The Second Noether Theorem on Time Scales

Abstract

We extend the second Noether theorem to variational problems on time scales. As corollaries we obtain the classical second Noether theorem, the second Noether theorem for the ℎ-calculus and the second Noether theorem for the 𝑞-calculus.

1. Introduction

In 1915, general relativity was almost a finished theory, but still there was a problem regarding the conservation of energy. David Hilbert asked for help Emmy Noether. She solved the problem proving two remarkable theorems that relate to the invariance of a variational integral with properties of its Euler-Lagrange equations. These results were published in 1918 in the paper Invariante Variations probleme [1]. Noether was described by many important scientists, such as Pavel Alexandrov, Albert Einstein, Jean Dieudonné, and David Hilbert, as the most important woman in the history of mathematics. In order to get a good exposition of the history of Emmy Noether and her important contributions to fundamental physics and mathematics, we refer the reader to the recent book [2]. This book explains very clearly that it took too much time before mathematicians and physicists began to recognize the importance of Noether’s theorems: until 1950 Noether’s theorems were poorly understood and Noether’s name disappeared almost entirely.

The first theorem in [1], usually known as Noether’s theorem, guarantees that the invariance of a variational integral with respect to continuous symmetry transformations that depend on 𝜌 parameters implies the existence of 𝜌 conserved quantities along the Euler-Lagrange extremals. Such transformations are global transformations. Noether’s theorem explains all conservation laws of mechanics: conservation of energy comes from invariance of the system under time translations; conservation of linear momentum comes from invariance of the system under spacial translations; conservation of angular momentum reflects invariance with respect to spatial rotations.

The first Noether theorem is nowadays a well-known tool in modern theoretical physics, engineering, and the calculus of variations [3–9]. Inexplicably, it is still not well known that the famous paper of Emmy Noether includes another important result, the second Noether theorem, which applies to variational problems that are invariant under a certain group of transformations, a so-called infinite continuous group, which depends on arbitrary functions and their derivatives (see also [10]). Such transformations are local transformations because they can affect every part of the system differently. Noether’s second theorem states that if a variational integral has an infinite-dimensional Lie algebra of infinitesimal symmetries parameterized linearly by 𝑟 arbitrary functions and their derivatives up to a given order 𝑚, then there are 𝑟 identities between Euler-Lagrange expressions and their derivatives up to order 𝑚. These identities Noether called them “dependencies.” For example, the Bianchi identities, in the general theory of relativity, are examples of such “dependencies.” Noether’s second theorem has applications in general relativity, electrodynamics, hydromechanics, quantum chromodynamics, and other gauge field theories. Motivated by the important applications of the second Noether theorem, our goal in this paper is to generalize this result proving that the second Noether theorem is valid for an infinite number of time scales. As we will see, in the particular case where the time scale 𝕋 is ℝ, we get from our result the classical second Noether theorem; when 𝕋=ℤ, we obtain the analogue of the second Noether theorem for the difference calculus of variations; when 𝕋=𝑞^ℕ0 (for some 𝑞>1), we obtain a new result: the second Noether theorem for the 𝑞-calculus (quantum calculus). For more on the theory of quantum calculus and quantum calculus of variation, we refer to [11–21].

The theory of time scales was introduced in 1988 by Hilger in his Ph.D. thesis [22] as a means of unifying theories of differential calculus and difference calculus into a single theory. With a short time this unification aspect has been supplemented by the extension and generalization features. The time scale calculus allows to consider more complex time domains, such as ℎℤ, 𝕋=𝑞^ℕ0, or hybrid domains. The study of the calculus of variations in the context of time scales had its beginning only in 2004 with the paper [23] of Bohner (see also [24]). Since then, the variational calculus on time scales advanced fairly quickly, as can be verified with the large number of published papers on the subject [25–35]. Noether’s first theorem has been extended to the variational calculus on time scales using several approaches [4, 9, 36], while the second Noether theorem on times scales is still not available in the literature. So, there is evidently a need for a time scale analogue of Noether’s second theorem.

The paper is organized as follows. In Section 2, we review some preliminaries about single-variable variational calculus on time scales; for example, we recall the Euler-Lagrange equation for a delta variational problem. Our main results are stated in Section 3. Namely, in Subsection 3.1, we prove Noether’s second theorem for variational problems involving a single delta integral (with and without transformation of time) and in Section 3.2 we prove Noether’s second theorem for variational problems involving multiple delta integrals (without transformation of time). Section 4 provides a concrete example of application of our results. Finally, in Section 5, we present some concluding remarks.

2. Preliminaries

In this paper, we assume the reader to be familiar with the calculus on time scales. For a good introduction to the theory of time scales, we refer to the well-known books in this field [37, 38]. The first developments on time scale calculus were done essentially using the delta-calculus. However, for some applications, in particular to solve problems of the calculus of variations and control theory in economics, it is often more convenient to work backwards in time, that is, using the nabla-calculus. In this paper, we are concerned with the delta-calculus. It is clear that all the arguments used in the proofs of our results can be modified to work for the nabla-calculus. In what follows, we review some preliminaries about the variational calculus on time scales needed in this paper.

Let 𝕋 be a given time scale, 𝑛∈ℕ, and let 𝐿∶𝕋×ℝ^𝑛×ℝ^𝑛→ℝ be continuous, together with its partial delta derivatives of first and second order with respect to 𝑡 and partial usual derivatives of the first and second order with respect to the other variables. Suppose that 𝑎,𝑏∈𝕋, and 𝑎<𝑏. We consider the following optimization problem on 𝕋: ℒ𝑦=^𝑏_𝑎𝐿𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡)Δ𝑡⟶extremize, (1) where the set of admissible functions are 𝒟=𝑦∣𝑦∶[𝑎,𝑏]∩𝕋⟶ℝ^𝑛,𝑦∈𝐶¹_rd[𝑎,𝑏]∩𝕋,ℝ^𝑛,𝑦(𝑎)=𝛼,𝑦(𝑏)=𝛽, (2) for some 𝛼,𝛽∈ℝ^𝑛, and where 𝜎 is the forward jump operator, 𝑦^Δ is the delta-derivative of 𝑦, and, for 𝑖∈ℕ, 𝐶^𝑖_rd[𝑎,𝑏]∩𝕋,ℝ^𝑛∶=𝑦∣𝑦∶[𝑎,𝑏]∩𝕋⟶ℝ^𝑛,𝑦^Δ𝑖isrd-continuouson[𝑎,𝑏]^𝜅𝑖. (3) As usual, 𝑦^𝜎(𝑡) denotes 𝑦(𝜎(𝑡)) and [𝑎,𝑏]^𝜅𝑖∶=[𝑎,𝜌^𝑖(𝑏)], where 𝜌 is the backward jump operator. By extremize, we mean maximize or minimize.

In what follows, all intervals are time scales intervals; that is, we simply write [𝑎,𝑏] to denote the set [𝑎,𝑏]∩𝕋. Let 𝑦∶=(𝑦₁,…,𝑦_𝑛) and denote by 𝜕𝐿/𝜕𝑦_𝑘 the partial derivative of 𝐿 with respect to 𝑦_𝑘.

Definition 1. We say that 𝑦_∗∈𝐶¹_rd([𝑎,𝑏],ℝ^𝑛) is a local minimizer (resp., local maximizer) for problem (1) if there exists 𝛿>0 such that ℒ𝑦_∗≤ℒ𝑦resp.,ℒ𝑦_∗≥ℒ𝑦, (4) for all 𝑦∈𝐶¹_rd([𝑎,𝑏],ℝ^𝑛) satisfying the boundary conditions 𝑦(𝑎)=𝛼, 𝑦(𝑏)=𝛽, and 𝑦−𝑦_∗∶=sup_{𝑡∈[𝑎,𝑏]𝜅}𝑦^𝜎(𝑡)−𝑦^𝜎_∗(𝑡)+sup_{𝑡∈[𝑎,𝑏]𝜅}𝑦^Δ(𝑡)−𝑦^Δ_∗(𝑡)<𝛿, (5) where |⋅| denotes a norm in ℝ^𝑛.

Definition 2. We say that 𝜂∈𝐶¹_rd([𝑎,𝑏],ℝ^𝑛) is an admissible variation for problem (1) provided that 𝜂(𝑎)=𝜂(𝑏)=0.

Definition 3. A function 𝑓∶[𝑎,𝑏]×ℝ→ℝ is called continuous in the second variable, uniformly in the first variable, if for each 𝜖>0, there exists 𝛿>0 such that |𝑥₁−𝑥₂|<𝛿 implies |𝑓(𝑡,𝑥₁)−𝑓(𝑡,𝑥₂)|<𝜖 for all 𝑡∈[𝑎,𝑏].

Lemma 4 (see [23]). Suppose that 𝜂∶=(𝜂₁,…,𝜂_𝑛) is an admissible variation for problem (1) and 𝑦∶=(𝑦₁,…,𝑦_𝑛)∈𝒟. Let 𝜙∶ℝ→ℝ and 𝑓∶[𝑎,𝑏]×ℝ→ℝ be defined, respectively, by 𝜙(𝜖)∶=ℒ[𝑦+𝜖𝜂] and 𝑓(𝑡,𝜖)∶=𝐿(𝑡,𝑦^𝜎(𝑡)+𝜖𝜂^𝜎(𝑡),𝑦^Δ(𝑡)+𝜖𝜂^Δ(𝑡)). If 𝜕𝑓/𝜕𝜖 is continuous in 𝜖, uniformly in 𝑡, then 𝜙̇(0)=^𝑏_𝑎^𝑛_𝑘=1𝜕𝐿𝜕𝑦^𝜎_𝑘𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡)𝜂^𝜎_𝑘(𝑡)+𝜕𝐿𝜕𝑦^Δ_𝑘𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡)𝜂^Δ_𝑘(𝑡)Δ𝑡. (6)

Next, we present the following result that is a fundamental tool in the calculus of variations on time scales.

Theorem 5 (Euler-Lagrange equation on time scales [23]). If 𝑦_∗ is a weak local extremizer for problem (1) and 𝐿 satisfies the assumption of Lemma 4, for every 𝑦 and 𝜂, then the components of 𝑦_∗ satisfy the 𝑛 Euler-Lagrange equations: ΔΔ𝑡𝜕𝐿𝜕𝑦^Δ_𝑘𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡)=𝜕𝐿𝜕𝑦^𝜎_𝑘𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡),𝑘=1,…,𝑛 (7) for 𝑡∈[𝑎,𝑏]^𝜅.

It is well known that the forward jump operator, 𝜎, is not delta differentiable for certain time scales. Also, the chain rule as we know it from the classical calculus (i.e., when 𝕋=ℝ) is not valid in general. For this reason, we suppose that the time scale 𝕋 satisfies the following condition(H)for each 𝑡∈𝕋, 𝜎(𝑡)=𝑏₁𝑡+𝑏₀ for some 𝑏₁∈ℝ⁺ and 𝑏₀∈ℝ.

Remark 6. Note that condition (H) implies that 𝜎 is delta differentiable and 𝜎^Δ(𝑡)=𝑏₁, 𝑡∈𝕋^𝜅. Also note that condition (H) describes, in particular, the differential calculus (𝕋=ℝ, 𝑏₁=1, 𝑏₀=0), the difference calculus (𝕋=ℤ, 𝑏₁=1, 𝑏₀=1), the ℎ-calculus (𝕋=ℎℤ∶={ℎ𝑧∶𝑧∈ℤ}, 𝑏₁=1, 𝑏₀=ℎ for some ℎ>0), and the 𝑞-calculus (𝕋=𝑞^ℕ0∶={𝑞^𝑘∶𝑘∈ℕ₀} for some 𝑞>1, 𝑏₁=𝑞, 𝑏₀=0).

Lemma 7 (see [27]). Let 𝕋 be a time scale satisfying condition (H). If 𝑓∶𝕋→ℝ is two times delta differentiable, then 𝑓^𝜎Δ(𝑡)=𝑏₁𝑓^Δ𝜎(𝑡),𝑡∈𝕋^𝜅2. (8)

The next result is also useful for the proofs of our main results.

Lemma 8 (cf. [35]). Assume that the time scale 𝕋 satisfies condition (H), 𝑚∈ℕ, and 𝜂∈𝐶^2𝑚_rd([𝑎,𝑏],ℝ) is such that 𝜂^Δ𝑖(𝑎)=0 for all 𝑖=0,1,…,𝑚. Then, 𝜂^{𝜎Δ𝑖−1}(𝑎)=0 for each 𝑖=1,…,𝑚.

3. Main Results

In this section, we formulate and prove the second Noether theorem for single and multiple integral variational problems.

3.1. Noether’s Second Theorem: Single Delta Integral Case

In this subsection, we suppose that the time scale 𝕋 satisfies condition (H) and that 𝐿 satisfies the assumption of Lemma 4, for every 𝑦 and 𝜂. As usual, 𝜂^Δ0 and 𝜂^𝜎0 denote 𝜂. Let 𝑚 be a fixed natural number. We also assume that the time scale interval [𝑎,𝑏] has, at least, 2𝑚+1 points.

We begin with some technical results that will be useful in the proofs of Theorems 16 and 22.

Lemma 9 (higher-order fundamental lemma of the calculus of variations). Let 𝕋 be a time scale satisfying condition (H) and 𝑓₀,𝑓₁,…,𝑓_𝑚∈𝐶_rd([𝑎,𝑏],ℝ). If ^𝜌𝑚−1_𝑎^(𝑏)𝑚_𝑖=0𝑓_𝑖(𝑡)𝜂^{𝜎𝑚−𝑖Δ𝑖}(𝑡)Δ𝑡=0, (9) for all 𝜂∈𝐶^2𝑚_rd([𝑎,𝑏],ℝ) such that 𝜂(𝑎)=0,𝜂𝜌^𝑚−1(𝑏⋮𝜂)=0,_Δ𝑚−1(𝑎)=0,𝜂^Δ𝑚−1𝜌^𝑚−1(𝑏)=0, (10) then ^𝑚_𝑖=0(−1)^𝑖1𝑏₁^{𝑖(𝑖−1)/2}𝑓^Δ𝑖_𝑖(𝑡)=0,𝑡∈[𝑎,𝑏]^𝜅𝑚. (11)

Proof. The proof is similar to the proof of Lemma 16 of [33].

Remark 10. We emphasize that the delta differentiability of the functions 𝑓₀,𝑓₁,…,𝑓_𝑚 was not assumed in advance.

Lemma 11. Assume that the time scale 𝕋 satisfies condition (H) and 𝜂∈𝐶^2𝑚_rd([𝑎,𝑏],ℝ) is such that, 𝜂^Δ𝑖(𝑎)=0,𝑖=0,1,…,𝑚. (12) Then, 𝜂^𝜎𝑖(𝑎)=0,𝑖=0,1,…,𝑚. (13)

Proof. If 𝑎 is right-dense, the result is trivial. Suppose that 𝑎 is right-scattered. Since 𝜂^Δ(𝑎)=(𝜂^𝜎(𝑎)−𝜂(𝑎))/(𝜎(𝑎)−𝑎)=0 and 𝜂(𝑎)=0, we conclude that 𝜂^𝜎(𝑎)=0. Since 𝜂^Δ2(𝑎)=(𝜂^Δ)^Δ(𝑎)=((𝜂^Δ)^𝜎(𝑎)−𝜂^Δ(𝑎))/(𝜎(𝑎)−𝑎)=0 and 𝜂^Δ(𝑎)=0, then (𝜂^Δ)^𝜎(𝑎)=0. Using Lemma 7, we get (𝜂^𝜎)^Δ(𝑎)=0. Since (𝜂^𝜎)^Δ(𝑎)=(𝜂^𝜎2(𝑎)−𝜂^𝜎(𝑎))/(𝜎(𝑎)−𝑎)=0 and 𝜂^𝜎(𝑎)=0, we conclude that 𝜂^𝜎2(𝑎)=0.
Since 𝜂^Δ3(𝑎)=(𝜂^Δ2)^Δ(𝑎)=((𝜂^Δ2)^𝜎(𝑎)−𝜂^Δ2(𝑎))/(𝜎(𝑎)−𝑎)=0 and 𝜂^Δ2(𝑎)=0, then 𝜂^Δ2𝜎(𝑎)=0. Using Lemma 7, we get (𝜂^𝜎)^Δ2(𝑎)=0. Since (𝜂^𝜎)^Δ2(𝑎)=(𝜂^𝜎Δ𝜎(𝑎)−𝜂^𝜎Δ(𝑎))/(𝜎(𝑎)−𝑎)=0 and 𝜂^𝜎Δ(𝑎)=0, then 𝜂^𝜎Δ𝜎(𝑎)=0. Lemma 7 proves that 𝜂^𝜎2Δ(𝑎)=0. Since 𝜂^𝜎2Δ(𝑎)=(𝜂^𝜎3(𝑎)−𝜂^𝜎2(𝑎))/(𝜎(𝑎)−𝑎)=0 and 𝜂^𝜎2(𝑎)=0, we get 𝜂^𝜎3(𝑎)=0. Repeating recursively this process, we conclude the proof.

Lemma 12. Assume that the time scale 𝕋 satisfies condition (H) and 𝜂∈𝐶^2(𝑚−1)_rd([𝑎,𝑏],ℝ) is such that, 𝜂^Δ𝑖(𝑎)=0,𝑖=0,1,…,𝑚−1. (14) Then, 𝜂^{𝜎Δ𝑚−2}(𝑎)=𝜂^{𝜎2Δ𝑚−3}(𝑎)=𝜂^{𝜎3Δ𝑚−4}(𝑎)=…=𝜂^{𝜎𝑚−2Δ}(𝑎)=0. (15)

Proof. If 𝑎 is right-dense, the result is trivial. Suppose that 𝑎 is right-scattered. Since 𝜂^Δ𝑚−1(𝑎)=(𝜂^{Δ𝑚−2𝜎}(𝑎)−𝜂^Δ𝑚−2(𝑎))/(𝜎(𝑎)−𝑎)=0 and 𝜂^Δ𝑚−2(𝑎)=0, then 𝜂^{Δ𝑚−2𝜎}(𝑎)=0. Using Lemma 7, we get 𝜂^{𝜎Δ𝑚−2}(𝑎)=0 (or use Lemma 8). Note that 𝜂^{𝜎Δ𝑚−2}(𝑎)=(𝜂^{𝜎Δ𝑚−3𝜎}(𝑎)−𝜂^{𝜎Δ𝑚−3}(𝑎))/(𝜎(𝑎)−𝑎)=0. Lemma 8 shows that 𝜂^{𝜎Δ𝑚−3}(𝑎)=0; hence, 𝜂^{𝜎Δ𝑚−3𝜎}(𝑎)=0. Using Lemma 7, we get 𝜂^{𝜎2Δ𝑚−3}(𝑎)=0. Next, we prove that 𝜂^{𝜎2Δ𝑚−4}(𝑎)=0. Since 𝜂^{𝜎Δ𝑚−3}(𝑎)=(𝜂^{𝜎Δ𝑚−4𝜎}(𝑎)−𝜂^{𝜎Δ𝑚−4}(𝑎))/(𝜎(𝑎)−𝑎)=0 and 𝜂^{𝜎Δ𝑚−4}(𝑎)=0 (by Lemma 8), then 𝜂^{𝜎Δ𝑚−4𝜎}(𝑎)=0. Lemma 7 shows that 𝜂^{𝜎2Δ𝑚−4}(𝑎)=0. Since 𝜂^{𝜎2Δ𝑚−3}(𝑎)=(𝜂^{𝜎2Δ𝑚−4𝜎}(𝑎)−𝜂^{𝜎2Δ𝑚−4}(𝑎))/(𝜎(𝑎)−𝑎)=0 and 𝜂^{𝜎2Δ𝑚−4}(𝑎)=0, then 𝜂^{𝜎2Δ𝑚−4𝜎}(𝑎)=0. Lemma 7 shows that 𝜂^{𝜎3Δ𝑚−4}(𝑎)=0. Repeating recursively this process, we prove the intended result.

Let 𝑦∶=(𝑦₁,𝑦₂,…,𝑦_𝑛). Firstly, we will prove the second Noether theorem without transformation of time. For that consider the following transformations that depend on arbitrary functions 𝑝₁,𝑝₂,…,𝑝_𝑟 and their delta-derivatives up to order 𝑚: 𝑡=𝑡𝑦_𝑘(𝑡)=𝑦_𝑘(𝑡)+^𝑟_𝑗=1𝑇^𝑘𝑗𝑝_𝑗(𝑡),𝑘=1,2,…,𝑛, (16) where, for each 𝑗=1,2,…,𝑟, 𝑝_𝑗∈𝐶^2𝑚_rd([𝑎,𝜎^𝑚(𝑏)],ℝ), 𝑇^𝑘𝑗𝑝_𝑗∶=^𝑚_𝑖=0𝑔^𝑘_𝑖𝑗𝑝^{𝜎𝑚−(𝑖+1)Δ𝑖}_𝑗, (17) and 𝑔^𝑘_𝑖𝑗∈𝐶¹_rd([𝑎,𝑏],ℝ).

Definition 13. Functional ℒ is invariant under transformations (16) if, and only if, for all 𝑦∈𝐶¹_rd([𝑎,𝑏],ℝ), we have ^𝑏_𝑎𝐿𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡)Δ𝑡=^𝑏_𝑎𝐿𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡)Δ𝑡. (18)

Remark 14. Note that the most common definition of invariance (with equality of the integrals for any subinterval [𝑡_𝑎,𝑡_𝑏]⊆[𝑎,𝑏] with 𝑎,𝑏∈𝕋) implies Definition 13.

Theorem 15 (necessary condition of invariance). If functional ℒ is invariant under transformations (16), then ^𝑛_𝑘=1^𝑏_𝑎𝜕𝐿𝜕𝑦^𝜎_𝑘𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡)⋅^𝑟_𝑗=1𝑇^𝑘𝑗𝑝_𝑗^𝜎(𝑡)+𝜕𝐿𝜕𝑦^Δ_𝑘𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡)⋅^𝑟_𝑗=1𝑇^𝑘𝑗𝑝_𝑗^Δ(𝑡)Δ𝑡=0. (19)

Proof. Using the definition of invariance and noting that the family of transformations (16) depends upon arbitrary functions 𝑝₁,𝑝₂,…,𝑝_𝑟, we conclude that, for any real number 𝜀, ^𝑏_𝑎𝐿𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡)=Δ𝑡_𝑏𝑎𝐿𝑡,𝑦^𝜎₁(𝑡)+𝜀^𝑟_𝑗=1𝑇^1𝑗(𝑝_𝑗)^𝜎(𝑡),…,𝑦^𝜎_𝑛(𝑡)+𝜀^𝑟_𝑗=1𝑇^𝑛𝑗(𝑝_𝑗)^𝜎(𝑡),𝑦^Δ₁(𝑡)+𝜀^𝑟_𝑗=1𝑇^1𝑗(𝑝_𝑗)^Δ(𝑡),…,𝑦^Δ_𝑛(𝑡)+𝜀^𝑟_𝑗=1𝑇^𝑛𝑗(𝑝_𝑗)^Δ(𝑡)Δ𝑡. (20) Differentiating with respect to 𝜀 (use Lemma 4) and taking 𝜀=0, we get equality (19).

Define 𝐸_𝑘(𝐿)∶=𝜕𝐿𝜕𝑦^𝜎_𝑘−ΔΔ𝑡𝜕𝐿𝜕𝑦^Δ_𝑘,𝑘=1,2,…,𝑛. (21) We will call 𝐸_𝑘(𝐿), 𝑘=1,2,…,𝑛, the Euler-Lagrange expressions associated to the Lagrangian 𝐿.

Theorem 16 (Noether’s second theorem without transforming time). If functional ℒ is invariant under transformations (16), then there exist the following identities: ^𝑛𝑚_𝑘=1𝑖=0(−1)^𝑖1𝑏₁^{𝑖(𝑖+1)/2}(𝑔^𝑘_𝑖𝑗)^𝜎𝐸_𝑘(𝐿)^Δ𝑖≡0,𝑗=1,2,…,𝑟. (22)

Proof. Using the necessary condition of invariance (Theorem 15), we conclude that ^𝑛_𝑘=1^𝑏_𝑎𝜕𝐿𝜕𝑦^𝜎_𝑘𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡)⋅^𝑟_𝑗=1𝑇^𝑘𝑗(𝑝_𝑗)^𝜎(𝑡)+𝜕𝐿𝜕𝑦^Δ_𝑘𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡)⋅^𝑟_𝑗=1𝑇^𝑘𝑗(𝑝_𝑗)^Δ(𝑡)Δ𝑡=0. (23) Fix 𝑗=1,2,…,𝑟. By the arbitrariness of 𝑝₁,𝑝₂,…,𝑝_𝑟, we can suppose that 𝑝_ℎ≡0 for ℎ≠𝑗. Therefore, ^𝑛_𝑘=1^𝑏_𝑎𝜕𝐿𝜕𝑦^𝜎_𝑘𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡)⋅𝑇^𝑘𝑗𝑝_𝑗^𝜎(𝑡)+𝜕𝐿𝜕𝑦^Δ_𝑘𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡)⋅𝑇^𝑘𝑗𝑝_𝑗^Δ(𝑡)Δ𝑡=0. (24) Integrating by parts, we obtain ^𝑛_𝑘=1^𝑏_𝑎𝜕𝐿𝜕𝑦^𝜎_𝑘𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡−Δ)Δ𝑡𝜕𝐿𝜕𝑦^Δ_𝑘𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡⋅𝑇_𝑘𝑗)(𝑝_𝑗)^𝜎(𝑡)+Δ𝑡𝜕𝐿𝜕𝑦^Δ_𝑘𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡)⋅𝑇^𝑘𝑗𝑝_𝑗(𝑡)^𝑏_𝑎=0. (25) By the arbitrariness of 𝑝_𝑗, we can restrict to those functions such that 𝑝_𝑗(𝑎)=0,𝑝_𝑗(𝑏)⋮𝑝=0,_{Δ𝑚−1𝑗}(𝑎)=0,𝑝^Δ𝑚−1_𝑗(𝑏)𝑝=0,_{𝜎−1Δ𝑚𝑗}(𝑎)=0,𝑝^{𝜎−1Δ𝑚}_𝑗(𝑏)=0. (26) Using Lemmas 8, 11, and 12, we conclude that 𝑇^𝑘𝑗(𝑝_𝑗)(𝑎)=0 and 𝑇^𝑘𝑗(𝑝_𝑗)(𝑏)=0, 𝑘=1,2,…,𝑛. Then, ^𝑛_𝑘=1^𝑏_𝑎𝜕𝐿𝜕𝑦^𝜎_𝑘𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡−Δ)Δ𝑡𝜕𝐿𝜕𝑦^Δ_𝑘𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡⋅𝑇_𝑘𝑗)𝑝_𝑗^𝜎(𝑡)Δ𝑡=0; (27) that is, ^𝑛_𝑘=1^𝑏_𝑎𝐸_𝑘(𝐿)𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡)⋅𝑇^𝑘𝑗(𝑝_𝑗)^𝜎(𝑡)Δ𝑡=0. (28) Hence, ^𝑛_𝑘=1^𝑏_𝑎𝐸_𝑘(𝐿)𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡)⋅^𝑚_𝑖=0𝑔^𝑘_𝑖𝑗𝑝^{𝜎𝑚−(𝑖+1)Δ𝑖}_𝑗^𝜎(𝑡)Δ𝑡=0. (29) Therefore, by Lemma 7, we get ^𝑏_𝑎^𝑚𝑛_𝑖=0𝑘=1𝐸_𝑘(𝐿)𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡⋅𝑔_{𝑘𝑖𝑗𝜎})(𝑡)1𝑏₁^𝑖𝑝^{𝜎𝑚−𝑖Δ𝑖}_𝑗(𝑡)Δ𝑡=0. (30) Applying the higher-order fundamental lemma of the calculus of variations (Lemma 9), we obtain ^𝑚𝑛_𝑖=0𝑘=1(−1)^𝑖1𝑏₁^{𝑖(𝑖−1)/2}𝐸_𝑘(𝐿)𝑔^𝑘_𝑖𝑗^𝜎1𝑏₁^𝑖Δ𝑖≡0, (31) which is equivalent to ^𝑛𝑚_𝑘=1𝑖=0(−1)^𝑖1𝑏₁^{𝑖(𝑖+1)/2}𝑔^𝑘_𝑖𝑗^𝜎𝐸_𝑘(𝐿)^Δ𝑖≡0 (32) proving the desired result.

We present some particular results that follow from Theorem 16 in the case where 𝕋=ℝ, 𝕋=ℎℤ (for some ℎ>0), and 𝕋=𝑞^ℕ0 (for some 𝑞>1). If 𝕋=ℝ, then 𝜎(𝑡)=𝑡, 𝑓^Δ(𝑡)=𝑓(̇𝑡), and we obtain the classical second Noether theorem without transformation of time.

Corollary 17 (cf. [1]). Let 𝐿∶ℝ×ℝ^𝑛×ℝ^𝑛→ℝ be a 𝐶² function. If functional ℒ defined by ℒ𝑦=^𝑏_𝑎𝐿𝑡,𝑦(𝑡),𝑦̇(𝑡)𝑑𝑡 (33) is invariant under transformations (16) (where 𝜎 denotes in this context the identity function and Δ denotes the usual derivative), then there exist the following identities: ^𝑛𝑚_𝑘=1𝑖=0(−1)^𝑖𝑔^𝑘_𝑖𝑗𝐸_𝑘(𝐿)^(𝑖)≡0,𝑗=1,2,…,𝑟. (34)

Choosing ℎℤ, we obtain Noether’s second theorem without transformation of time for the ℎ-calculus.

Corollary 18. Let ℎ>0, 𝐿∶ℎℤ×ℝ^𝑛×ℝ^𝑛→ℝ and 𝑎,𝑏∈ℎℤ, 𝑎<𝑏. If functional ℒ defined by ℒ𝑦=^𝑏−ℎ_𝑡=𝑎𝐿𝑡,𝑦(𝑡+ℎ),Δ_ℎ𝑦(𝑡) (35) is invariant under transformations (16) (where 𝜎 denotes in this context the function 𝜎(𝑡)=𝑡+ℎ and Δ denotes the ℎ-difference, that is, 𝑦^Δ(𝑡)=Δ_ℎ[𝑦](𝑡)=(𝑦(𝑡+ℎ)−𝑦(𝑡))/ℎ), then there exist the following identities: ^𝑛𝑚_𝑘=1𝑖=0(−1)^𝑖𝑔^𝑘_𝑖𝑗(𝑡+ℎ)⋅𝐸_𝑘(𝐿)𝑡,𝑦(𝑡+ℎ),Δ_ℎ𝑦(𝑡)^Δ𝑖ℎ[]=0,𝑡∈𝑎,𝑏−𝑚ℎ,𝑗=1,2,…,𝑟. (36)

For ℎ=1, we obtain the analogue of Noether’s second theorem for the difference calculus of variations recently proved in [39]. In the case 𝕋=𝑞^ℕ0, we obtain the new result.

Corollary 19. Let 𝑞>1, 𝐿∶𝑞^ℕ0×ℝ^𝑛×ℝ^𝑛→ℝ and 𝑎,𝑏∈𝑞^ℕ0, 𝑎<𝑏. If functional ℒ defined by ℒ𝑦=^𝑏/𝑞_𝑡=𝑎𝐿𝑡,𝑦𝑞𝑡,Δ_𝑞𝑦(𝑡) (37) is invariant under transformations (16) (where 𝜎 denotes in this context the function 𝜎(𝑡)=𝑞𝑡 and Δ denotes the 𝑞-derivative, that is, 𝑦^Δ(𝑡)=Δ_𝑞[𝑦](𝑡)=(𝑦(𝑞𝑡)−𝑦(𝑡))/(𝑞−1)𝑡), then there exist the following identities: ^𝑛𝑚_𝑘=1𝑖=0(−1)^𝑖1𝑞^{𝑖(𝑖+1)/2}𝑔^𝑘_𝑖𝑗𝑞𝑡⋅𝐸_𝑘(𝐿)𝑡,𝑦𝑞𝑡,Δ_𝑞𝑦(𝑡)^Δ𝑖𝑞𝑏=0,𝑡∈𝑎,𝑞^𝑚,𝑗=1,2,…,𝑟. (38)

In order to prove the second Noether theorem with transformation of time, we will consider that the Lagrangian 𝐿 is defined for all 𝑡∈ℝ (not only for 𝑡 from the initial time scale 𝕋), 𝐿∶ℝ×ℝ^𝑛×ℝ^𝑛→ℝ. Consider the following transformations that depend on arbitrary functions 𝑝₁,𝑝₂,…,𝑝_𝑟 and their delta-derivatives up to order 𝑚: 𝑡=𝑡+^𝑟_𝑗=1𝐻^𝑗𝑝_𝑗(𝑡),𝑦_𝑘𝑡=𝑦_𝑘(𝑡)+^𝑟_𝑗=1𝑇^𝑘𝑗𝑝_𝑗(𝑡),𝑘=1,2,…,𝑛, (39) where, for each 𝑗=1,2,…,𝑟, 𝑝_𝑗∈𝐶^2𝑚_rd([𝑎,𝜎^𝑚(𝑏)],ℝ), 𝐻^𝑗𝑝_𝑗∶=^𝑚_𝑖=0𝑓_𝑖𝑗𝑝^{𝜎𝑚−(𝑖+1)Δ𝑖}_𝑗,𝑇^𝑘𝑗𝑝_𝑗∶=^𝑚_𝑖=0𝑔^𝑘_𝑖𝑗𝑝^{𝜎𝑚−(𝑖+1)Δ𝑖}_𝑗, (40)𝑓_𝑖𝑗∈𝐶¹_rd([𝑎,𝑏],ℝ) and 𝑔^𝑘_𝑖𝑗∈𝐶¹_rd([𝑎,𝑏],ℝ).

Moreover, we assume that the map 𝑡↦𝛼(𝑡)∶=𝑡+^𝑟_𝑗=1𝐻^𝑗𝑝_𝑗(𝑡) (41) is a strictly increasing 𝐶¹_rd function and its image is again a time scale, 𝕋. We denote the forward shift operator relative to 𝕋 by 𝜎 and the delta derivative by Δ. We remark that the following holds [40]: 𝜎∘𝛼=𝛼∘𝜎. (42)

Definition 20. Functional ℒ is invariant under transformations (39) if, and only if, for all 𝑦∈𝐶¹_rd([𝑎,𝑏],ℝ), we have ^𝑏_𝑎𝐿𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡)Δ𝑡=^𝑏_𝑎𝐿𝑡,𝑦^𝜎𝑡,𝑦^Δ𝑡Δ𝑡. (43)

We recall the following results that will be very useful in the proof of Theorem 22.

Theorem 21 (see [37]). Assume that 𝜈∶𝕋→ℝ is strictly increasing and 𝕋∶=𝜈(𝕋) is a time scale. (1)Chain rule: let 𝜔∶𝕋→ℝ. If 𝜈^Δ(𝑡) and 𝜔Δ(𝜈(𝑡)) exist for all 𝑡∈𝕋^𝜅, then (𝜔∘𝜈)^Δ=𝜔Δ∘𝜈𝜈^Δ. (44)(2)Substitution in the integral: if 𝑓∶𝕋→ℝ is a 𝐶_rd function and 𝜈 is a 𝐶¹_rd function, then for 𝑎,𝑏∈𝕋, ^𝑏_𝑎𝑓(𝜈(𝑡))𝜈^Δ(𝑡)Δ𝑡=_𝜈(𝑎)^𝜈(𝑏)𝑓(𝑠)Δ𝑠. (45)

Now, we are ready to state and prove Noether’s second theorem with transformation of time.

Theorem 22 (Noether’s second theorem with transformation of time). If functional ℒ is invariant under transformations (39), then there exist the following identities: ^𝑛𝑚_𝑘=1𝑖=0(−1)^𝑖1𝑏₁^{𝑖(𝑖+1)/2}𝑔^𝑘_𝑖𝑗^𝜎𝐸_𝑘(𝐿)^Δ𝑖+𝑓_𝑖𝑗^𝜎𝜕𝐿−Δ𝜕𝑡Δ𝑡𝐿−𝑦^Δ_𝑘𝜕𝐿𝜕𝑦^Δ_𝑘−𝜇𝜕𝐿𝜕𝑡^Δ𝑖≡0 (46) for 𝑗=1,2,…,𝑟.

Proof. The idea of the proof is to reduce the statement of this result to the one of Theorem 16 using a technique of reparametrization of time: artificially, we will consider 𝑡 as a dependent variable of the same footing with 𝑦.
Let 𝑟≠0 and define 𝐿(𝑡)𝑣𝑡,𝑠,𝑦,𝑟,𝑣∶=𝐿𝑠−𝜇𝑟,𝑦,𝑟𝑟. (47) Note that, for 𝑠(𝑡)=𝑡 and any 𝑦∈𝐶¹_rd([𝑎,𝑏],ℝ^𝑛), we have 𝐿𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡)=𝐿𝑡,𝑠^𝜎(𝑡),𝑦^𝜎(𝑡),𝑠^Δ(𝑡),𝑦^Δ(𝑡). (48) Therefore, for 𝑠(𝑡)=𝑡, ℒ𝑦∶=^𝑏_𝑎𝐿𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡)=Δ𝑡_𝑏𝑎𝐿𝑡,𝑠^𝜎(𝑡),𝑦^𝜎(𝑡),𝑠^Δ(𝑡),𝑦^Δ(𝑡)Δ𝑡∶=ℒ𝑠,𝑦. (49) Note that, for 𝑠(𝑡)=𝑡, ℒ𝑠(⋅)(⋅𝑦(⋅=,𝑦)=ℒ)_𝑏𝑎𝐿𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡)=Δ𝑡_{𝛼(𝑎)𝛼(𝑏)}𝐿𝑡,𝑦∘𝜎𝑡,𝑦^Δ𝑡Δ𝑡=^𝑏_𝑎𝐿𝛼(𝑡),𝑦∘𝜎∘𝛼(𝑡),𝑦^Δ(𝛼(𝑡))𝛼^Δ(𝑡)=Δ𝑡_𝑏𝑎𝐿𝛼(𝑡),𝑦∘𝛼∘𝜎(𝑡),𝑦∘𝛼^Δ(𝑡)𝛼^Δ(𝑡)𝛼^Δ(𝑡)=Δ𝑡_𝑏𝑎𝐿𝛼^𝜎(𝑡)−𝜇(𝑡)𝛼^Δ(𝑡),𝑦∘𝛼^𝜎(𝑡),𝑦∘𝛼^Δ(𝑡)𝛼^Δ(𝑡)⋅𝛼^Δ(𝑡)=Δ𝑡_𝑏𝑎𝐿𝑡,𝛼^𝜎(𝑡),𝑦∘𝛼^𝜎(𝑡),𝛼^Δ(𝑡),𝑦∘𝛼^Δ(𝑡)=ℒ𝛼(⋅),Δ𝑡𝑦∘𝛼(⋅). (50) Let 𝐻(𝑡,𝑦(𝑡))∶=𝛼(𝑡) and 𝑇=(𝑇¹,𝑇²,…,𝑇^𝑛) where 𝑇^𝑘𝑡,𝑦(𝑡)∶=𝑦_𝑘(𝑡)+^𝑟_𝑗=1𝑇^𝑘𝑗𝑝_𝑗(𝑡),𝑘=1,2,…,𝑛. (51) Then, for 𝑠(𝑡)=𝑡, 𝛼(𝑡),𝑦∘𝛼(𝑡)=𝑡,𝑦𝑡=𝐻(𝑡)(𝑡)=𝐻𝑠(𝑡)(𝑡)𝑠(𝑡)(𝑡).𝑡,𝑦,𝑇𝑡,𝑦,𝑦,𝑇,𝑦 (52) Hence, using (50) and (52), we get ℒ𝑠(⋅),𝑦(⋅)=ℒ𝐻𝑠(⋅),𝑦(⋅),𝑇𝑠(⋅),𝑦(⋅). (53) This means that ℒ is invariant on 𝑈=𝑠,𝑦∣𝑠(𝑡)=𝑡∧𝑦∈𝐶¹_rd[𝑎,𝑏],ℝ^𝑛 (54) under the group of state transformations 𝑠,𝑦=𝐻𝑠,𝑦,𝑇𝑠,𝑦 (55) in the sense of Definition 13.
Using Theorem 16, we can conclude that there exist the following 𝑟 identities (𝑗=1,2,…,𝑟)^𝑛𝑚_𝑘=1𝑖=0(−1)^𝑖1𝑏₁^{𝑖(𝑖+1)/2}𝑔^𝑘_𝑖𝑗^𝜎𝐸_𝑘(𝐿)^Δ𝑖+^𝑚_𝑖=0(−1)^𝑖1𝑏₁^{𝑖(𝑖+1)/2}𝑓_𝑖𝑗^𝜎𝐸_𝑠𝐿^Δ𝑖≡0, (56) where we denote 𝐸_𝑠(𝐿)∶=(𝜕𝐿/𝜕𝑠^𝜎)−(Δ/Δ𝑡)(𝜕𝐿/𝜕𝑠^Δ).
Note that, for 𝑠(𝑡)=𝑡, 𝜕𝐿𝜕𝑠^𝜎𝑡,𝑠^𝜎(𝑡),𝑦^𝜎(𝑡),𝑠^Δ(𝑡),𝑦^Δ(𝑡)=𝜕𝐿𝑠𝜕𝑡^𝜎(𝑡)−𝜇(𝑡)𝑠^Δ(𝑡),𝑦^𝜎(𝑡),𝑦^Δ(𝑡)𝑠^Δ(𝑡)𝑠^Δ(𝑡),𝜕𝐿𝜕𝑠^Δ𝑡,𝑠^𝜎(𝑡),𝑦^𝜎(𝑡),𝑠^Δ(𝑡),𝑦^Δ(𝑡)=𝐿𝑠^𝜎(𝑡)−𝜇(𝑡)𝑠^Δ(𝑡),𝑦^𝜎(𝑡),𝑦^Δ(𝑡)𝑠^Δ(𝑡)−^𝑛_𝑘=1𝑦^Δ_𝑘(𝑡)𝑠^Δ(𝑡)𝜕𝐿𝜕𝑦^Δ_𝑘𝑠^𝜎(𝑡)−𝜇(𝑡)𝑠^Δ(𝑡),𝑦^𝜎(𝑡),𝑦^Δ(𝑡)𝑠^Δ(𝑡)−𝜕𝐿𝑠𝜕𝑡^𝜎(𝑡)−𝜇(𝑡)𝑠^Δ(𝑡),𝑦^𝜎(𝑡),𝑦^Δ(𝑡)𝑠^Δ(𝑡)⋅𝜇(𝑡)𝑠^Δ(𝑡). (57) Hence, for 𝑠(𝑡)=𝑡, 𝐸_𝑠𝐿𝑡,𝑠^𝜎(𝑡),𝑦^𝜎(𝑡),𝑠^Δ(𝑡),𝑦^Δ(𝑡)=𝜕𝐿𝜕𝑡𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡)−Δ𝐿Δ𝑡𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡)−^𝑛_𝑘=1𝑦^Δ_𝑘(𝑡)𝜕𝐿𝜕𝑦^Δ_𝑘𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡)−𝜇(𝑡)𝜕𝐿𝜕𝑡𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡). (58) Also note that, for 𝑠(𝑡)=𝑡 and 𝑘=1,2,…,𝑛, 𝐸_𝑘𝐿𝑡,𝑠^𝜎(𝑡),𝑦^𝜎(𝑡),𝑠^Δ(𝑡),𝑦^Δ(𝑡)=𝐸_𝑘(𝐿)𝑡,𝑦^𝜎(𝑡),𝑦^Δ(𝑡). (59) Substituting the above equalities into (56), we conclude the desired result: ^𝑛𝑚_𝑘=1𝑖=0(−1)^𝑖1𝑏₁^{𝑖(𝑖+1)/2}𝑔^𝑘_𝑖𝑗^𝜎𝐸_𝑘(𝐿)^Δ𝑖+𝑓_𝑖𝑗^𝜎𝜕𝐿−Δ𝜕𝑡Δ𝑡𝐿−𝑦^Δ_𝑘𝜕𝐿𝜕𝑦^Δ_𝑘−𝜇𝜕𝐿𝜕𝑡^Δ𝑖≡0 (60) for 𝑗=1,2,…,𝑟.

Remark 23. Define 𝐸^𝑠_𝑘(𝐿)∶=𝜕𝐿−Δ𝜕𝑡Δ𝑡𝐿−^𝑛_𝑘=1𝑦^Δ_𝑘𝜕𝐿𝜕𝑦^Δ_𝑘−𝜇𝜕𝐿,𝜕𝑡 (61) for 𝑘=1,2,…,𝑛. Then, 𝐸^𝑠_𝑘(𝐿)=0 are the second Euler-Lagrange equations for problem (1) [36]. Therefore, expression (46) provides “dependencies” between two types of the Euler-Lagrange expressions.

Note that if 𝕋=ℝ, Noether’s identity (46) simplifies because 𝜕𝐿−Δ𝜕𝑡Δ𝑡𝐿−^𝑛_𝑘=1𝑦^Δ_𝑘𝜕𝐿𝜕𝑦^Δ_𝑘−𝜇𝜕𝐿=𝜕𝑡𝜕𝐿−𝑑𝜕𝑡𝑑𝑡𝐿−^𝑛_𝑘=1𝑦̇_𝑘𝜕𝐿𝜕𝑦̇_𝑘=−^𝑛_𝑘=1𝑦̇_𝑘𝐸_𝑘(𝐿), (62) and we obtain the following corollary.

Corollary 24 (classical Noether’s second theorem, cf. [1]). If functional ℒ defined by ℒ𝑦=^𝑏_𝑎𝐿𝑡,𝑦(𝑡),𝑦̇(𝑡)𝑑𝑡 (63) is invariant under transformations (39) (where 𝜎 denotes in this context the identity function and Δ denotes the usual derivative), then there exist the following identities: ^𝑛𝑚_𝑘=1𝑖=0(−1)^𝑖𝑔^𝑘_𝑖𝑗𝐸_𝑘(𝐿)^(𝑖)−𝑓_𝑖𝑗⋅𝑦̇_𝑘𝐸_𝑘(𝐿)^(𝑖)≡0,𝑗=1,2,…,𝑟. (64)

In the case 𝕋=ℎℤ (for some ℎ>0), we obtain from Theorem 22 the second Noether theorem for the ℎ-calculus; whereas if 𝕋=𝑞^ℕ0 (for some 𝑞>1), we get the second Noether theorem for the 𝑞-calculus.

3.2. Noether’s Second Theorem: Multiple Delta Integral Case

In this subsection, we extend the second Noether theorem (without transformation of time) to multiple integral variational problems in the time scale setting. For simplicity of presentation, we prove the result for the case of two independent variables and transformations that depend on an arbitrary function and its first-order partial delta derivatives. Clearly, our result can be generalized for 𝑛 independent variables and 𝑟 arbitrary functions and their higher-order partial delta derivatives.

For the convenience of the reader, we recall notions and results that are needed in the sequel. A general introduction to differential calculus and integration theory for multivariable functions on time scales is presented, respectively, in [41] (see also [42]) and [43]. For the double integral calculus of variations on time scales, we refer the reader to [44].

Let 𝕋₁ and 𝕋₂ be two given time scales. For 𝑖=1,2, denote by 𝜎_𝑖 and Δ_𝑖 the forward jump operator and the delta derivative on 𝕋_𝑖, respectively. Let 𝐶⁽¹⁾_rd denote the set of all continuous functions defined on 𝕋₁×𝕋₂ for which both the Δ₁-partial derivative and the Δ₂-partial derivative exist and are of class 𝐶_rd (for a definition see [44]).

Let Ω⊆𝕋₁×𝕋₂ be an 𝜔-type set and let Γ be its positively fence (see [44]). Denote Ω^∘∶=𝑥,𝑦∈Ω∶𝜎₁(𝑥),𝜎₂𝑦∈Ω. (65) Let a function 𝐿(𝑥,𝑦,𝑢,𝑝,𝑞), where (𝑥,𝑦)∈Ω∪Γ and (𝑢,𝑝,𝑞)∈ℝ^3𝑛, be given. We will suppose that 𝐿 is continuous, together with its partial delta derivatives of first and second order with respect to 𝑥,𝑦 and partial usual derivatives of the first and second order with respect to 𝑢,𝑝,𝑞. In what follows, 𝑢^Δ1 and 𝑢^Δ2 denote, respectively, 𝜕𝑢/Δ₁𝑥 and 𝜕𝑢/Δ₂𝑦.

Consider the following optimization problem: ℒ[𝑢]=_Ω𝐿𝑥,𝑦,𝑢𝜎₁(𝑥),𝜎₂𝑦,𝑢^Δ1𝑥,𝜎₂𝑦,𝑢^Δ2𝜎₁(𝑥),𝑦Δ₁𝑥Δ₂𝑦⟶extremize, (66) where the set of admissible functions are 𝒟=𝑢∣𝑢∶Ω∪Γ⟶ℝ^𝑛,𝑢∈𝐶⁽¹⁾_rd,𝑢=𝑔onΓ, (67) where 𝑔 is a fixed function defined and continuous on the fence Γ of Ω.

As noticed in [44], for the variational problem (66) being well posed, we have to assume that there exists at least one admissible function 𝑢₀∈𝒟 because it is possible to choose a continuous function 𝑔 such that no function 𝑢 is admissible. Note that if there exists an admissible function 𝑢₀, then the set 𝒟 contains a set of functions of the form 𝑢=𝑢₀+𝜂, where 𝜂∶Ω∪Γ→ℝ^𝑛 is 𝐶⁽¹⁾_rd and 𝜂=0 on Γ. Any such 𝜂 is called an admissible variation for problem (66).

Definition 25. We say that 𝑢_∗∈𝒟 is a local minimizer (resp., local maximizer) for problem (66) if there exists 𝛿>0 such that ℒ𝑢_∗≤ℒ[𝑢]resp.,ℒ𝑢_∗≥ℒ[𝑢], (68) for all 𝑢∈𝒟 with 𝑢−𝑢_∗∶=sup_{𝑥,𝑦∈Ω∪Γ}𝑢𝑥,𝑦−𝑢_∗𝑥,𝑦+sup_{𝑥,𝑦∈Ω}𝑢^Δ1𝑥,𝜎₂𝑦−𝑢^Δ1_∗𝑥,𝜎₂𝑦+sup_{𝑥,𝑦∈Ω}𝑢^Δ2𝜎₁(𝑥),𝑦−𝑢^Δ2_∗𝜎₁(𝑥),𝑦<𝛿, (69) where |⋅| denotes a norm in ℝ^𝑛.

We recall the following results which will play an important role in the proofs of our results.

Theorem 26 (Green’s theorem [45]). If the functions 𝑀 and 𝑁 are continuous and have continuous partial delta derivatives 𝜕𝑀/Δ₂𝑦 and 𝜕𝑁/Δ₁𝑥 on Ω∪Γ, then _Ω𝜕𝑁Δ₁𝑥−𝜕𝑀Δ₂𝑦Δ₁𝑥Δ₂𝑦=_Γ𝑀𝑑^∗𝑥+𝑁𝑑^∗𝑦, (70) where the “star line integrals" on the right side in (70) denote the sum of line delta integrals taken over the line segment constituents of Γ directed to the right or upwards and line nabla integrals taken over the line segment constituents of Γ directed to the left or downwards.

Lemma 27 (fundamental lemma of the double variational calculus [44]). If 𝑀 is continuous on Ω∪Γ with _Ω𝑀𝑥,𝑦𝜂𝜎₁(𝑥),𝜎₂𝑦Δ₁𝑥Δ₂𝑦=0 (71) for any admissible variation 𝜂, then 𝑀𝑥,𝑦=0∀𝑥,𝑦∈Ω^∘. (72)

Theorem 28 (Euler-Lagrange equation of the double variational calculus [44]). Suppose that an admissible function 𝑢_∗ provides a local minimum for ℒ and that 𝑢_∗ has continuous partial delta derivatives of the second order. Then 𝑢_∗ satisfies the Euler-Lagrange equation 𝜕𝐿(⋅)−𝜕𝜕𝑢Δ₁𝑥𝜕𝐿(⋅)−𝜕𝜕𝑝Δ₂𝑦𝜕𝐿(⋅)𝜕𝑞=0, (73) where (⋅)=(𝑥,𝑦,𝑢(𝜎₁(𝑥),𝜎₂(𝑦)), 𝑢^Δ1(𝑥,𝜎₂(𝑦)), 𝑢^Δ2(𝜎₁(𝑥), 𝑦)) for (𝑥,𝑦)∈Ω^∘.

Let us denote by 𝜌₁ and 𝜌₂ the backward jump operator of 𝕋₁ and 𝕋₂, respectively. In what follows, we will suppose that 𝕋₁ and 𝕋₂ are such that 𝜎₁𝜌₁(𝑥)=𝑥,∀𝑥∈𝕋_1𝜅,𝜎₂𝜌₂𝑦=𝑦,∀𝑦∈𝕋_2𝜅, (74) where 𝕋_𝜅∶=𝕋⧵{𝑚} if 𝕋 has a right-scattered minimum 𝑚; otherwise, 𝕋_𝜅=𝕋. We recall the fact that ℝ, ℎℤ (for some ℎ>0), 𝑞^ℕ0 (for some 𝑞>1), and many other interesting time scales satisfy property (74).

Let 𝑢(𝑥,𝑦)=(𝑢₁(𝑥,𝑦),𝑢₂(𝑥,𝑦),…,𝑢_𝑛(𝑥,𝑦)) and consider the following transformations that depend on an arbitrary continuous function 𝑝 and the partial delta derivatives of 𝑝: 𝑥=𝑥,𝑦=𝑦,𝑢_𝑘𝑥,𝑦=𝑢_𝑘𝑥,𝑦+𝑇^𝑘𝑝𝑥,𝑦, (75) where, for each 𝑘=1,2,…,𝑛, 𝑇^𝑘𝑝𝑥,𝑦∶=𝑎^𝑘₀𝑥,𝑦𝑝𝑥,𝑦+𝑎^𝑘₁𝜕𝑥,𝑦Δ₁𝑥𝑝𝜌₁(𝑥),𝑦+𝑎^𝑘₂𝜕𝑥,𝑦Δ₂𝑦𝑝𝑥,𝜌₂𝑦, (76)𝑎₀, 𝑎₁, and 𝑎₂ are 𝐶¹ functions and we assume that 𝑝 has continuous partial delta derivatives of the first and second order.

Definition 29. Functional ℒ is invariant under transformations (75) if, and only if, for all 𝑢∈𝒟, we have _Ω𝐿𝑥,𝑦,𝑢𝜎₁(𝑥),𝜎₂𝑦,𝑢^Δ1𝑥,𝜎₂𝑦,𝑢^Δ2𝜎₁(𝑥),𝑦Δ₁𝑥Δ₂𝑦=_Ω𝐿𝑥,𝑦,𝑢𝜎₁(𝑥),𝜎₂𝑦,𝑢^Δ1𝑥,𝜎₂𝑦,𝑢^Δ2𝜎₁(𝑥),𝑦Δ₁𝑥Δ₂𝑦. (77)

In what follows, we use the notations 𝑇^𝑘𝑝^𝜎𝑥,𝑦∶=𝑇^𝑘𝜎₁(𝑥),𝜎₂𝑦,𝑇^𝑘𝑝^𝜎1𝑥,𝑦∶=𝑇^𝑘𝜎₁(𝑥),𝑇,𝑦_𝑘𝑝^𝜎2𝑥,𝑦∶=𝑇^𝑘𝑥,𝜎₂𝑦. (78) Using similar arguments as the ones used in the proof of Theorem 15, we can prove the following result.

Theorem 30 (necessary condition of invariance). If functional ℒ is invariant under transformations (75), then ^𝑛_𝑘=1Ω𝜕𝐿𝜕𝑢^𝜎_𝑘⋅𝑇^𝑘𝑝^𝜎+𝜕𝐿𝜕𝑢^Δ1_𝑘𝜕Δ₁𝑥𝑇^𝑘𝑝^𝜎2+𝜕𝐿𝜕𝑢^Δ2_𝑘𝜕Δ₂𝑦𝑇^𝑘𝑝^𝜎1Δ₁𝑥Δ₂𝑦=0. (79)

Define 𝐸_𝑘(𝐿)∶=𝜕𝐿𝜕𝑢^𝜎_𝑘−𝜕Δ₁𝑥𝜕𝐿𝜕𝑢^Δ1_𝑘−𝜕Δ₂𝑦𝜕𝐿𝜕𝑢^Δ2_𝑘,𝑘=1,2,…,𝑛. (80) We call 𝐸_𝑘(𝐿), 𝑘=1,2,…,𝑛, the Euler-Lagrange expressions associated to the Lagrangian 𝐿 relative to problem (66).

The following lemmas will be used in the proof of Theorem 34.

Lemma 31. If ℒ is invariant under transformations (75), then ^𝑛_𝑘=1Ω𝐸_𝑘(𝐿)⋅𝑇^𝑘𝑝^𝜎Δ₁𝑥Δ₂𝑦=0. (81)

Proof. Fix 𝑖∈{1,2,…,𝑛}. Observe that _Ω𝜕𝐿𝜕𝑢^Δ1_𝑖𝜕Δ₁𝑥𝑇^𝑖𝑝^𝜎2+𝜕𝐿𝜕𝑢^Δ2_𝑖𝜕Δ₂𝑦𝑇^𝑖𝑝^𝜎1Δ₁𝑥Δ₂𝑦=_Ω𝜕Δ₁𝑥𝜕𝐿𝜕𝑢^Δ1_𝑖⋅𝑇^𝑖𝑝^𝜎2+𝜕Δ₂𝑦𝜕𝐿𝜕𝑢^Δ2_𝑖⋅𝑇^𝑖𝑝^𝜎1Δ₁𝑥Δ₂𝑦−_Ω𝜕Δ₁𝑥𝜕𝐿𝜕𝑢^Δ1_𝑖⋅𝑇^𝑖𝑝^𝜎+𝜕Δ₂𝑦𝜕𝐿𝜕𝑢^Δ2_𝑖⋅𝑇^𝑖𝑝^𝜎Δ₁𝑥Δ₂𝑦. (82) Using Green’s theorem we get _Ω𝜕Δ₁𝑥𝜕𝐿𝜕𝑢^Δ1_𝑖⋅𝑇^𝑖𝑝^𝜎2+𝜕Δ₂𝑦𝜕𝐿𝜕𝑢^Δ2_𝑖⋅𝑇^𝑖𝑝^𝜎1Δ₁𝑥Δ₂𝑦=_Γ𝜕𝐿𝜕𝑢^Δ1_𝑖⋅𝑇^𝑖𝑝^𝜎2𝑑^∗𝑦−𝜕𝐿𝜕𝑢^Δ2_𝑖⋅𝑇^𝑖𝑝^𝜎1𝑑^∗𝑥. (83) Since 𝑝 is arbitrary, we can choose 𝑝 such that 𝑝𝑥,𝜎₂𝑦_Γ𝑝𝜎=0,₁(𝑥),𝑦_Γ𝜕=0,Δ₁𝑥𝑝𝜌₁(𝑥),𝜎₂𝑦_Γ𝜕=0,Δ₁𝑥𝑝𝑥,𝑦_Γ𝜕=0,Δ₂𝑦𝑝𝑥,𝑦_Γ𝜕=0,Δ₂𝑦𝑝𝜎₁(𝑥),𝜌₂𝑦_Γ=0, (84) and therefore, _Ω𝜕Δ₁𝑥𝜕𝐿𝜕𝑢^Δ1_𝑖⋅𝑇^𝑖𝑝^𝜎2+𝜕Δ₂𝑦𝜕𝐿𝜕𝑢^Δ2_𝑖⋅𝑇^𝑖𝑝^𝜎1Δ₁𝑥Δ₂𝑦=0. (85) By Theorem 30, we obtain ^𝑛_𝑘=1Ω𝜕𝐿𝜕𝑢^𝜎_𝑘⋅𝑇^𝑘𝑝^𝜎−𝜕Δ₁𝑥𝜕𝐿𝜕𝑢^Δ1_𝑘⋅𝑇^𝑘𝑝^𝜎−𝜕Δ₂𝑦𝜕𝐿𝜕𝑢^Δ2_𝑘⋅𝑇^𝑘𝑝^𝜎Δ₁𝑥Δ₂𝑦=0, (86) which proves that ∑^𝑛_𝑘=1∫∫_Ω𝐸_𝑘(𝐿)⋅𝑇^𝑘(𝑝^𝜎)Δ₁𝑥Δ₂𝑦=0.

Lemma 32. For each 𝑘=1,2,…,𝑛, _Ω𝑞⋅𝑇^𝑘𝑝^𝜎Δ₁𝑥Δ₂𝑦=_Ω𝑞𝑎^𝑘₀−𝜕Δ₁𝑥𝑞𝑎^𝑘₁−𝜕Δ₂𝑦𝑞𝑎^𝑘₂⋅𝑝^𝜎Δ₁𝑥Δ₂𝑦 (87) holds.

Proof. Note that _Ω𝑞⋅𝑇^𝑘𝑝^𝜎Δ₁𝑥Δ₂𝑦=_Ω𝑞𝑎^𝑘₀𝑥,𝑦𝑝^𝜎𝑥,𝑦+𝑞𝑎^𝑘₁𝜕𝑥,𝑦Δ₁𝑥𝑝𝑥,𝜎₂𝑦+𝑞𝑎^𝑘₂𝜕𝑥,𝑦Δ₂𝑦𝑝𝜎₁(𝑥),𝑦Δ₁𝑥Δ₂𝑦,_Ω𝑞𝑎^𝑘₁𝜕𝑥,𝑦Δ₁𝑥𝑝𝑥,𝜎₂𝑦+𝑞𝑎^𝑘₂𝜕𝑥,𝑦Δ₂𝑦𝑝𝜎₁(𝑥),𝑦Δ₁𝑥Δ₂𝑦=_Ω𝜕Δ₁𝑥𝑞𝑎^𝑘₁𝑥,𝑦𝑝𝑥,𝜎₂𝑦+𝜕Δ₂𝑦𝑞𝑎^𝑘₂𝑥,𝑦𝑝𝜎₁(𝑥),𝑦Δ₁𝑥Δ₂𝑦−_Ω𝜕Δ₁𝑥𝑞𝑎^𝑘₁𝑥,𝑦⋅𝑝𝜎₁(𝑥),𝜎₂𝑦+𝜕Δ₂𝑦𝑞𝑎^𝑘₂𝑥,𝑦⋅𝑝𝜎₁(𝑥),𝜎₂𝑦Δ₁𝑥Δ₂𝑦. (88)(89) Using Green’s theorem, we can conclude that _Ω𝜕Δ₁𝑥𝑞𝑎^𝑘₁𝑥,𝑦𝑝𝑥,𝜎₂𝑦+𝜕Δ₂𝑦𝑞𝑎^𝑘₂𝑥,𝑦𝑝𝜎₁(𝑥),𝑦Δ₁𝑥Δ₂𝑦=0. (90) Hence, _Ω𝑞⋅𝑇^𝑘𝑝^𝜎Δ₁𝑥Δ₂𝑦=_Ω𝑞𝑎^𝑘₀⋅𝑝^𝜎−𝜕Δ₁𝑥𝑞𝑎^𝑘₁⋅𝑝^𝜎−𝜕Δ₂𝑦𝑞𝑎^𝑘₂⋅𝑝^𝜎Δ₁𝑥Δ₂𝑦 (91) proving the desired result.

Remark 33. Lemma 32 shows that we can define an adjoint operator of 𝑇^𝑘, 𝑇^𝑘, by 𝑇^𝑘𝑞=𝑞𝑎^𝑘₀−𝜕Δ₁𝑥𝑞𝑎^𝑘₁−𝜕Δ₂𝑦𝑞𝑎^𝑘₂. (92)

We are now ready to state and prove the Noether second theorem without transformation of time for multiple integral problems on time scales.

Theorem 34 (Noether’s second theorem without transforming time). If functional ℒ is invariant under transformations (75), then ^𝑛_𝑘=1𝑇^𝑘𝐸_𝑘(𝐿)≡0𝑜𝑛Ω^∘, (93) where 𝐸_𝑘(𝐿) are the 𝑛 Euler-Lagrange expressions and 𝑇^𝑘 is the adjoint operator of 𝑇^𝑘.

Proof. Using Lemmas 31 and 32, we conclude that if ℒ is invariant under transformations (75), then ^𝑛_𝑘=1Ω𝐸_𝑘(𝐿)⋅𝑇^𝑘𝑝^𝜎Δ₁𝑥Δ₂𝑦=^𝑛_𝑘=1Ω𝑇^𝑘𝐸_𝑘(𝐿)⋅𝑝^𝜎Δ₁𝑥Δ₂𝑦=0, (94) where 𝑇^𝑘 is the adjoint operator of 𝑇^𝑘. Applying the fundamental lemma of the double variational calculus (Lemma 27), we get ^𝑛_𝑘=1𝑇^𝑘𝐸_𝑘(𝐿)≡0onΩ^∘, (95) proving the desired result.

Corollary 35 (classical Noether’s second theorem for double integrals problems, cf. [1]). Let Ω⊆ℝ² be an 𝜔-type set and let Γ be its positive fence. Let 𝐿(𝑥,𝑦,𝑢,𝑝,𝑞) be a function of class 𝐶², (𝑥,𝑦)∈Ω∪Γ, 𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑛). If functional ℒ defined by ℒ𝑦=_Ω𝐿𝑥,𝑦,𝑢𝑥,𝑦,𝜕𝑢,𝜕𝑥𝑥,𝑦𝜕𝑢𝜕𝑦𝑥,𝑦𝑑𝑥𝑑𝑦 (96) is invariant under transformations (75) (where 𝜌₁ and 𝜌₂ denote in this context the identity function, and Δ₁ and Δ₂ denote the usual derivative), then ^𝑛_𝑘=1𝑇^𝑘𝐸_𝑘(𝐿)≡0𝑜𝑛Ω, (97) where 𝐸_𝑘(𝐿)∶=𝜕𝐿𝜕𝑢_𝑘−𝜕𝜕𝑥𝜕𝐿𝜕𝑝_𝑘−𝜕𝜕𝑦𝜕𝐿𝜕𝑞_𝑘,𝑘=1,2,…,𝑛 (98) and 𝑇^𝑘 is the adjoint operator of 𝑇^𝑘.

Similarly to the single delta integral case choosing 𝕋=ℎℤ (for some ℎ>0), we obtain from Theorem 34 the second Noether theorem for the double variational ℎ-calculus; whereas choosing 𝕋=𝑞^ℕ0 (for some 𝑞>1), we obtain the second Noether theorem for the double variational 𝑞-calculus.

4. Example

In order to illustrate the second Noether Theorem for the multiple integral case, we will present the following example. Let 𝕋₀, 𝕋₁, 𝕋₂, and 𝕋₃ be time scales and let Ω⊆𝕋₀×𝕋₁×𝕋₂×𝕋₃ be an 𝜔-type set. For 𝑖=0,1,2,3, denote by 𝜎_𝑖, 𝜌_𝑖, and Δ_𝑖 the forward jump operator, the backward jump operator, and the delta derivative on 𝕋_𝑖, respectively.

Let 𝑡∶=(𝑡₀,𝑡₁,𝑡₂,𝑡₃)∈Ω and consider the following real functions defined on Ω: 𝐴₀,𝐴₁,𝐴₂,𝐴₃. Let 𝐀∶=(𝐴₁,𝐴₂,𝐴₃) and denote ∇𝐴₀(𝑡)∶=𝜕𝐴₀Δ₁𝑡₁𝜎₀𝑡₀,𝑡₁,𝜎₂𝑡₂,𝜎₃𝑡₃,𝜕𝐴₀Δ₂𝑡₂𝜎₀𝑡₀,𝜎₁𝑡₁,𝑡₂,𝜎₃𝑡₃,𝜕𝐴₀Δ₃𝑡₃𝜎₀𝑡₀,𝜎₁𝑡₁,𝜎₂𝑡₂,𝑡₃,𝜕𝐀Δ₀𝑡₀(𝑡)∶=𝜕𝐴₁Δ₀𝑡₀𝑡₀,𝜎₁𝑡₁,𝜎₂𝑡₂,𝜎₃𝑡₃,𝜕𝐴₂Δ₀𝑡₀𝑡₀,𝜎₁𝑡₁,𝜎₂𝑡₂,𝜎₃𝑡₃,𝜕𝐴₃Δ₀𝑡₀𝑡₀,𝜎₁𝑡₁,𝜎₂𝑡₂,𝜎₃𝑡₃,curl𝐀(𝑡)∶=𝜕𝐴₃Δ₂𝑡₂𝜎₀𝑡₀,𝜎₁𝑡₁,𝑡₂,𝜎₃𝑡₃−𝜕𝐴₂Δ₃𝑡₃𝜎₀𝑡₀,𝜎₁𝑡₁,𝜎₂𝑡₂,𝑡₃,𝜕𝐴₁Δ₃𝑡₃𝜎₀𝑡₀,𝜎₁𝑡₁,𝜎₂𝑡₂,𝑡₃−𝜕𝐴₃Δ₁𝑡₁𝜎₀𝑡₀,𝑡₁,𝜎₂𝑡₂,𝜎₃𝑡₃,𝜕𝐴₂Δ₁𝑡₁𝜎₀𝑡₀,𝑡₁,𝜎₂𝑡₂,𝜎₃𝑡₃−𝜕𝐴₁Δ₂𝑡₂𝜎₀𝑡₀,𝜎₁𝑡₁,𝑡₂,𝜎₃𝑡₃. (99) We will consider the following Lagrangian function: 1𝐿=2∇𝐴₀−𝜕𝐀Δ₀𝑡₀²−12‖curl𝐀‖² (100) that is the time scale version of the Lagrangian density for the electromagnetic field (see, e.g., [46]).

It can be proved that the functional ℒ=⋯_Ω𝐿Δ₀Δ₁Δ₂Δ₃ (101) is invariant under the gauge transformations: 𝐴_𝑘=𝐴_𝑘+𝜕Δ_𝑘𝑡_𝑘𝑝^𝜌𝑘,𝑘=0,1,2,3, (102) where 𝑝∶Ω→𝑅 is an arbitrary continuous function that has continuous partial delta derivatives of the first and second order (hence, we have equality of mixed partial delta derivatives; see [41]).

Since, for each 𝑘=0,1,2,3, 𝑇^𝑘𝑝=𝜕Δ_𝑘𝑡_𝑘𝑝^𝜌𝑘, (103) then, by Lemma 32, we conclude that 𝑇^𝑘𝑞𝜕=−Δ_𝑘𝑡_𝑘𝑞. (104) Hence, from the second Noether theorem (Theorem 34), we get ³_𝑘=0𝜕Δ_𝑘𝑡_𝑘𝐸_𝑘(𝐿)≡0onΩ^∘, (105) where 𝐸_𝑘(𝐿), 𝑘=0,1,2,3 are the Euler-Lagrange expressions associated to functional ℒ.

If we suppose that, for each 𝑘=0,1,2,3, 𝐴_𝑘 has continuous partial delta derivatives of the first and second order and that 𝐴₀ and the vector field 𝐀 satisfy the so-called Lorentz conditions on time scales: div𝐀|_{(𝑡0,𝜎1(𝑡1),𝜎2(𝑡2),𝜎3(𝑡3))}=𝜕𝐴₀Δ₀𝑡_{0(𝑡0,𝜎1(𝑡1),𝜎2(𝑡2),𝜎3(𝑡3))}div𝐀|_{(𝜎0(𝑡0),𝑡1,𝜎2(𝑡2),𝜎3(𝑡3))}=𝜕𝐴₀Δ₀𝑡_{0(𝜎0(𝑡0),𝑡1,𝜎2(𝑡2),𝜎3(𝑡3))}div𝐀|_{(𝜎0(𝑡0),𝜎1(𝑡1),𝑡2,𝜎3(𝑡3))}=𝜕𝐴₀Δ₀𝑡_{0(𝜎0(𝑡0),𝜎1(𝑡1),𝑡2,𝜎3(𝑡3))}div𝐀|_{(𝜎0(𝑡0),𝜎1(𝑡1),𝜎2(𝑡2),𝑡3)}=𝜕𝐴₀Δ₀𝑡_{0(𝜎0(𝑡0),𝜎1(𝑡1),𝜎2(𝑡2),𝑡3)}, (106) where div𝐀 denotes the divergence of a vector field 𝐀, that is, div𝐀∶=𝜕𝐴₁Δ₁𝑡₁+𝜕𝐴₂Δ₂𝑡₂+𝜕𝐴₃Δ₃𝑡₃, (107) then the Euler-Lagrange expressions can be written in the following way: 𝐸_𝑘(𝐿)=𝜕²𝐴_𝑘Δ₀𝑡²₀𝑡₀,𝜎₁𝑡₁,𝜎₂𝑡₂,𝜎₃𝑡₃−∇²𝐴_𝑘𝑡₀,𝑡₁,𝑡₂,𝑡₃,𝑘=0,1,2,3, (108) where ∇²𝐴_𝑘(𝑡)𝜕∶=²𝐴_𝑘Δ₁𝑡²₁𝜎₀𝑡₀,𝑡₁,𝜎₂𝑡₂,𝜎₃𝑡₃+𝜕²𝐴_𝑘Δ₂𝑡²₂𝜎₀𝑡₀,𝜎₁𝑡₁,𝑡₂,𝜎₃𝑡₃+𝜕²𝐴_𝑘Δ₃𝑡²₃𝜎₀𝑡₀,𝜎₁𝑡₁,𝜎₂𝑡₂,𝑡₃. (109) Hence, under these assumptions, we can conclude that ³_𝑘=0𝜕Δ_𝑘𝑡_𝑘𝜕²𝐴_𝑘Δ₀𝑡²₀𝑡₀,𝜎₁𝑡₁,𝜎₂𝑡₂,𝜎₃𝑡₃−∇²𝐴_𝑘𝑡₀,𝑡₁,𝑡₂,𝑡₃=0onΩ^∘. (110)

5. Concluding Remarks

We proved that the important Noether’s second theorem is valid not only for the continuous and discrete calculus but also for the quantum calculus. Moreover, in our opinion, the proofs presented in this paper are elegant and clear to follow.

The question of obtaining Noether’s second theorem for multiple integrals with transformation of time in the time scale setting remains an interesting open question. To the best of the authors’ knowledge, to extend the second Noether theorem to multiple integrals with transformation of time, substitution in the multiple integral is a fundamental tool and this result is not yet available in the literature.

For other generalizations of the second Noether theorem, we refer the reader to [47] (in the context of the fractional calculus of variations) and [48] (in the context of the optimal control).

Acknowledgments

This work was supported by FEDER funds through COMPETE-Operational Programme Factors of Competitiveness (“Programa Operacional Factores de Competitividade”) and by Portuguese funds through the Center for Research and Development in Mathematics and Applications (University of Aveiro) and the Portuguese Foundation for Science and Technology (“FCT-Fundação para a Ciência e a Tecnologia”), within Project PEst-C/MAT/UI4106/2011 with COMPETE no. FCOMP-01-0124-FEDER-022690. Agnieszka B. Malinowska was also supported by Bialystok University of Technology Grant S/WI/02/2011.