On a Non-Newtonian Calculus of Variations

Abstract

The calculus of variations is a field of mathematical analysis born in 1687 with Newton’s problem of minimal resistance, which is concerned with the maxima or minima of integral functionals. Finding the solution of such problems leads to solving the associated Euler–Lagrange equations. The subject has found many applications over the centuries, e.g., in physics, economics, engineering and biology. Up to this moment, however, the theory of the calculus of variations has been confined to Newton’s approach to calculus. As in many applications negative values of admissible functions are not physically plausible, we propose here to develop an alternative calculus of variations based on the non-Newtonian approach first introduced by Grossman and Katz in the period between 1967 and 1970, which provides a calculus defined, from the very beginning, for positive real numbers only, and it is based on a (non-Newtonian) derivative that permits one to compare relative changes between a dependent positive variable and an independent variable that is also positive. In this way, the non-Newtonian calculus of variations we introduce here provides a natural framework for problems involving functions with positive images. Our main result is a first-order optimality condition of Euler–Lagrange type. The new calculus of variations complements the standard one in a nontrivial/multiplicative way, guaranteeing that the solution remains in the physically admissible positive range. An illustrative example is given.

On the reachability tube of non-Newtonian first-order linear differential equations

“…for each x ∈ R α , and if we use the relation (23), then we can rewrite the non-Newtonian first-order differential equation (23) as…”

“…Non-Newtonian calculus was developed in the works of Grossman and Katz in a series of papers which are summarized in [1]. Recently, non-Newtonian calculus has provided a wide variety of mathematical tools for use in science, engineering, and mathematics, and appears to have considerable potential for use as an alternative to the calculus of Newton and Leibniz; see [20,23,24]. The basic principles are summarized as follows.…”

“…We should mention that this is probably one of the most popular non-Newtonian calculus due to the amount of research and applications reported in the literature. Some applications can be consulted in [12,19,20,23].…”

Non-Newtonian Pell and Pell-Lucas numbers

“…The non-Newtonian calculi provide diverse mathematical tools and are widely used in various fields, including science, mathematics, and engineering. The use of non-Newtonian calculus has numerous practical applications in multiple areas, such as quantum calculus, functional analysis, complex analysis, fractal geometry, differential equations, calculus of variations, image analysis, signal processing, and economics, for instance, see [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21].…”

Taylor’s Formula for Generalized Weighted Fractional Derivatives with Nonsingular Kernels

“…On the other hand, Taylor's formulas play a crucial role in mathematical analysis, e.g., in asymptotic methods, nonlinear programming, and the calculus of variations and optimal control [12][13][14]. Different forms of Taylor's formulas can be found in the literature, covering both classical and smooth one-dimensional cases as well as multi-dimensional, non-smooth, and non-Newtonian cases [15][16][17].…”