Problem 83

Let the coordinates of point $M$ be $(x,y)$, and the coordinates of point $P$ be $(x_{0},y_{0})$.
Then, we have $x= \dfrac {x_{0}+8}{2}$, $y= \dfrac {y_{0}}{2}$. That is, $x_{0}=2x-8$, $y_{0}=2y$.
Since point $P(x_{0},y_{0})$ is on the circle $x^{2}+y^{2}=4$, we have $x_{0}^{2}+y_{0}^{2}=4$.
That is, $(2x-8)^{2}+(2y)^{2}=4$, which simplifies to $(x-4)^{2}+y^{2}=1$. This is the equation of the trajectory of the moving point $M$.
Therefore, the answer is: $\boxed{(x-4)^{2}+y^{2}=1}$
To find the coordinates of point $M$, we use the midpoint formula to write a system of equations, solve for $x_{0}$ and $y_{0}$, and substitute into the equation of the given circle.
This method of finding the trajectory equation is known as the related point method.
This problem examines the related point method for finding the trajectory equation. When using this method, it is important to set the coordinates of the moving point as $(x,y)$, and the relationship between $x$ and $y$ obtained at the end is the desired equation.