Lagrangian Mechanics and Differential Equations

[exmachina]

The Wikipedia article regarding Lagrangian Mechanics mentions that we can essentially derive a new set of equations of motion, thought albeit non-linear ODEs, using Lagrangian Mechanics.

My question is: how difficult is it usually to solve these non-linear ODEs? What are the usual numerical methods? Things like Velocity Verlet seem to really only apply to the system of linear ODEs.

(eg. http://en.wikipedia.org/wiki/Lagrangian_mechanics#Pendulum_on_a_movable_support the non-linear ODEs given here)

 

[Thaakisfox]

It is generally impossible to solve non-linear ODEs analytically excepting a few special cases.
Usual numerical methods include the euler method, and the more advanced Runge-Kutta methods etc. Check the book "Numerical Recipes" (I don't remember the authors oops) and you will find a tonne...

 

[bigfooted]

Well, actually ODE's arising from a Lagrangian are a special case.

For second order nonlinear ODE's a systematic solution method exists based on point symmetries. When you find two point symmetries, you can solve the ODE.

A second order ODE arising from a Lagrangian is special because you only need one point symmetry to solve it. For a system of two ODE's you would need two symmetries. However, finding the point symmetries of systems of ODE's can be quite difficult. Most existing algorithms are not guaranteed to terminate in a finite time.

But for the example of the nonlinear pendulum, you might try your luck with maple and look into Lie's symmetry methods for obtaining symmetries and solving ODE's.

If you just want a numerical solution, I recommend the Runge-Kutta method.