The calculus of variations deals with “virtual” displacements, rather than “differential” displacements, and “variations of functions”. For example, when considering the time-evolution of some coordinate, \(q(t)\), one can write the differential, \(dq\), which is a measure of the infinitesimal change in \(q\) for a given change, \(dt\), in time.
A virtual displacement, denoted \(\delta q\), is a change in \(q(t)\), without the corresponding change in time. That is, the value of \(q(t)\) is infinitesimally displaced from the value that it should have at a particular time. In some respects, this is an un-physical change in \(q\).
As will be seen later, it is very useful to look at these un-physical changes in quantities to set restriction on how they can change physically, thus determining their equations of motion. Another example is the potential energy of a marble at the bottom of the bowl; it may be interesting for us to consider how the potential energy would change if the marble were to be (un-physically) displaced horizontally away from the bottom of the bowl. Those considerations will lead us to understanding how the marble can behave physically.