Scaling/nondimensionalization for numerical optimization

Question

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 simply stated the following:

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

The nondimensionalization constants are given by

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

where $R_{P}$ is the planet's radius, $\mu=GM$ is the planet's gravitational parameter and $m_{0}$ is the spacecraft's initial 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:

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

I'm a little confused how I should go about actually implementing the nondimensionalization in order to discretize the equations in their scaled/nondimensionalized form. From my simple understanding of the topic, my nondimensionalized equations would look as follows:

EDIT: In response to Kirill's answer, would the transformed system of equations instead look as follows?

$$\dot{\tilde{r_{x}}} = \frac{n_{length}}{n_{time}}v_{x}$$ $$\dot{\tilde{r_{y}}} = \frac{n_{length}}{n_{time}}v_{y}$$ $$\dot{\tilde{v}_{x}} = \frac{n_{force}}{n_{mass}}\left(-\frac{GMr_{x}}{||\mathbf{r}||^{3}}+\frac{T_{x}}{m}\right)$$ $$\dot{\tilde{v}_{y}} = \frac{n_{force}}{n_{mass}}\left(-\frac{GMr_{y}}{||\mathbf{r}||^{3}}+\frac{T_{y}}{m}\right)$$ $$\dot{\tilde{m}} = \frac{n_{mass}}{n_{time}}\left(-\frac{||\mathbf{T}||}{g_{0}I_{sp}}\right)$$

EDIT 2: Since I'm not really understanding the answers given (apologies, I'm a bit slow), and can't seem to find any well explained examples online, I thought it may help if I show some of the code from the optimization problem I'm working on. Given the original system of ODEs, they are discretized using $N$ nodes, where the final time is given by $t_{f}$ and the time-step between nodes is given by $dt=t_{f}/N$. Thus, using a trapezoidal integration scheme, the system of ODEs is discretized as follows in MATLAB,

for t = 2:N
   y(:,t) = (I-0.5*dt*A(:,:,t))\((I+0.5*dt*A(:,:,t-1))*y(:,t-1)+0.5*dt*B(u(:,t)+u(:,t-1)));
   .
   .
   .
end

where the state vector is given by $y = [r_{x},r_{y},v_{x},v_{y},m]^{T}$ and the control vector is given by $u = [T_{x},T_{y},||\mathbf{T}||]^{T}$, with $A$ given by

A = [0 0 1 0 0;
     0 0 0 1 0;
     a 0 0 0 0;
     0 a 0 0 0;
     0 0 0 0 0]

wherea = -(G*M)/(norm(r(:,t)).^3), and

B = [0 0 0;
     0 0 0;
     1/m 0 0;
     0 1/m 0;
     0 0 -1/(g0*Isp)];

How could the above system be scaled? I thought about simply applying the following diagonal preconditioner to both the left and right-hand sides of the discretized equations

M_scale = [
    n_length/n_time 0 0 0 0;
    0 n_length/n_time 0 0 0;
    0 0 n_force/n_mass 0 0;
    0 0 0 n_force/n_mass 0;
    0 0 0 0 n_mass/n_time];

where I then have

for t = 2:N
   M_scale*y(:,t) = M_scale*(I-0.5*dt*A(:,:,t))\((I+0.5*dt*A(:,:,t-1))*y(:,t-1)+0.5*dt*B(u(:,t)+u(:,t-1)));
   .
   .
   .
end

but unfortunately this doesn't seem to work.

Answer 1

One thing that makes your non-dimensionalized ODE confusing is that you use the same symbols for dimensionalized and nondimensionalized variables, even though they are different variables.

Consider writing it in a different way: let there be a new set of variables, $\tilde t, \tilde x,\ldots$, all unitless, defined by $$ \tilde t = n_{\mathrm{time}}t, \qquad \tilde x = n_{\mathrm{length}} x, \qquad \ldots $$ Then it's a straightforward change of variables, from $(x(t),\ldots)$ to $(\tilde x(\tilde t), \ldots)$, applied to the system of ODEs. You get $$ \frac{\mathrm{d}\tilde x}{\mathrm{d}\tilde t} = \frac{n_{\mathrm{length}}}{n_{\mathrm{time}}} \dot x = \frac{n_{\mathrm{length}}}{n_{\mathrm{time}}} v_x = \frac{n_{\mathrm{length}}}{n_{\mathrm{time}}} \frac{\tilde v_x}{n_{\mathrm{length}}/n_{\mathrm{time}}} = \tilde v_x. $$ The slight danger in not doing this explicitly is that you might accidentally produce an inequivalent system of ODEs. The transformation between the two sets of variables must be an equivalence transformation, and so it should not change the actual system dynamics. Simply multiplying the r.h.s. by the factors is not valid for this reason.

The reason it's called nondimensionalization is because you change the units in which you measure all the quantities that enter the ODE, but the ODE itself remains the same. This is not the case for the ODE you gave, because if you look at $\dot x = \frac{n_l}{n_t}v_x$, this contradicts even the very definition of velocity.

When your quote says "In order to nondimensionalize a quantity, just multiply by the appropriate constant", it means precisely that: you get a new variable by multiplying an original variable by a constant, but this is not quite the same as multiplying the r.h.s. of the ODE by that constant.