Noether-Type First Integrals Associated with Autonomous Second-Order Lagrangians

: In this paper, the analysis is centered on Noether-type ﬁrst integrals in Lagrange-Hamilton dynamics based on autonomous second-order Lagrangians. More precisely, by using the classical Noether’s theorem and a non-standard Legendrian duality, the single-time and multi-time versions of Noether’s result are investigated for autonomous second-order Lagrangians. A correspondence is established between the invariances under ﬂows and the ﬁrst integrals for autonomous second-order Lagrangians. In this way, our results extend, unify and improve several existing theorems in the current literature.

In this paper, inspired and motivated by the ongoing research in this field, we consider multi-time evolutions and the notion multi-time is regarded as multiple parameter of evolution. For the multi-time case, the multi-index notation introduced by Saunders [14] is used. Throughout the paper, we develop our points of view, by developing new concepts and methods for a theory that involves single-time and multi-time second-order Lagrangians. More exactly, by using a non-standard Legendrian duality, the main aim of this work is to study the single-time and multi-time versions of Noether's result for autonomous second-order Lagrangians. We prove that there exists a correspondence between the invariances under flows and the first integrals for autonomous second-order Lagrangians. This actually reflects Noether type theorems between symmetries and conservation laws for dynamical systems. This work can be an important source for many research problems and it should be of interest to engineers and applied mathematicians. For other different but connected ideas to this subject, the reader is directed to Ma [17,18].
The present paper is structured as follows. Section 2 contains some auxiliary results including the classical Noether's theorem. Section 3 introduces the main results of this paper. More exactly, Noether-type first integrals are investigated for autonomous second-order Lagrangians. Finally, Section 4 concludes the paper.
The next result formulates the classical Noether's theorem (for autonomous first-order Lagrangians).

Theorem 1.
[Noether] Let T(t, x) be the flow generated by the C 1 -class vector field X(x) = X i (x) , i = 1, n. If the , then the function is a first integral associated with the movement generated by the Lagrangian L.
Proof. Denote x s (t) := T (s, x(t)). The invariance of L means . Consequently, using derivation formulas and Euler-Lagrange equations, we get Further, in order to formulate the multi-time version for Noether-type first integrals (associated with autonomous first-order Lagrangians), in accordance with Treanţȃ [5][6][7][8][9][10] and following Udrişte and Matei [4], consider Ω t 0 ,t 1 ⊂ R m a hyper-parallelepiped determined by diagonally opposite points t 0 , t 1 from R m . If we define the partial order product on R m , then Ω t 0 ,t 1 is equivalent with the closed In the case of several variables of evolution (that is, the multi-time version), the Hamiltonian H (x, p, t) = .., m}, t ∈ Ω t 0 ,t 1 ) does not conserve, that is D α H = 0, where D α is the total derivative operator, even in autonomous case.
In the following, let L (x(t), x γ (t)) be an autonomous C 2 -class Lagrangian. Introducing the multi-time anti-trace Euler-Lagrange PDEs (for more details, see Udrişte and Matei [4]) and using Legendrian duality, we derive the multi-time anti-trace Hamilton PDEs (see δ α β as Kronecker's symbol) A direct computation gives us D γ H = 0. Consequently, H is a first integral both for multi-time anti-trace Euler-Lagrange PDEs and multi-time anti-trace Hamilton PDEs.
On the other hand, if we introduce energy-impulse tensor T of components we get the following results: (1) Div(T) := D α T α β = 0 (conservation law); (2) the trace of Hamilton tensor is the Hamiltonian Next, consider the Hamiltonian tensor field H α β defined by where x and p are the canonical variables. Here, L α β (x(t), x γ (t)) represents an extension of the Lagrangian L (x(t), x γ (t)), satisfying: (a) the trace of tensor field L α β is the Lagrangian L; (b) the Lagrangian 1-forms L α β (x(t), x γ (t)) dt β are completely integrable; (c) the functions L α β (x(t), x γ (t)) determine the multi-time anti-trace Euler-Lagrange PDEs Further, we assume p α x γ (t)) and that these nm 3 equations define the following m functions x γ = x γ (x, p). If L α β = Lδ α β , then H α β is exactly the classical energy-impulse tensor. The following result formulates the generalized Hamilton PDEs governed by first-order Lagrangians.

Theorem 2. (Generalized Hamilton PDEs, [4]) If x(·) is solution in
(1) and p = (p α i (·)) is defined as above, then the pair (x(·), p(·)) is solution of the following generalized Hamilton PDEs: . Moreover, if the Lagrangian tensor field L α β is autonomous, then the divergence of the transposed Hamilton The next theorem formulates the multi-time version for Noether-type first integrals associated with autonomous first-order Lagrangians.

Theorem 3. ([4]) Let T(t,
x) be the m-flow generated by the C 1 -class vector fields X α (x) = X i α (x) , α = 1, m, i = 1, n. If the Lagrangian L (x(t), x γ (t)) is invariant under this flow, then the function is a first integral of the movement generated by the Lagrangian L via multi-time anti-trace Euler-Lagrange PDEs.
Proof. The invariance of L means By direct computation and taking into account the multi-time anti-trace Euler-Lagrange PDEs, we get and the proof is now complete.

Main Results
This section, taking into account the aforementioned auxiliary results, introduces the main results of this paper. More exactly, the single-time and multi-time versions of Noether's result are investigated for autonomous second-order Lagrangians. Theorem 4. Let T(t, x) be the flow generated by the C 2 -class vector field X(x) = X i (x) , i = 1, n. If the autonomous second-order Lagrangian L (x(t),ẋ(t),ẍ(t)) is invariant under this flow, then the function is a first integral of the movement generated by the Lagrangian L.
Proof. The invariance of L means By using the associated Euler-Lagrange ODEs formulated as follows In consequence, the function I(x,ẋ,ẍ) is a first integral of the movement generated by the autonomous second-order Lagrangian Lagrangian L. The proof is complete.
The following two corollaries establish some more restrictive results. Corollary 1. Let T(t, x) be the flow generated by the C 2 -class vector field X(x) = X i (x) , i = 1, n. If the autonomous second-order Lagrangian L (x(t),ẋ(t),ẍ(t)) is invariant under this flow and d dt ∂L ∂ẍ i (x,ẋ,ẍ) X i (x) is constant, then the function is a first integral of the movement generated by the Lagrangian L.
Proof. The invariance of L means Consequently, we get Thus, the function I(x,ẋ,ẍ) is a first integral and the proof is complete.