Note on a Noetherian conservation law and its corresponding general class of nonlinear second‐order ordinary differential equations

By virtue of their fundamental nature and intrinsic elegance, there has long been a great interest in searching for the conservation laws of the problem at hand. In such field of research, the famous Noether's theorem has been recognized as a highlight. The objective of our note is to provide the reader with an original contribution to the study of Noetherian conservation laws. Namely, we aim at connecting a general class of nonlinear second‐order ordinary differential equations with a Noetherian conservation law. Our analysis will be considered within the known structure of Noether's theorem. An aspect to be here emphasized is that, particularly, we will use the following simplifying assumptions: the invariance condition holds absolutely, the transformation generators do not depend on the derivative of the generalized coordinate with respect to the independent variable. We will present our contribution via a theorem and a corollary. First, our theorem will establish a general condition to obtain Noetherian conservation law for a certain type of variational problem. Then, the corollary will demonstrate that such Noetherian conservation law is a first integral for a general class of nonlinear second‐order ordinary differential equations. This class of differential equations means the Lagrange's equation directly resulting from that type of variational problem. Last, a case of the Emden‐Fowler equation will be presented as a testing example.