Consider a particle with coordinates $(x(t),y(t))$ on a smooth curve $\phi(x,y)=0$. If the particle moves from $(x(0),y(0))$ to $(x(\tau),y(\tau))$ for $\tau >0$ such that its kinetic energy is minimized, then
$(a)$ $\frac{\ddot{x}}{\phi_x}=\frac{\ddot{y}}{\phi_y}$.
$(b)$ $\dot{x}^2(0)+\dot{y}^2(0)=\dot{x}^2(\tau)+\dot{y}^2(\tau)$.
$(c)$ $\dot{x}\phi_x+\dot{y}\phi_y=0$.
$(d)$ $\dot{x}^2(0)=\dot{x}^2(\tau)$.
Now, if we consider this problem as minimizing a functional $J[x(t),y(t)]=\int_0^\tau F(x,\dot{x},y,\dot{y},t)dt$ in two dependent variables representing coordinates and one independent variable representing time, then Euler-Lagrange equation will give a family of extremals from which we can conclude the answer. But I am unable to find a way to relate the K.E. as a functional as written above, and include the curve $\phi(x,y)=0$ in the same. So is there another method for this problem or am I on the right track? Any help will be appreciated.