Sophus Lie's contribution to solution of problems of variational type as in Euler and Lagrange The original impetus for Sophus Lie's work was apparently to streamline the solution of certain problems of variational type such as those treated in the work of Euler and Lagrange.  This presumably involves solution of differential equations involving groups of symmetries.  Is there a reference that explains this historical background?
 A: In 

Vladimir Itskov: Orbit Reduction of Contact Ideals. Contemporary Mathematics. Vol. 285, 2001

one reads on page 172:

As first observed by Sophus Lie [8], the Euler-Lagrange equations of every invariant variational problem can be written in terms of the differential invariants of the group action. In other words, the Euler-Lagrange equations of a group-invariant variational problem can be pushed forward to the orbit space. Surprisingly, up to date there was no general understanding of the meaning of the pushed forward equations on the orbit space, nor there was a general algorithm of producing the group-invariant Euler-Lagrange equations. 

In the above, the "nor there was a" seems not quite correct English and should be 'nor was there a'; the reference '[8]' is:

Sophus Lie Über Integralinvarianten und ihre Verwertung für die Theorie der Differentialgleichungen, Leipziger Berichte 49, 369-410, 1897.

and a slightly commented/edited/expanded/corrected version of the above seems to be available here; (I don't know what the referent of the initials 'G.S.' in that document is.)
