Physics+ Functional, Variation: UNIZOR.COM - Physics+ 4 All - Variations

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Functional and Variation

To introduce concepts of Functional (a noun, not an adjective) and Variation which happen to be very important mathematical tools of Physics, let's consider the following problem.

Imagine yourself on a river bank at point A.
River banks are two parallel straight lines with distance d between them.
You have a motor boat that can go with some constant speed V relative to water.
The river has a uniform current with known speed v which we assume to be less than the speed of a boat V.
You want to cross a river to get to point B exactly opposite to point A, so segment AB is perpendicular to the river's current.

Problem:

How should you navigate your boat from point A to point B to reduce the time to cross the river to a minimum?

It sounds like a typical problem to find a minimum of a function (to minimize time). But this resemblance is only on a surface.

In Calculus we used to find minimum or maximum of a real function of real argument by differentiating it and checking when its first derivative equals to zero.

In our case the problem is much more complex, because we are not dealing with a function (time to cross the river) whose argument is a real number. The argument to our function (time to cross the river) is a trajectory of a boat from point A to point B.
And what is a trajectory of a boat?

Trajectory is a set of positions of a boat, which is, in its own rights, can be a function of some argument (trajectory can be a function of time, of an angle with segment AB or a distance from line AB in a direction of a river's current).
Trajectory is definitely not a single real number.