Final Answer:
a) For the choice of F(p,u,x) = (√1+p²)/u, the first variation and Euler-Lagrange equations yield the following extremal equation: d/dx(F_p) - F_u = 0, where F_p and F_u denote the partial derivatives of F with respect to p and u, respectively.
b) For the choice of F(p,u,x) = (√p²+u²), the extremal equation is d/dx(F_p) - F_u = 0.
c) For the choice of F(p,u,x) = ((1/2)u²)-((1/2)p²), the extremal equation is d/dx(F_p) - F_u = 0.
Step-by-step explanation:
For the choice of F(p,u,x) = (√1+p²)/u, the first variation and Euler-Lagrange equations are applied. The extremal equation, d/dx(F_p) - F_u = 0, is derived by taking the partial derivatives of F with respect to p (F_p) and u (F_u). This equation sets the stage for finding extremals for the given functional I(u).
Similarly, for F(p,u,x) = (√p²+u²), the extremal equation is obtained using the Euler-Lagrange approach. This equation, d/dx(F_p) - F_u = 0, is crucial in determining the extremals that minimize or maximize the given functional I(u).
For F(p,u,x) = ((1/2)u²)-((1/2)p²), the Euler-Lagrange equation is applied, resulting in the extremal equation d/dx(F_p) - F_u = 0. This equation, derived from the calculus of variations, guides the search for the general solution for extremals of I(u) for the specified choice of F.
In summary, the Euler-Lagrange equations for each choice of F(p,u,x) provide the necessary conditions for finding extremals of the given functional I(u). These equations serve as a foundation for solving and obtaining a general solution with two constants of integration for each case.