newtonian mechanics - Is there work being done if no displacement occurs?

So the definition of work is $W = \vec{F}\cdot\vec{s}$. Say I have a point mass which is being pushed on both sides by equal forces and therefore does not move. Does this mean that no work is being done by any force? It's apparent that there is no net force, but could I calculate the work done by each side to be the work that would have been done absent the other?

For example, assuming our point mass has a mass of 1 kg and would have been moved 1 m in a direction by our 1 N force if an equal and opposite force did not counteract it. Would our force have exerted $1N \cdot 1m = 1J$ of work, or did it not perform any work since our object didn't actually move?

Answer

If the displacement of the object is zero, then one can calculate the work done by each individual force, the work done by each force is zero.

Why? Work is not defined in terms of what would have happened to the object in the absence of other forces; it is defined in terms of the motion that actually occurred.

More concretely, if from time $t_a$ to time $t_b$ an object moves along a curve $\vec x(t)$, and if it is acted on by a force $\vec F(t)$, then (regardless of whether $\vec F$ here denotes the net force, or a single force acting on the object, or some other combination), the work done by the force $\vec F$ is defined as follows: \begin{align} W(t_b, t_a) = \int_{t_a}^{t_b} \vec F(t) \cdot \frac{d \vec x}{dt}(t) \,dt. \end{align} If the object doesn't move during its trip, then $d\vec x/dt = 0$, and the integral vanishes, so we obtain \begin{align} W(t_b, t_a) = 0. \end{align}