Chapter 9.8, Problem 1MP

(a) In Section 3, we showed how to obtain a first integral of the Euler equation when $F=F\left(y,y^{\prime }\right).$ There is an alternative method of handling this case. You can show that if $F=F\left(y,y^{\prime }\right),$ then $F-y^{\prime }\partial F/\partial y^{\prime }=$ const. To prove this, differentiate the left-hand side with respect to $x,$ and show that the result is zero if $F$ satisfies the Euler equation. Note that what you have is a first integral of the Euler equation.

(b) Use the method of (a) to do the problems at the end of Section 3.

(c) Consider the motion of a particle along the $x$ axis; then $L=T-V=$ $-\frac{1}{2}m{\stackrel{\dot{}}{x}}^2-V(x).$ Note that $L$ does not contain the independent variable $t$; this corresponds to the case $F=F\left(y,y^{\prime }\right)$ in (a). Show that the first integral found in (a) is just the equation of conservation of energy for the mechanics problem.