8/6/2019 Dido' Problem Contromechanics

1/3

THE DIDOS PROBLEM

Possible Solution

1 | P a g e

Given the length of a line u, calculate a function y(x) such that the enclosedarea is maximum.

Goal

This problem is dealt in two parts. The first part is to formulate the function ofthe given line and the area it will enclose, and the second step will beoptimizing the area with the constraint on the perimeter.

Solution

If we have an enclosed curve, we can say that the curve can be described interms of a particle tracing the curve. If the position of the particle is given byx(t) and y(t), where both are a function of time.

Part 1

As length can be stated as the product of velocity with the time, we can find

the perimeter by integrating the velocity function with respect to time. We canexpress the velocity as follows

= () 2 + () 2dtTo get the complete length, that is, the perimeter of the enclosed area we canintegrate the above equation with respect of t. In shorter for we may write as

= 2 + 2dtNow that we have the perimeter, we need a relation between the perimeter

and the area it encloses. This relation is given by the Greens theorem.

+ =

Here R and S are any functions that are smooth and are in the enclosed are.Another important property that they should have is that they have todifferentiable. p and e represent the perimeter, or the length if the lineenclosing the area, and the enclosed area respectively.Now if we choose

= 2 = 2The equation becomes as followsWe chose the functions as simple as possible and the negative is made sothat the RHS term does not go to zero.

2

+ 2 =

1

2 +

2

+2

=

8/6/2019 Dido' Problem Contromechanics

2/3

THE DIDOS PROBLEM

Possible Solution

2 | P a g e

Now the RHS represents the area. We can thus say that

=

1

2(

)dt

With this equation the first part of the question is completed. We move to theoptimization now

Step 2For the second step we need formulate the Lagrangian function and thensolve it using the Euler-Lagrangian formulation.The Lagrangian is a formulation that is used to find the maximum or minimumof any function when there is a given constraint. In our case the maximizationis of the area function and the constraint is that of the length, or the perimeter.

According to the Lagrangian functionIf we have functions such that we needed to optimize Q such that

(, , ) With respect to

(, ,) We define the Legrangian as

(, , ) = (, , ) + (,, ) In our case the formulation is as

=1

2( ) + 2 + 2

The solution of this formulation satisfies the Euler-Lagrange Equations

= 0For our formulation the above equation will look like

= 0

= 0

This gives us

8/6/2019 Dido' Problem Contromechanics

3/3

THE DIDOS PROBLEM

Possible Solution

3 | P a g e

1

2 + 2 + 2

1

2 = 0

and

1

2 + 2 + 2 +1

2 = 0If we integrate both sides we get

12 + 2 + 2

1

2 = 1

1

2 + 2 + 2 +

1

2 = 2

If we square these equations and add them we get

( 2)2 + ( 1)2 = 2This as we know is the equation of a circle

Thus the function that maximizes the area for a fixed perimeter is that of acircle.

RESULT