# Why is the gravitational potential energy (or electrical) the work done to move a point mass $m$ from infinity to that point in a gravitational field?

To be specific, I'm asking why does the mass need to be moved "at a constant speed"?

My textbook says it is so that no kinetic energy is involved. But shouldn't kinetic energy be involved in order to move the point mass from infinity to a point in a gravitational field? Since kinetic energy is converted to gravitational potential energy.

If you consider a generic stationary mass distribution, it generates a gravitational stationary field in all the space around: $$\mathbf G=\mathbf G(x,y,z)$$ You can demonstrate that always exists a function called potential (not potential energy) for which: $$grad(V(x,y,z))=\mathbf G(x,y,z)$$ in every point of the space. Also you can demonstrate that: $$\int_{\beta} \mathbf G \cdot \mathbf {dr} =V(\mathbf b)-V(\mathbf a)$$ Where $\beta$ is a line which starts in $\mathbf a$ and ends in $\mathbf b$. You can obtain also that: $$\int_{\beta} m\mathbf G \cdot \mathbf {dr} =mV(\mathbf b)-mV(\mathbf a)$$ If you define $$U=Vm$$ the potential energy own by a mass, finaly you have: $$W_{\beta}=\int_{\beta} \mathbf F \cdot \mathbf {dr} =U(\mathbf b)-U(\mathbf a)=\Delta k_{a-b}$$ However the kinetic energy is not involved in the defintion of the potential energy.