For my Variational Calculus course we have been given the following problem: Given the following Euler Lagrange for the path P(t(s),s) $\frac{d}{dr}(\frac{((1-a/r)\frac{dt}{dr}}{\sqrt{(1-a/r)(dt/dr)^2-1/(1-a/r)}})=0$ rewrite to a function in the form of $t(B)-t(A)=\int_A^BC^2/(1-a/r)\sqrt{\frac{1}{a/r+C^2-1}}dr$
How would I do this? I have tried writing it as: $\frac{((1-a/r)\frac{dt}{dr}}{\sqrt{(1-a/r)(dt/dr)^2-1/(1-a/r)}})=N$ N a constant and then taking the dt to one side and the dr to the other and integrating over these differentials, however this has yielded some unwieldy expressions.
You want to solve an equation $\frac{cv}{\sqrt{cv^2-1/c}}=N$ for $v$. This is a purely algebraic exercise $$ c^2v^2=N^2(cv^2-1/c)=\frac{N^2}{c}(c^2v^2-1)\\ c^2v^2\left(\frac{N^2}{c}-1\right)=\frac{N^2}{c}\\ v^2=\frac{N^2}{c^2(N^2-c)}\\ v=\frac{N}{c\sqrt{N^2-c}} $$ Note that the sign choice in the last square root is determined by the original equation. Now re-insert back $v=\frac{dt}{dt}$ and $c=1-\frac ar$.