# How to get the Lagrangian that generates certain diff equation ?

1. Sep 14, 2008

### smallphi

how to get the Lagrangian that generates certain diff equation ???

I have an ordinary differential equation of second order. I am looking for the Lagrangian(s) for which this equation is the Euler-Lagrange equation. I need a practical reference specifically written for one degree of freedom (one equation), I'm not interested in general theorems of existence or highly geometrical language. I found some reference by Darboux in 1894 but its in French. Any english reference?

Last edited: Sep 14, 2008
2. Sep 18, 2008

### smallphi

Re: how to get the Lagrangian that generates certain diff equation ???

Got it. The problem is known as 'inverse problem of variational calculus'. For future references:

N. Akhiezer, "The Calculus of Variations", 1962:

page 166: example of 1D problem

R. Santilli, "Foundations of Theoretical Mechanics II", Springer-Verlag, 1983:

page 353: reworking the same example as Akhiezer
page 315: by knowing one first integral, Kobussen's method

R. Santilli, "Foundations of Theoretical Mechanics I", Springer-Verlag, 1978:

page 201: Douglas approach for n-dimension, easily applied to n=1
page 208: first obtaining Hamiltonian and then get the Lagrangian through inverse Legendre transformation

Last edited: Sep 18, 2008
3. Sep 18, 2008

### atyy

Re: how to get the Lagrangian that generates certain diff equation ???

Thanks for the pointers!