  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.
