Noether's theorem on time scales

Abstract

We show that for any variational symmetry of the problem of the calculus of variations on time scales there exists a conserved quantity along the respective Euler-Lagrange extremals.

The variational calculus on time scales

“…(16) We now split the proof in two parts: (i) we prove (13) transforming the delta integral in (16) to a nabla integral by means of (10); (ii) we prove (14) transforming the nabla integral in (16) to a delta integral by means of (11). (i) By (10) the necessary optimality condition (16) is equivalent to…”

“…The calculus of variations on time scales was introduced in 2004 by M. Bohner using the delta derivative and integral [12], and has been since then further developed by several different authors in several different directions [2,11,21,22,33,34]. In all these works, the integral functional to be extremized has the form…”

A unified approach to the calculus of variations on time scales

“…Given u ∈ R \ {0}, find y that is a solution to [6,7]), while if f is ∇-differentiable, then for u = −1 (22) reduces to a problem of the calculus of variations with ∇ derivative (see [2,15]). …”

“…It has found applications in several different fields that require simultaneous modeling of discrete and continuous data, in particular in the calculus of variations. There are two approaches that are followed in the literature of the calculus of variations on time scales: one is concerned with the minimization of delta integrals with a Lagrangian depending on delta derivatives [1,6,7,11,14]; the other with minimization of nabla integrals with integrands that involve nabla derivatives [2,4]. Both formulations of the problems of the calculus of variations give results that are similar among them and similar to the classical results of the calculus of variations (see, e.g., [16]) but are obtained independently.…”

Hamilton‐Jacobi Method for Mechanical Systems on Time Scales

“…Inspired by these works, the Euler-Lagrange equations, the Hamilton canonical equations [36][37][38], and the Birkhoff's equations [39] were established for mechanical systems on time scales. Besides, the Noether's theorems [37][38][39][40][41][42] were established in finding conserved quantities for mechanical systems on time scales. However, it is worth mentioning that the famous Hamilton-Jacobi method also represents an important integration method.…”