What conditions are necessary to guarantee uniform circular motion? Suppose an object is subjected to a force of constant magnitude, which is  always directed to the origin. And suppose we know the initial position of the object relative to the origin, and the initial velocity of the object, can we determine if the object will perform uniform circular motion? If so, what conditions are necessary? Can we determine its position as a function of time from these givens?
I know that if we know that an object performs uniform circular motion, and we have the equations which describe its motion, for example $$ \mathbf r= \begin{bmatrix} \cos(t)\\ \sin(t)\\ \end{bmatrix} $$
we can find the velocity, and acceleration simply by taking derivitives. But can we go the other way around and deduce the equation of motion as I described above? Perhaps by solving the differential equation $$ m \ddot{\mathbf r} = - \lVert \mathbf F \rVert \frac{\mathbf{r}}{\lVert \mathbf r \rVert}$$
where $\lVert \mathbf F \rVert$ is constant?
 A: 
But can we go the other way around and deduce the equation of motion as I described above? Perhaps by solving the differential equation
$$m \ddot{\mathbf r} = - \lVert \mathbf F \rVert \frac{\mathbf{r}}{\lVert \mathbf r \rVert}$$
where $||\mathbf F||$ is constant

This would not be the correct equation of motion.
You can just apply Newton's second law in Polar Coordinates:
$$\mathbf F=m\mathbf a=m(\ddot r-r\dot\theta^2)\,\hat r+m(r\ddot\theta+2\dot r\dot\theta)\,\hat\theta$$
For a force of constant magnitude always pointing towards the origin we have $\mathbf F=-F\,\hat r$, and so the equations of motion become
$$m\ddot r-mr\dot\theta^2+F=0$$
$$r\ddot\theta+2\dot r\dot\theta=0$$
which hold for any initial conditions.
In order to have uniform circular motion, we need

*

*$\dot r$, $\ddot r$, and $\ddot\theta$ to all be $0$ and,

*$r$ and $\dot \theta$ to be non-zero (they also need to be constant, but that follows from point 1).

This occurs when
$$F=mr\dot\theta^2$$
$$\dot r=0$$
So this shows that in order to have uniform circular motion we need for our initial conditions

*

*The force magnitude is equal to $mr(0)\cdot(\dot\theta(0))^2$

*$\dot r(0)=0$
If these two properties are not true of the initial conditions then you will not get uniform circular motion. You can determine what the motion will be from the general equations of motion we obtained above.
A: These are the equations of motion
$${\ddot{r}}\,m-m{\dot\varphi }^{2}r+F=0\tag 1$$
$$ \ddot\varphi \,r+2\,{\dot r}\,\dot\varphi=0\tag 2   $$
equation (2) is also;
$$\frac{d}{dt}\left(\,r^2\dot\varphi\right)=0$$
thus :
$$\dot{\varphi}=\frac{L}{r^2}$$
where L is a constant.
substitute $\dot\varphi$ in equation (1)
$$m\,\ddot{r}-m\,\frac{L^2}{r^3}+F=0\tag 3$$
the conditions for a uniform circular motion are: $~\ddot r=0$ and $r=\text{constant}=r_0~\Rightarrow $ the initial conditions are
$$r(0)=r_0,~D(r)(0)=0$$
and the force F is:
$$F=m\,\frac{L^2}{\,r_0^3}$$
