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Probelm in calculating a First Variation 
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#1
May112, 10:23 AM

P: 152

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 selfcontained between equations (4) and (10), quick to read for somebody with a clue... Thank you! 


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