(16) |

(17) | |||

(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:

(19) | |||

(20) |

where is the acceleration of the th mass point and is its velocity. Initial conditions must specify and .

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 . The leapfrog idea works with estimated
velocities at the odd multiples of : , , ,
etc and with estimated positions at the even multiples , ,
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 , so we do a half-step:

(21) |

(22) | |||

(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 (15) and (17), 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.