# Dynamical Systems

Dynamical systems are a very important part of Theoretical Mechanics. In our mechanics class, we use Lagrange's equations to find the equations of motion of a diversity of systems. Nevertheless, most of these equations are very hard or impossible to solve analytically. In this section, we are going to learn how to use the Runge-Kutta method to aproimate numerically the behaviour of a variety of dynamical systems.

## Hamilton's equations

As we know, one can perform a Legendre transformation to the Lagrangian of a system in order to obtain the Hamiltonian. With it, we can obtain a new set of first order differential equations that describes the evolution of the system: 

$$
\dot{q\_i} = \frac{\partial H}{\partial p\_i},\~\~\dot{p\_i} = - \frac{\partial H}{\partial q\_i}
$$

This is a system of equations that, with given initial conditions, can be solved using the Runge-Kutta method. To do this, we define a discrete time domain. Instead of having a set of continuos values in an interval, we define the following set of points: 

$$
{t\_i| t\_i = t\_0+i\Delta t\~\text{for}\~ i =1,2,...,Nt},\~\~\Delta t = \frac{tf}{Nt}
$$

The Runge-Kutta method states that, given the following initial value problem:

$$
\frac{d\vec{x}(t)}{dt} = \vec{F}(\vec{x}),\~\~\vec{x}(0) = \vec{x}\_0
$$

then, the function can be approximated by the following formula:

$$
\begin{eqnarray}
\vec{x}*i=\vec{x}(t\_i) &=& \vec{x}(t*{i-1}) + \frac{\vec{k\_1} + 2\vec{k\_2} + 2\vec{k\_3} + \vec{k\_4}}{6}\\
\vec{k\_1} &=& \vec{F}(\vec{x}(t\_{i-1})) \Delta t \\
\vec{k\_2} &=& \vec{F}(\vec{x}(t\_{i-1})+\frac{1}{2}\vec{k\_1}) \Delta t \\
\vec{k\_3} &=& \vec{F}(\vec{x}(t\_{i-1})+\frac{1}{2}\vec{k\_2}) \Delta t \\
\vec{k\_4} &=& \vec{F}(\vec{x}(t\_{i-1})+\vec{k\_3}) \Delta t \\
\end{eqnarray}
$$

We are going to use this method to solve a lot of sets of differential equations in the following sections.


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