$(30$ marks$)$ The Potential Energy $(,)$ of a dynamic problem, which consists of two

wheeled carts connected to each other is given by the equation below.

$(,)=.$

$+.$

$$

In order to determine the displacements $(,)$ of the carts, the potential energy needs to

be minimised. Minimize this function using the Nelder$-$Mead simplex method, starting with

$$

$=[$

$]=[..],$ an initial step size c$=2,=.,=.,=.,=.$ and

tol $=0\mathrm{.}001.$

Perform the Nelder$\_$Mead search until the convergence criterion

$(()())$

$+(()())$

$+(()())$

$$

$$

$<$

is satisfied.