How to Invert the Euler Lagrange Equations?

Question

Suppose I have a functional L. For example $L = y+3y'$. Where y is itself a function of real variable x

It's easy for me to evaluate the Functional Derivative of L via the Euler Lagrange Equations:

$$ \frac{\partial L}{\partial y} - \frac{d}{dx} \frac{\partial L}{\partial y'}$$

But how does one do the inverse? Given a functional L, find the functional G that when applied by the above gives L.

The d/dx term makes thing very hard as the two partial derivatives are independent of each other but this differentiation forces a relationship between the two.

Answer 1

What you are looking for is called the inverse Helmotz problem I think : given a differential operator of second order, at what condition is there a corresponding lagrangian ?

You can learn more about this on the web (for example https://en.wikipedia.org/wiki/Inverse_problem_for_Lagrangian_mechanics) but I never found out that much references about it.

For what I know, there exists a sine qua none condition on the operator for being an Euler-Lagrange derivative of some Lagrangian, called the Helmoltz condition.

But I think that in practice, it's not always easy to use. However, if we are considering a problem with only one degree of liberty, which seems to be the case in your question, there is a more explicit condition that is equivalent, sometimes called the explicit Helmoltz condition. I think that in this case, there is even an explicit construction of the Lagrangian.

I'll see if I can find out where I've read that.

Update : I found it, it is http://www.unilim.fr/pages_perso/loic.bourdin/Documents/bourdin-thesis2013.pdf. It is half in french, half in english but the part you're looking for is in english. It is the section IV.2. page 65.

Answer 2

Your problem is the classical inverse problem formulated by Helmholtz in 1887. This problem is, as far as I know, completely solved by Mayer and Hirsch in 1897. If you want a complete and modern study of this problem, you can read the book of Olver "Applications of Lie groups to differential equations". In this book, the multidimensional case (PDE) with general order is studied.

In my thesis (above cited by Sylvain L.), I gave an explicit formulation of the Helmholtz condition in the unidimensional case (ODE) with second order. In this particular (and simpler) case, the Helmholtz condition can be written in a very simple way. You will find it in Equation (IV.2.12) p.67. This condition is sufficient and necessary in order to ensure that a second order differential operator $O$ is a (second-order) Euler-Lagrange operator.

Now, if you consider a second order differential operator $O$ that satisfies the Helmholtz condition, and if you want to find a Lagrangian $L$ such that $EL[L]=O$ (where $EL$ is the Euler-Lagrange operator), then you can follow the step of the proof of Theorem IV-2 p.67. This proof gives an explicit way to construct such a Lagrangian $L$.

Remark: Note that $L$ is not unique since there exist some "null Lagrangian", see Section IV-2-3 p.69.