonian Helmholtz Conditions. The Helmholtz obtain the adjoint operator associated with 𝑂◻,𝑋. conditions in the Hamiltonian case are given by the following 3.12.1 result (see Theorem , p. 176-177 in [28]). Proposition 27. Let 𝛽 be such that 𝛼+𝛽.Let >1 (𝑢, V)∈ 𝛽 𝑑 𝛽 𝑑 𝛼 𝑑 𝐻 (𝐼, R )×𝐻(𝐼, C ),suchthat◻𝑢/◻𝑡∈𝐻(𝐼, C ) and Theorem 24 (Hamiltonian Helmholtz theorem). Let 𝑋(𝑞, 𝑝) 0 0 ⊤ ◻V/◻𝑡 ∈𝛼 𝐻 (𝐼, C𝑑) 𝐷𝑂 beavectorfielddefinedby𝑋(𝑞, 𝑝) =(𝑋𝑞(𝑞, 𝑝),𝑝 𝑋 (𝑞, 𝑝)). .TheFrechet´ derivative ◻,𝑋 of (33) at (𝑞, 𝑝) (𝑢, V) The differential equation (27) is Hamiltonian if and only if the along is then given by associated differential operator 𝑂𝑋 given by (28) has a self- 2 adjoint Frechet´ derivative with respect to the 𝐿 symplectic ◻𝑢 𝜕𝑋 𝜕𝑋 − 𝑞 ⋅𝑢− 𝑞 ⋅ V scalar product. In this case the Hamiltonian is given by ◻𝑡 𝜕𝑞 𝜕𝑝 𝐷𝑂◻,𝑋 (𝑞, 𝑝) (𝑢, V) =( ). (35) 1 ◻V 𝜕𝑋 𝜕𝑋 − 𝑝 ⋅𝑢− 𝑝 ⋅ V 𝐻(𝑞,𝑝)=∫ [𝑝 ⋅𝑞 𝑋 (𝜆𝑞, 𝜆𝑝)𝑝 −𝑞⋅𝑋 (𝜆𝑞, 𝜆𝑝)] 𝑑𝜆. (30) 0 ◻𝑡 𝜕𝑞 𝜕𝑝 6 Discrete Dynamics in Nature and Society

Assume that 𝑢⋅ℎand V ⋅ℎsatisfy condition (9) for any ℎ∈ derivative with respect to the symplectic scalar product if and 𝛽 𝑑 ∗ 𝐻0 (𝐼, C ).Inconsequence,theadjoint𝐷𝑂◻,𝑋 of 𝐷𝑂◻,𝑋(𝑞, 𝑝) only if 2 with respect to the 𝐿 symplectic scalar product is given by ⊤ 𝜕𝑋𝑞 𝜕𝑋𝑝 ∗ +( ) =0, (HC1) 𝐷𝑂◻,𝑋 (𝑞, 𝑝) (𝑢, V) 𝜕𝑞 𝜕𝑝 ⊤ ⊤ ◻𝑢 𝜕𝑋 𝜕𝑋 𝜕𝑋 𝜕𝑋 +( 𝑝 ) ⋅𝑢−( 𝑞 ) ⋅ V 𝑞 , 𝑝 𝑎𝑟𝑒 𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐. ( 2) ◻𝑡 𝜕𝑝 𝜕𝑝 (36) 𝜕𝑝 𝜕𝑞 HC =( ⊤ ⊤ ). ◻V 𝜕𝑋 𝜕𝑋 −( 𝑝 ) ⋅𝑢+( 𝑞 ) ⋅ V Remark 29. One can see that the Helmholtz conditions are ◻𝑡 𝜕𝑞 𝜕𝑞 thesameasintheclassical,discrete,time-scale,andstochastic cases. We expected such a result because Cresson’s quantum Proof. The expression for the Frechet´ derivative of (33) at (𝑞, 𝑝) (𝑢, V) (𝑤, 𝑥) ∈ calculus provides a quantum Leibniz rule and a quantum along is a simple computation. Let version of the fundamental theorem of calculus. If such 𝐻𝛽(𝐼, R𝑑)×𝐻𝛽(𝐼, C𝑑) ◻𝑤/◻𝑡𝛼 ∈𝐻 (𝐼, C𝑑) 0 0 be such that and properties of an underlying calculus exist, then the Helmholtz 𝛼 𝑑 ◻𝑥/◻𝑡 ∈𝐻 (𝐼, C ). By definition, we have conditions will always be the same up to some conditions on the working space of functions. 𝑏 ◻𝑢 ⟨𝐷𝑂 (𝑞, 𝑝) (𝑢, V) , (𝑤, 𝑥)⟩ = ∫ [ ⋅𝑥 ◻,𝑋 𝐿2,𝐽 𝑎 ◻𝑡 We now obtain the main result of this paper, which is the Helmholtz theorem for nondifferentiable Hamiltonian systems. 𝜕𝑋 𝜕𝑋 ◻V −( 𝑞 ⋅𝑢)⋅𝑥−( 𝑞 ⋅ V)⋅𝑥− ⋅𝑤 (37) Theorem 30 (nondifferentiable Hamiltonian Helmholtz the- 𝜕𝑞 𝜕𝑝 ◻𝑡 ⊤ orem). Let 𝑋(𝑞, 𝑝) beavectorfielddefinedby𝑋(𝑞, 𝑝) = 𝜕𝑋 𝜕𝑋 (𝑋𝑞(𝑞, 𝑝),𝑝 𝑋 (𝑞, 𝑝)).Thenondifferentiablesystemof(32)is +( 𝑝 ⋅𝑢)⋅𝑤+( 𝑝 ⋅ V)⋅𝑤]𝑑𝑡. 𝜕𝑞 𝜕𝑝 Hamiltonian if and only if the associated quantum differential operator 𝑂◻,𝑋 given by (33) has a self-adjoint Frechet´ derivative 𝛽 𝑑 𝐿2 As 𝑢⋅ℎand V ⋅ℎsatisfy condition (9) for any ℎ∈𝐻0 (𝐼, C ), with respect to the symplectic scalar product. In this case, the using the quantum Leibniz rule and the quantum version of Hamiltonian is given by the fundamental theorem of calculus, we obtain 1 𝑏 ◻𝑢 𝑏 ◻𝑏 𝐻(𝑞,𝑝)=∫ [𝑝 ⋅𝑞 𝑋 (𝜆𝑞, 𝜆𝑝)𝑝 −𝑞⋅𝑋 (𝜆𝑞, 𝜆𝑝)] 𝑑𝜆. (40) ∫ ⋅𝑏𝑑𝑡=∫ −𝑢 ⋅ 𝑑𝑡, 0 𝑎 ◻𝑡 𝑎 ◻𝑡 (38) Proof. If 𝑋 is Hamiltonian, then there exists a function 𝐻: 𝑏 ◻V 𝑏 ◻𝑏 𝑑 𝑑 ∫ ⋅𝑎𝑑𝑡=∫ −𝑢 ⋅ 𝑑𝑡. R × C → C such that 𝐻(𝑞, 𝑝) is holomorphic with respect 𝑎 ◻𝑡 𝑎 ◻𝑡 to V and differentiable with respect to 𝑞 and 𝑋𝑞 =𝜕𝐻/𝜕𝑝and 𝑋𝑝 =−𝜕𝐻/𝜕𝑞. The nondifferentiable integrability conditions Then, are clearly verified using Schwarz’s lemma. Reciprocally, we 𝑏 assume that 𝑋 satisfies the nondifferentiable integrability ⟨𝐷𝑂 (𝑞, 𝑝) (𝑢, V) , (𝑤, 𝑥)⟩ = ∫ [−𝑢 ◻,𝑋 𝐿2,𝐽 conditions. We will show that 𝑋 is Hamiltonian with respect 𝑎 to the Hamiltonian ⊤ ⊤ ◻𝑥 𝜕𝑋 𝜕𝑋 1 ⋅( −( 𝑝 ) ⋅𝑤+( 𝑞 ) ⋅𝑥)+V (39) 𝐻(𝑞,𝑝)=∫ [𝑝⋅𝑋𝑞 (𝜆𝑞, 𝜆𝑝)𝑝 −𝑞⋅𝑋 (𝜆𝑞, 𝜆𝑝)] 𝑑𝜆; (41) ◻𝑡 𝜕𝑞 𝜕𝑞 0

⊤ ⊤ ◻𝑤 𝜕𝑋 𝜕𝑋 that is, we must show that ⋅( +( 𝑝 ) ⋅𝑤−( 𝑞 ) ⋅𝑥)]𝑑𝑡. ◻𝑡 𝜕𝑝 𝜕𝑝 𝜕𝐻 (𝑞, 𝑝) 𝑋 (𝑞, 𝑝) = , ∗ 𝑞 𝜕𝑝 By definition, we obtain the expression of the adjoint 𝐷𝑂◻,𝑋 2 (42) of 𝐷𝑂◻,𝑋(𝑞, 𝑝) with respect to the 𝐿 symplectic scalar 𝜕𝐻 (𝑞, 𝑝) 𝑋 (𝑞, 𝑝) =− . product. 𝑝 𝜕𝑞 In consequence, from a direct identification, we obtain the nondifferentiable self-adjointess conditions called Helm- We have holtz’s conditions. As in the classical case, we call these con- 1 𝜕𝐻 (𝑞, 𝑝) 𝜕𝑋𝑞 (𝜆𝑞, 𝜆𝑝) ditions nondifferentiable integrability conditions. = ∫ [𝑝⋅𝜆 −𝑋𝑝 (𝜆𝑞, 𝜆𝑝) 𝜕𝑞 0 𝜕𝑞 Theorem 28 (nondifferentiable integrability conditions). Let ⊤ 𝑋(𝑞, 𝑝) =(𝑋(𝑞, 𝑝), 𝑋 (𝑞, 𝑝)) 𝜕𝑋𝑝 (𝜆𝑞, 𝜆𝑝) 𝑞 𝑝 be a vector field. The differ- −𝑞⋅𝜆 ]𝑑𝜆, ential operator 𝑂◻,𝑋 given by (33) has a self-adjoint Frechet´ 𝜕𝑞 Discrete Dynamics in Nature and Society 7

1 + 𝑑 𝜕𝐻 (𝑞, 𝑝) 𝜕𝑋𝑞 (𝜆𝑞, 𝜆𝑝) with 𝑚∈R and 𝑞, 𝑝 ∈ R .Thisequationpossessesanatural = ∫ [𝑋𝑝 (𝜆𝑞, 𝜆𝑝) +𝑝⋅𝜆 𝜕𝑝 0 𝜕𝑝 Hamiltonian structure with the Hamiltonian given by

𝜕𝑋 (𝜆𝑞, 𝜆𝑝) 1 2 −𝑞⋅𝜆 𝑝 ]𝑑𝜆. 𝐻(𝑞,𝑝)= 𝑝 +𝑈(𝑞). (49) 𝜕𝑝 2𝑚 (43) Using Cresson’s quantum calculus, we obtain a natural nondifferentiable system given by Using the nondifferentiable integrability conditions, we obtain ◻𝑞 𝑝 = , 1 𝜕𝐻 (𝑞, 𝑝) 𝜕(𝜆𝑋𝑝 (𝜆𝑞, 𝜆𝑝)) ◻𝑡 𝑚 = ∫ − 𝑑𝜆 = −𝑋 (𝑞, 𝑝) , (50) 𝜕𝑞 𝜕𝜆 𝑝 ◻𝑝 0 =−𝑈󸀠 (𝑞) . (44) ◻𝑡 1 𝜕𝐻 (𝑞, 𝑝) 𝜕(𝜆𝑋𝑞 (𝜆𝑞, 𝜆𝑝)) = ∫ 𝑑𝜆𝑞 =𝑋 (𝑞, 𝑝) , The Hamiltonian Helmholtz conditions are clearly satisfied. 𝜕𝑞 0 𝜕𝜆 which concludes the proof. Remark 31. It must be noted that Hamiltonian (49) associated with (50) is recovered by formula (40). 5. Applications 6. Conclusion We now provide two illustrative examples of our results: one with the formulation of dynamical systems with linear parts We proved a Helmholtz theorem for nondifferentiable equa- and another with Newton’s equation, which is particularly tions, which gives necessary and sufficient conditions for the useful to study partial differentiable equations such as the existence of a Hamiltonian structure. In the affirmative case, Navier-Stokes equation. Indeed, the Navier-Stokes equation the Hamiltonian is given. Our result extends the results of can be recovered from a Lagrangian structure with Cresson’s the classical case when restricting attention to differentiable quantum calculus [17]. For more applications see [34]. functions. An important complementary result for the non- 𝛼 𝑑 𝛼 𝑑 Let 0<𝛼<1and let (𝑞, 𝑝) ∈𝐻 (𝐼, R )×𝐻 (𝐼, C ) be differentiable case is to obtain the Helmholtz theorem in the 𝛼 𝑑 𝛼 𝑑 such that ◻𝑞/◻𝑡 ∈𝐻 (𝐼, C ) and ◻𝑞/◻𝑡∈𝐻 (𝐼, C ). Lagrangian case. This is nontrivial and will be subject of future research. 5.1. The Linear Case. Let us consider the discrete nondifferentiable system Competing Interests ◻𝑞 =𝛼𝑞+𝛽𝑝, The authors declare that they have no competing interests. ◻𝑡 ◻𝑝 (45) =𝛾𝑞+𝛿𝑝, Acknowledgments ◻𝑡 where 𝛼, 𝛽, 𝛾,and𝛿 are constants. The Helmholtz condition This work was supported by FCT and CIDMA through (HC2) is clearly satisfied. However, system (45) satisfies the project UID/MAT/04106/2013. The first author is grateful to condition (HC1) if and only if 𝛼+𝛿=0.Asaconsequence, CIDMA and DMat-UA for the hospitality and good working linear Hamiltonian nondifferentiable equations are of the conditions during his visit at University of Aveiro. The form authors would like to thank an anonymous referee for careful ◻𝑞 reading of the submitted paper and for useful suggestions. =𝛼𝑞+𝛽𝑝, ◻𝑡 ◻𝑝 (46) References =𝛾𝑞−𝛼𝑝. ◻𝑡 [1] V. Kac and P. Cheung, Quantum Calculus, Universitext, Sprin- Using formula (40), we compute explicitly the Hamiltonian, ger, New York, NY, USA, 2002. which is given by [2] N. Martins and D. F. M. Torres, “Higher-order infinite horizon variational problems in discrete quantum calculus,” Computers 1 2 2 𝐻(𝑞,𝑝)= (𝛽𝑝 −𝛾𝑞 )+𝛼𝑞⋅𝑝. (47) & Mathematics with Applications,vol.64,no.7,pp.2166–2175, 2 2012. [3] A.M.C.BritodaCruz,N.Martins,andD.F.M.Torres,“Higher- 5.2. Newton’s Equation. Newton’s equation (see [38]) is given order Hahn’s quantum variational calculus,” Nonlinear Analysis: by Theory, Methods & Applications, vol. 75, no. 3, pp. 1147–1157, 𝑝 2012. 𝑞=̇ , 𝑚 [4] A. B. Malinowska and N. Martins, “Generalized transversality (48) conditions for the Hahn quantum variational calculus,” Opti- 󸀠 𝑝=−𝑈̇ (𝑞) , mization,vol.62,no.3,pp.323–344,2013. 8 Discrete Dynamics in Nature and Society

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