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\title{Mathematical Modeling, 2010}
\maketitle
\begin{itemize}
\item Introduction. Great Models (Copenicus, Newton, Lagrange, Einstein)


Simplicity of a model, non-uniqueness

Mental experiment versus real experiment

Drawing,algebra, ODE, PDE


\item Dynamical models:

Population growth, equation with delay, simulation, age groups.

Parashute 

\item Curve fitting model
\item Biological model.
\end{itemize}



\section{Models and optimization}

\subsection{Cleat hitch}

1. Consider a round surface and a rope around it. The rope is stretched at one free end by the force $F_0$, and is fasten at the other end, $360^\circ$ from the other end. Assume that the tension force is proportional to the normal force at the surface (Coulomb/Amonton friction law).  Compute the force in the rope along it.

The normal tension $N$ is 
$$
N(s)= k(s) F(s) $$
where $k(s)$ is curvature, and $s$ is the arch length. 

The friction force $FF$ is proportional to the normal tension
$$ d\,FF(s)=-\gamma k(s) F(s) \, ds$$ and it is directed along the tangent.
The tangent component satisfies the equation
$$
\frac{d F}{d  s}= -\gamma k(s) F(s)
$$
If $K$ is constant (round) the force decrease exponentially.

~

2. Design a surface that correspond to constant normal force in the rope along its path.
We find $k(s)$ from the condition 
$$
k(s)F(s)=c={\rm const}
$$ 
then 
$$
\frac{d F}{d  s}= -\gamma c, \quad F(s)=F_0 -\gamma c \, s
$$
and 
$$
k(s)=\frac{c}{F(s)}=\frac{c}{ F_0 -\gamma c \, s }
$$
\subsection{Isochrone}

\section{Nonstandard equations}



\subsection{"Population dynamics"}

Three basis models

Delay: birthrate depends on the size f population in the past.
$$
\frac{d \, u}{d\, t}= u(t)  ( a-  b\, u(t-\tau))
$$
discretize, model. 

Sensus linaer model, add nonlinearity, play with


Disease spread. Equatons:

http://plus.maths.org/issue14/features/diseases/

\subsection{Domino}

Boundary condition. Slinky

Chain dynamics. Geisers. 
\subsection{Traffic wave. Continuum limit}
Chain dymanics. Dis
 Delay
 
\section{Instabilities}


\subsection{box bumping}



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