Nonlinear Problem

The nonlinear system with a quadratic force term satisfies
\frac{d^2x_i(t)}{dt^2} = x_{i+1}(t) + x_{i-1}(t) - 2x_i(t) +
\alpha\{[x_{i+1}(t) - x_i(t)]^2 - [x_i(t) - x_{i-1}(t)]^2\}
\end{displaymath} (16)

for $i = 1,\ldots{},N-2$ and at the ends we have
$\displaystyle \frac{d^2x_0(t)}{dt^2}$ $\textstyle =$ $\displaystyle x_1(t) - 2x_0(t)
+ \alpha\{[x_1(t) - x_0(t)]^2 - [x_0(t)]^2\}$ (17)
$\displaystyle \frac{d^2x_{N-1}(t)}{dt^2}$ $\textstyle =$ $\displaystyle x_{N-2}(t) - 2x_{N-1}(t)
+ \alpha\{[x_{N-1}(t)]^2 - [x_{N-1}(t) - x_{N-2}(t)]^2\}$ (18)

This problem can't be solved using normal modes. Instead we do a direct numerical integration. To simplify notation and prepare for numerical integration, we rewrite the system of equations in two steps:
$\displaystyle \dot v_i(t)$ $\textstyle =$ $\displaystyle a_i(t,{\bf x})$ (19)
$\displaystyle \dot x_i(t)$ $\textstyle =$ $\displaystyle v_i(t)$ (20)

where $a_i(t,{\bf x})$ is the acceleration of the $i$th mass point and $v_i(t)$ is its velocity. Initial conditions must specify $v_i(0)$ and $x_i(0)$.

Here we discuss the leapfrog method, but you may use any method of numerical integration you like. With the leapfrog method we advance the time by small steps $dt$. The leapfrog idea works with estimated velocities at the odd multiples of $dt/2$: $0.5dt$, $1.5dt$, $2.5dt$, etc and with estimated positions at the even multiples $dt$, $2dt$, etc. We leap the velocity over the position, using the acceleration evaluated at that position. Then we leap the position over the velocity. To get the process started we need the velocity estimate at the first odd time $0.5dt$, so we do a half-step:

v_i(0.5dt) = v_i(0) + 0.5 a_i[0,{\bf x}(0)]   dt
\end{displaymath} (21)

Then we are ready to roll:
$\displaystyle x_i(t + dt)$ $\textstyle =$ $\displaystyle x_i(t) + v_i(t + 0.5dt)   dt$ (22)
$\displaystyle v_i(t + 1.5dt)$ $\textstyle =$ $\displaystyle v_i(t + 0.5dt) + a_i[t,{\bf x}(t + dt)]   dt$ (23)

This process is repeated for many steps. At the end of the integration process, we should bring the velocity up to the final even time by taking a final half-step.

At regular intervals we want to measure the energy in the first few normal modes. This is done using Eqs (16) and (18), which require knowledge of the eigenvalues and eigenenergies. To be precise, before measuring the energies one should bring the velocity up to the even time by taking a half-step. Then another half-step to resume integration.