Change the value of a variable at each time step in matlab?

[patric44]

TL;DR Summary

is it possible to Change value of variable at each time step?

hi guys
I am trying to implement the a second order differential equation tat contains a time dependent term, the equation looks something like
r'' = -\mu/r^3+(g/g^3-m/m^3)
the idea is that i want to calculate 'r' the position vector of a point, that is dependent on vectors g and m, I tired to crate an simple ode integrator to get the g vector as

function dx = themoonpos(t,x)
R = 6378.137;
u = 398600.441800; % km/s2
% J2 = 0.0010826269;
% b = -1.5*J2*u*R^2 ;
r = sqrt(x(1)^2+x(3)^2+x(5)^2);
%% the propagator
    dx=[
    x(2)
    ;-u*x(1)/(r)^3
    ;x(4)
    ;-u*x(3)/(r)^3
    ;x(6);
    -u*x(5)/(r)^3];
end

than i tried to implement that in another function dx that will be integrated as

function dx = thethirdp(t,x,yk)
 k0 = [
  -120901.611171555;
  89921.008706315/(24*60*60);
-298392.399263114;
-25049.485694581/(24*60*60);
-162652.182605121;
-12799.914191783/(24*60*60)
];

opts = odeset; opts.RelTol = 10^-9;
[t2,yk] = ode45(@themoonpos,[0:60:5000],k0,opts);

R = 6378.137;
u = 398600.432896939; % km/s2
um = 4902.80058214776; % km/s2
J2 = 0.0010826269;
b = -1.5*J2*u*R^2 ;
r = sqrt(x(1)^2+x(3)^2+x(5)^2);
xm = (yk(1)-x(1));
ym = (yk(3)-x(3));
zm = (yk(5)-x(5));
rd = sqrt(yk(1)^2+yk(3)^2+yk(5)^2);
rm = sqrt(xm^2+ym^2+zm^2);
%% the perturbation
apx = um*((xm/((rm))^3)-yk(1)/(rd)^3);
apy = um*((ym/((rm))^3)-yk(3)/(rd)^3);
apz = um*((zm/((rm))^3)-yk(5)/(rd)^3);

%% the Integrator
    dx=[
    x(2)
    ;-u*x(1)/(r)^3+apx
    ;x(4)
    ;-u*x(3)/(r)^3+apy
    ;x(6);
    -u*x(5)/(r)^3+apz];
 end

then this function is integrated using the ode45.
the thing is the values yk i don't know how to assign it to each time step that the integrator uses, is this the right thing to do it
I hope anybody help i tried to post this on mathwork but no answer

 

[f95toli]

Maybe I am misunderstanding your question, but can't you just use the "usual" trick of introducing a dummy variable equal to r' (e.g. set x=r')and then solve the resulting system of two equations?
This should be well described in the manual

Generally speaking, when solving a 2nd order ODE you always end up solving a system of two equations (for 3rd order you would get 3 etc)

 

[patric44]

Maybe I am misunderstanding your question, but can't you just use the "usual" trick of introducing a dummy variable equal to r' (e.g. set x=r')and then solve the resulting system of two equations?
This should be well described in the manual

Generally speaking, when solving a 2nd order ODE you always end up solving a system of two equations (for 3rd order you would get 3 etc)

the problem is that my 2nd order ode has a time dependent term that comes from another solution of a 2nd order ode, so I am trying to solve first for the first one and get a column vector of solutions, then plug that back into my other integrator to get the original "r" I want. but it doesn't seem to work
what is the usual way to do that?

 

[f95toli]

Sorry, I still don't get it
Are you saying you have two coupled 2nd order ODEs?
I initially assumed that you should be able to rewrite this as a system of 1st order ODEs
I.e. similar to what is done in the example of the van der Pol equation in the Matlab documentation

https://uk.mathworks.com/help/matlab/ref/ode45.html

Are you saying you can't re-write your problem as a system of 1st order ODEs for some reason?

 

[pasmith]

the problem is that my 2nd order ode has a time dependent term that comes from another solution of a 2nd order ode, so I am trying to solve first for the first one and get a column vector of solutions, then plug that back into my other integrator to get the original "r" I want. but it doesn't seem to work
what is the usual way to do that?

Isn't that just the single system
$$\begin{align*} \ddot g &= F_g(g) \\ \ddot x &= F_x(x,g) \end{align*}$$

 

[patric44]

i am trying to solve this equation which take the third body perturbation into account

separating each compoent I am left with 3 second order equations for each component with rm as a time dependent variable also, what is the right way to solve this ?

 

[f95toli]

If you have 3 2nd order ODEs you should probably re-write this as a system of 6 1st order ODEs
When you say that rm is a time dependent variable; do you mean that it is an independent variable (no dependence on r) which changes over time?

 

[patric44]

If you have 3 2nd order ODEs you should probably re-write this as a system of 6 1st order ODEs
When you say that rm is a time dependent variable; do you mean that it is an independent variable (no dependence on r) which changes over time?

i meant that the position of the moon is dependent on time, but I guess I can solve them as you guys said as a whole 6 second order coupled equations. I will try that hope it works

 

[pasmith]

If you work in the inertial frame in which the centre of mass is fixed at the origin, then you can reduce the number of equations by a third since one of the positions can be determined from $$0 = m_1\mathbf{r}_1 + m_2\mathbb{r}_2 + m_3\mathbf{r}_3.$$