Is it true that an object, in course of its motion, always seeks the position of minimum potential energy?

Question

I have heard the statement, in a Classical mechanics course, that the motion of an object is always toward the position of minimum potential energy. I don't think that this statement correct because I can immediately think of a counter-example. Consider the swing of the bob of a pendulum from the lowest position to one extreme position. During this part of its trajectory, the bob is moving to positions of higher potential energy than the minimum (at the lowest point) due to inertia.

Am I correct? Isn't this statement wrong?

Answer 1

The force on the object is related to the gradient of the potential it moves in by:

$$ \mathbf F = -\nabla V $$

For many smooth potentials this force will point towards the minimum, and your example of a pendulum is one such case. However even in this case it only means the object accelerates towards the minimum, not that its velocity is in the direction of the minimum.

In one dimension, as long as there aren't any local minima, the force and therefore the acceleration always points towards the minimum. However in two and three dimensions even this isn't necessarily true and at some positions the object might accelerate away from the minimum.

Answer 2

In the generality that you stated it, this statement is false. What is true is that on an object always acts a force $F = - \nabla \Phi$ that is the gradient of the Potential. For an object at point $\vec{p}$, the negative gradient points towards a local minimum of the potential. This minimum isn't neccesarily the nearest minimum of the potential, and it's not even given that the potential has a minimum.

However, there is a case in which the statement is true: If you have a dissipative system, that means, the kinetic energy of your object can not only be transfered to potential energy as your object moves in a conservative force-field, but it can also dissipate and just dissappear (due to friction with the ground or surrounding objects or whatever). In all this cases, your object will eventually come to halt at the position with the lowest potential energy: For example, your pendulum is able to move away from the lowest position, but if you let it swing, it will somewhen stop exactly at the position of minimum potential. In that sense you could say that "the motion is toward the position of minimum potential energy".