— 1. Variational Calculus —
|Definition 1 A functional is a mapping from vector spaces to into real numbers.|
Let . Suppose that and are constants, the functional form of is known.
According to definition 1 is a functional and the goal of the Calculus of Variations is to determine such that the value of is an extremum.
Let be a parametric representation of such that is the function that makes an extremum.
We can write , where is a function of of the class (that means that is a continuous function whose derivative is also continuous) with .
Now is of the form
Therefore the condition for to be an extremum is
|Example 1 Let . Take as a parametric representation of . Let , and . Given the previous parametric equation find such that is a minimum.
Now and .
By the previous expression it is trivial to see that the minimum value is reached when
|Exercise 1 Given the points and , calculate the equation of the curve that minimizes the distance between the points.
It is ,
And it is with .
The rest is left as an exercise for the reader.
— 2. Euler Equations —
In the following section we’ll analyze the condition for to be an extremum:
Since it is and it follows
For the first term it is since by hypothesis.
Remembering that and taking into account the fact that is an arbitrary function one can conclude that
The previous equation is known as the Euler’s Equation
To close our thoughts on the Euler equation let us say that there also is a second form for the Euler equation. The second form is
and is used in the cases where doesn’t depend explicitly on .
— 3. Euler Equation for variables —
Let be of the form .
Now we have and for each of the values of . Since are independent functions it follows that for
That is to say we have independent Euler equations.