Potential energy and equilibrium points

Question

At an equilibrium point of a system we consider that if we place the object (or in general the objects) there with zero Kinetic energy ,the object will stay there . The fact that the object stays there means that it will not change its position , thus the Potential energy will remain the same . Is this the intuitive interpretation of why the partial derivative of the potential energy with respect to the position has to be equal to zero at the equilibrium point (in other words the condition θV/θq = 0 must be true at the equilibrium point ) ?

Answer 1

Yes. $F = - \frac {\delta V }{\delta q}$ so if $\frac {\delta V }{\delta q}$ were not zero there would be a force on a stationary object which would move it away from that point.