  Through duality it is possible to transform left fractional operators into
right fractional operators and vice versa. In contrast to existing literature,
we establish integration by parts formulas that exclusively involve either left
or right operators. The emergence of these novel fractional integration by
parts formulas inspires the introduction of a new calculus of variations, where
only one type of fractional derivative (left or right) is present. This applies
to both the problem formulation and the corresponding necessary optimality
conditions. As a practical application, we present a new Lagrangian that relies
solely on left-hand side fractional derivatives. The fractional variational
principle derived from this Lagrangian leads us to the equation of motion for a
dissipative/damped system.
