Probelm in calculating a First Variation

muzialis

Hello,

I am reading a paper on evolution problems of the type

u(0) = 0
u' (t) = v (t)
v(t) = argmin {G (t, u(t), v(t)}

The authour claims that the Euler Lagrange equation related to the last equation is

dG / dv = 0 (1)

I can not understand this.

The Euler Equation formally can be written as

dG / du - d (dG/dv) / dx (2)

and this equation will be satisifed by u.

But as the authour says, v satisfies
dG / dv = 0 (3)

u is the primitive of u, hence it should satisfy
d (dG/dv) / dx (4)

and this is different from (2)!

Any help please? I am confused. I attach the paper for completeness, equation (10) in the paper's introduction is the problem, which is self-contained between equations (4) and (10), quick to read for somebody with a clue...

Thank you!

Attached Files

Larsen Fracture Paths.pdf (557.1 KB, 6 views)