Calculus of variation for geodesic

Question

I need to minimize $$J[v]=\int\sqrt{P(x)+R(x)(v')^2}dx$$ By Euler equation, I get $$\frac{d}{dx}\frac{Rv'}{\sqrt{P+Rv^{'2}}}=0$$ Then I need to solve a complex ODE, but I don't know how to deal with it.

Answer 1

The differential equation is,

$$ \frac{\mathrm d }{\mathrm d x }\left( \frac{Rv'}{\sqrt{P+Rv'^2}}\right) = 0.$$

From elementary calculus we have that if the derivative of a function is zero then it is a constant function,

$$ \frac{Rv'}{\sqrt{P+Rv'^2}} = A,$$

solving this equation for $v'$ we get,

$$ v' =\pm\sqrt{ \frac{PA^2}{R^2-A^2R}},$$

which can be integrated to yield,

$$ v(x) = \pm \int^x \sqrt{ \frac{PA^2}{R^2-A^2R}} \ \mathrm{d}x. $$

Decisions regarding the constant $A$ and the lower bound of integration are determined by the boundary conditions of the problem. Notice that $R=0$ wouldn't make sense in this situation because if it were $J[\cdot]$ would be a constant.