Given a vector field $\vec{F}(P(x,y),Q(x,y))$ and both starting point $A$ and endpoint $B$ find the trajectory where the work of vector field is the lowest asuming that trajectory starts at $A$ and ends in $B$. Generally saying find such $\overparen{AB}$ that $\int_{\overparen{AB}}{Pdy+Qdx}$ is minimal.
If that is not possible then, more particularly saying, we can assume that $\vec{F}(x,y)=<x^2+y, 3y^2+x>$ and starpoint and endpoint are $(-3, 0)$ and $(3, 0)$. Would there be any solution?
$$ (\text{curl } F)_3 = \partial_x F_2 - \partial_y F_1 = 1 - 1 = 0 $$ So $F$ is conservative. This has the nice consequence that all closed curves will give zero as integral value and this means that the integral value (for open curves) is curve independent.