Interpreting the L2 Norm of Force on a Path

[maze]

The path integral of force is the work, something that has a clear physical meaning we can relate to. My question is, what is the physical interpretation for the L² norm of the force along a path? (integral of the force squared, basically):

$$\left(\int_\Gamma F \cdot F \right)^{\frac{1}{2}}$$

If a particle takes the path from point A to B which minimizes the work, then the least amount of external energy was expended moving it from point A to B. Can we analogously characterize the type of paths between 2 points that minimize the L² norm of the force.

Thanks!

 

[gabbagabbahey]

The path integral of force is the work, something that has a clear physical meaning we can relate to. My question is, what is the physical interpretation for the L² norm of the force along a path? (integral of the force squared, basically):

$$\left(\int_\Gamma F \cdot F \right)^{\frac{1}{2}}$$

Do you mean:

$$\left(\int_\Gamma (\mathbf{F} \cdot \mathbf{F})ds \right)^{\frac{1}{2}}$$

?

 

[maze]

Yes, of course. F:R³->R³, $\Gamma:[0,1]->\textbf{R}^3$