# Scaling/nondimensionalization for numerical optimization

I have a numerical optimization problem that I am trying to scale appropriately, in order to allow for the solver to achieve faster and more accurate results. I found a paper here that had a short section on nondimensionalization on page 44, that stated the following:

In order to nondimensionalize a quantity, just multiply by the appropriate constant.

$$n_{length}=\frac{1}{R_{P}}$$ $$n_{velocity}=\sqrt{\frac{R_{P}}{\mu}}$$ $$n_{mass}=\frac{1}{m_{0}}$$ $$n_{time}=\frac{n_{length}}{n_{velocity}}$$ $$n_{force}=\frac{n_{mass}n_{length}}{n_{time}^{2}}$$

Where $R_{P}$ is the planet's radius, $\mu=GM$ is the planet's gravitational parameter and $m_{0}$ is the initial spacecraft's mass. I have very similar equations to those used in the thesis, where the following system of first-order ODEs describes the spacecraft's dynamics:

$$\dot{x} = v_{x}$$ $$\dot{y} = v_{y}$$ $$\dot{v_{x}} = -\frac{GMx}{||\mathbf{r}||^{3}}-\frac{D_{x}}{m}+\frac{T_{x}}{m}$$ $$\dot{v_{y}} = -\frac{GMy}{||\mathbf{r}||^{3}}-\frac{D_{y}}{m}+\frac{T_{y}}{m}$$ $$\dot{m} = -\frac{||\mathbf{T}||}{g_{0}I_{sp}}$$

So, since the right-hand side of the first two ODEs contain velocity terms, should I just multiply each term by $n_{velocity}$, and since the 3rd and 4th ODEs describe accelerations, do I just multiply each term by $n_{force}/n_{mass}$? But, since each derivative is with respect to time, do I need to multiply the left-hand side of each equation by $n_{time}$? As you can see, I'm a little confused how I should go about actually implementing the nondimensionalization so I can discretize the equations in their scaled/nondimensionalized form. From my simple understanding of the topic, my nondimensionalized equations would look as follows:

$$\dot{x} = \frac{n_{velocity}}{n_{time}}v_{x}$$ $$\dot{y} = \frac{n_{velocity}}{n_{time}}v_{y}$$ $$\dot{v_{x}} = \frac{n_{force}}{n_{time}n_{mass}}\left(-\frac{GMx}{||\mathbf{r}||^{3}}-\frac{D_{x}}{m}+\frac{T_{x}}{m}\right)$$ $$\dot{v_{y}} = \frac{n_{force}}{n_{time}n_{mass}}\left(-\frac{GMy}{||\mathbf{r}||^{3}}-\frac{D_{y}}{m}+\frac{T_{y}}{m}\right)$$ $$\dot{m} = \frac{n_{mass}}{n_{time}^{2}}\left(-\frac{||\mathbf{T}||}{g_{0}I_{sp}}\right)$$

And I would then discretize these equations in order to perform the numerical optimization. Would such a process work, or have I gone about doing it in the wrong way?