[AntonioLao]

The variation of a function analogous to the variation of the position of a point whose derivative must be zero is characterized by two fundamental changes: (1) it is an infinitesimal change since epsilon approaches zero, and (2) it is also a virtual change which means the changed function is chosen arbitrarily as long as it satisfies all conditions of continuity and differentiability.

There are fundamental differences between a variation dE and differential ¶E. Although both signify infinitesimal change, dE produces a new energy function E+dE while ¶E implies that the independent variable has a limit as it approaches zero. The new function dE, on the other hand is chosen arbitrarily. This is not a real change but a virtual change. If the independent variable is time then its variation can serve no useful purposes and it is agreed that the variation of time is always zero: dt=0. Moreover, if the two limiting ordinates E(t1) and E(t2) are known then they also cannot be varied which means dE(t1)=0 and dE(t2)=0. Therefore the study of calculus of variations is equivalent to the study of virtual changes for arbitrarily chosen continuously differentiable functions. This is truly mathematics of choice.