Lagrangian method for nonlinear ODE, 2nd order ?

[smallphi]

Lagrangian method for nonlinear ODE, 2nd order ??

I have to solve non-linear ODE of 2nd order. The Maple routines can't find integrating factor. I think that's connected to Lie symmetries that can't be found.

I'm thinking of getting a Lagrangian for which that equation is the Euler-Lagrange equation and somehow guess a symmetry of the Lagrangian or perhaps choose more appropriate variables in the equation to obtain partial if not general solutions. Essentially that is looking for a Noether symmetry (Lagrangian method) vs Lie symmetry (integrating factor method).

Has anyone seen something like that? Give references if you can.

 

[jvicens]

Hello,

Yes, the Lagrangian method can be used to solve nonlinear ODEs of 2nd order. This method is also known as the inverse variational problem or the Noether symmetry approach. It involves finding a Lagrangian function that satisfies the Euler-Lagrange equation, which is equivalent to the original ODE.

To find the Lagrangian, you can start by rewriting the ODE in the form of a second-order Lagrangian system, where the dependent variable is the first derivative of the original dependent variable. Then, using the standard procedure for finding a Lagrangian, you can solve for the Lagrangian function.

Once you have the Lagrangian, you can use the Noether theorem to find symmetries of the Lagrangian, which can then be used to obtain solutions to the ODE. However, it should be noted that this method may not always provide a complete solution and may only give partial solutions.

There are a few references that you can look into for more information on this method. Some of these include the book "Applications of Lie Groups to Differential Equations" by Peter J. Olver and the paper "The inverse variational problem in classical mechanics" by Boris A. Kupershmidt.

I hope this helps. Best of luck with your research!