Necessary condition for an Euler-Lagrange equation on time scales

We prove a necessary condition for a dynamic integro-differential equation to be an Euler-Lagrange equation. New and interesting results for the discrete and quantum calculus are obtained as particular cases. An example of a second order dynamic equation, which is not an Euler-Lagrange equation on an arbitrary time scale, is given.


Introduction
The time-scale calculus is a unification of the theories of difference and differential equations, unifying integral and differential calculus with the calculus of finite differences, and offering a formalism for studying hybrid discrete-continuous dynamical systems [1,2]. It has applications in any field that requires simultaneous modeling of discrete and continuous data [3,4,5].
The study of optimal control problems on arbitrary time scales is a subject under strong current research [6,7]. This is particularly true for the particular, but rich case, of the calculus of variations on time scales [8,9,10]. Compared with the direct problem, that establish dynamic equations of Euler-Lagrange type to the time-scale variational problems, the inverse problem has not yet been studied in the framework of time scales. It turns out that there is a simple explanation for the absence of such an inverse general theory for the time-scale variational calculus: the classical approach relies on the use of the chain rule, which is not valid in the general context of time scales [2]. To address the problem, a different approach to the subject is needed.
In this paper we introduce a completely different approach to the inverse problem of the calculus of variations, using an integral perspective instead of the classical differential point of view [11,12]. The differential form of equations is often related to dynamics via the time derivative. The integral form has proved to be successful for proving the existence and uniqueness of solutions, to study analytical properties of solutions, and to prove coherence of variational embeddings [13]. Here we show its usefulness with respect to the inverse problem of the calculus of variations. We prove a necessary condition for an integro-differential equation on an arbitrary time scale T to be an Euler-Lagrange equation, related with a property of self-adjointness (Definition 3.1) of the equation of variation (Definition 3.2) of the given dynamic integro-differential equation.
The text is organized as follows. Section 2 provides all the necessary definitions and results of the delta-calculus on time scales, which will be used throughout the text. The main results are proved in Section 3. We present a sufficient condition of self-adjointness for an integro-differential equation (Lemma 3.4). Using this property, we prove a necessary condition for a general (non-classical) inverse problem of the calculus of variations on an arbitrary time scale (Theorem 3.5). As a result, we obtain a useful tool to identify integro-differential equations which are not Euler-Lagrange equations (Remark 3.6). To illustrate the method, we give a second order dynamic equation on time scales which is not an Euler-Lagrange equation (Example 3.8). Next we apply Theorem 3.5 to the particular cases of time scales T ∈ {R, hZ, q Z }, h > 0, q > 1 (Corollaries 3.9, 3.10, and 3.11). In Section 4 some final remarks are presented. We begin by proving the equivalence between an integro-differential equation and a second order dynamic equation (Proposition 4.1). Then we show that, due to lack of a chain rule in an arbitrary time scale, it is impossible to obtain an equivalence between equations of variation in integral and differential forms. This is in contrast with the classical case T = R, where such equivalence holds (Proposition 4.2).

Preliminaries
In this section we introduce basic definitions and theorems that will be useful in the sequel. For more results concerning the theory of time scales we refer the reader to the books [2,4].
The delta derivative is defined for points from the set Definition 2.2 (Section 1.1 of [2]). Let f : T → R and t ∈ T κ . We define f ∆ (t) to be the number (provided it exists) with the property that given any ε > 0, there is a neighborhood U of t such that Theorem 2.3 (Theorem 1.16 of [2]). Let f : T → R and t ∈ T κ . If f is continuous at t and t is right-scattered, then f is delta differentiable at t with Theorem 2.4 (Theorem 1.20 of [2]). Let f, g : T → R be delta differentiable at t ∈ T κ . Then, Theorem 2.5 (Theorem 1.16 from [2]).
Definition 2.6 (Definition 1.58 of [2]). A function f : T → R is called rdcontinuous provided it is continuous at right-dense points in T and its left-sided limits exist (finite) at all left-dense points in T.
The set of all rd-continuous functions f : The set of functions f : T → R that are delta differentiable and whose derivative is rd-continuous is denoted by Theorem 2.9 (Theorem 1.74 of [2]). Every rd-continuous function f has an antiderivative F . In particular, if t 0 ∈ T, then F defined by is an antiderivative of f .
where the integral on the right side is the usual Riemann integral.
Theorem 2.11 (Theorem 1.77 from [2]). If a, b, c ∈ T, α ∈ R, and f, g ∈ C rd (T), then For more properties of the delta derivative and delta integral we refer the reader to [2,4].

Main results
Our main result (Theorem 3.5) provides a necessary condition for an integrodifferential equation on an arbitrary time scale to be an Euler-Lagrange equation. For that the notions of self-adjointness (Definition 3.1) and equation of variation (Definition 3.2) are essential. These definitions, in integro-differential form, are new (cf. the notion of self-adjointness for a dynamic time-scale equation of second order in [2, Sec. 4.1] and the notion of equation of variation for a second order differential equation in [12]). Definition 3.1 (First order self-adjoint integro-differential equation). A first order integro-differential dynamic equation is said to be self-adjoint if it has the form with p, q ∈ C rd , p = 0 for all t ∈ T, and t 0 ∈ T.
Let D be the set of all functions y : T −→ R such that y ∆ : T κ −→ R is continuous. A function y ∈ D is said to be a solution of (1) provided Ly(t) = const holds for all t ∈ T κ . Along the text we use the operators [·] T and · T defined by [y] T (t) := (t, y σ (t), y ∆ (t)) and y T (t) := (t, y σ (t), y ∆ (t), y ∆∆ (t)). (2)

Definition 3.2 (Equation of variation). Let
be an integro-differential equation on time scales with ∂ 3 H = 0 and t −→ The equation of variation associated with (3) is given by Remark 3.3. The equation of variation (4) can be interpreted in the following way. Assuming y = y(t, b), b ∈ R, is a one-parameter solution of a given integrodifferential equation (3), then Let u(t) be a particular solution, that is, u(t) = y(t,b) for a certainb. Differentiating (5) with respect to the parameter b and then putting b =b, we obtain equation (4).
then its equation of variation (4) is self-adjoint.
Proof. Let us consider a given equation of variation (4). Using Theorem 2.5 and third item of Theorem 2.11, we expand the two components of the given equation: Hence, equation of variation (4) can be written in the form If (6) holds, then (7) is a particular case of (1) with This concludes the proof. L(t, y σ (t), y ∆ (t))∆t, where L ∈ C 2 . Since the Euler-Lagrange equation in integral form of (9) is given by [13,15,16]
Remark 3.6. In practical terms, Theorem 3.5 is useful to identify equations which are not Euler-Lagrange equations: if the equation of variation (4) of a given dynamic equation (3) is not self-adjoint, then we conclude that (3) is not an Euler-Lagrange equation.
Remark 3.7 (Self-adjointness for a second order differential equation). Let p be delta-differentiable in Definition 3.1 and u ∈ C 2 rd . Then, by differentiating (1), one obtains a second-order self-adjoint dynamic equation with q ∈ C rd and p ∈ C 1 rd and p = 0 for all t ∈ T. Now we present an example of a second order differential equation on time scales which is not an Euler-Lagrange equation.
We may write this equation (10) in integro-differential form (3): where H[y] T (t) = y ∆ (t) and G[y] T (t) = y ∆ (t) − t. Because then the equation of variation associated with (11) is given by We may notice that equation (12) cannot be written in form (1), hence, it is not self-adjoint. Indeed, notice that (12) is a first-order dynamic equation while from Remark 3.7 one obtains a second-order dynamic equation. Following Theorem 3.5 (see Remark 3.6), we conclude that equation (10) is not an Euler-Lagrange equation.
Now we consider the particular case of Theorem 3.5 when T = R and y ∈ C 2 ([t 0 , t 1 ]; R). In this case our operator [·] T of (2) has the form [y] R (t) = (t, y(t), y ′ (t)), while condition (1) can be written as Corollary 3.9. If a given integro-differential equation G(s, y(s), y ′ (s))ds = const is to be the Euler-Lagrange equation of a variational problem L(t, y(t), y ′ (t))dt (cf., e.g., [17]), then its equation of variation must be self-adjoint, in the sense of Definition 3.1 with (1) given by (13).
Proof. Follows from Theorem 3.5 with T = R.
Now we consider the particular case of Theorem 3.5 when T = hZ, h > 0. In this case our operator [·] T of (2) has the form For T = hZ, h > 0, condition (1) can be written as Corollary 3.10. If a given difference equation is to be the Euler-Lagrange equation of a discrete variational problem (cf., e.g., [18]), then its equation of variation is self-adjoint, in the sense of Definition 3.1 with (1) given by (14).
Proof. Follows from Theorem 3.5 with T = hZ.
Finally, let us consider the particular case of Theorem 3.5 when T = q Z = q Z ∪ {0}, where q Z = q k : k ∈ Z, q > 1 . In this case operator [·] T of (2) has the form [y] q Z (t) = (t, y(qt), ∆ q y(t)) =: For T = q Z , q > 1, condition (1) can be written as (cf., e.g., [19]), where we use notation r(t) instead of q(t) in order to avoid confusion between the q = const that defines the time scale and function q(t) of (1).
Corollary 3.11. If a given q-equation sG(s, y(qs), ∆ q y(s)) = const, q > 1, is to be the Euler-Lagrange equation of a variational problem tL(t, y(qt), ∆ q y(t)), t 0 , t 1 ∈ q Z , then its equation of variation is self-adjoint, in the sense of Definition 3.1 with (1) given by (15).
Proof. Choose T = q Z in Theorem 3.5.
The reader interested in the study of Euler-Lagrange equations for problems of the q-variational calculus is referred to [16,20,21] and references therein.

Discussion
In an arbitrary time scale T, it is easy to show equivalence between the integrodifferential equation (3) and the second order differential equation (16) below (Proposition 4.1). However, when we consider equations of variations of them, we notice that it is impossible to prove an equivalence between them in an arbitrary time scale. This impossibility is true even in the discrete time scale Z. The main reason is the lack of chain rule on time scales ([2, Example 1.85]). However, in T = R we can present this equivalence (Proposition 4.2).
Proposition 4.1. The integro-differential equation (3) is equivalent to the second order delta differential equation Proof. Let (16) be a given second order differential equation. We may write it as a sum of two components Let F y T = H ∆ [y] T . Then, H ∆ (t, y σ (t), y ∆ (t)) + G(t, y σ (t), y ∆ (t)) = 0.
Integrating both sides of equation (18) from t 0 to t, we obtain the integrodifferential equation (3).
Let T be a time scale such that µ is delta differentiable. The equation of variation of a second order differential equation (16) is given by Equation (19) is obtained by using the method presented in Remark 3.3. In an arbitrary time scale it is impossible to prove the equivalence between the equation of variation (4) and (19). Indeed, after differentiating both sides of equation (4) and using the product rule given by Theorem 2.4, we have The direct calculations allow us to write the equation (20) in form that is, using Theorem 2.5, We are not able to prove that the coefficients of equation (21) are the same as in (19), respectively. This is due to the fact that we cannot find the partial derivatives of (16), that is, ∂ 4 W u T (t), ∂ 3 W u T (t) and ∂ 2 W u T (t), from equation (18) because of lack of chain rule in an arbitrary time scale. The equivalence, however, is true for T = R. is equivalent to the second order differential equation Proof. We show that coefficients of equations (22) and (23) are the same, respectively. Let T = R. From equation (17) and relation F u R = d dt H[u] R we have W (t, u(t), u ′ (t), u ′′ (t)) = d dt H(t, u(t), u ′ (t)) + G(t, u(t), u ′ (t)).
Using notation (2) and chain rule (that is valid for T = R only) we can calculate the partial derivatives: • ∂ 4 W u R (t) = ∂ 3 H[u] R (t).
After differentiation both sides of equation (22) we obtain Hence, the intended equivalence is proved.
Proposition 4.2 allows us to obtain the classical result of [12, Theorem II] as a corollary of our Theorem 3.5. The absence of a chain rule on time scales (even for T = Z) implies that the classical approach of [12] fails on time scales. This is the reason why here we introduced a completely different approach to the subject based on the integro-differential form. The case T = Z was recently investigated in [11]. However, similarly to [12], the approach of [11] is based on the differential form and cannot be extended to general time scales.