A colleague of mine scrawled out some equations on a whiteboard a few days ago. As far as I can tell, they are a set of cross-coupled, vector-valued, two-dimensional, first-order nonlinear differential-algebraic equations.
I don't know what context exactly they are from, but IIRC they have something to do with the intercept of ballistic trajectories.
I've tried pretty much every trick in the book to sort this out. They're not seperable, they don't match any well-studied nonlinear ODE, and none of the integral transforms are of any help.
Let's see if anyone has any ideas. Equations are spread across three pictures.
For clarification, [math]x[/math],[math]y[/math],[math]x_1[/math], and [math]y_1[/math] all depend on [math]t[/math], but [math]v_0[/math], [math]\vec{V}_0[/math], [math]\theta[/math], [math]\vec{n}[/math], and [math]a[/math] are all constants.
I'm not looking for an analytic solution, and I really doubt that a meaningful one exists at all, but what I'd like to do is seperate out all the symbolic and functional components so I can have something numerically solvable.
Also, here's a table of variable dependencies I scribbled.
just use euler lagrange doe
>>7700981
How is that even applicable? It's not an optimization problem.
Here's what I've gathered so far:
https://www.scribd.com/doc/292305422/Diff-Eqs
Wait, what if it's not [math]\vec{V}_0 = \hat{n}v_0[/math], but rather [math]v_0 = \sqrt{(x')^2+(y')^2}[/math]? Have I been reading this wrong the entire time?
>>7701863
Damn, that can't be it, because that would just result in [math]v_0 = \sqrt{v_0^2}[/math]
>>7701863
how could that be?
also isn't sqrt( (x')^2 + (y')^2 ) written in the first equation? so why wouldn't v_0 be written there instead?
>>7701901
My thought was perhaps the writer noticed that scalar term kept coming up in the equations, so he then defined [math]v_0[/math] to be that term.
>>7701907
I think I understand but just to clarify
you wrote vector V_0 but isn't is V_D? that's a D not a 0 right?
sorry I haven't figured out mathjax yet
>>7701922
I think it's [math]V_0[/math], but I can kinda see why it might look like a D. Nevertheless, they refer to the same thing.
Also, I renamed [math]a[/math] as [math]p_0[/math] since the author told me it was the initial position.
>>7701926
sorry to go off topic but the letter circled in yellow is what you're saying is a 0?
who writes their 0's like that
please tell me we're talking about the same letter here
>>7701928
We are talking about the same thing. It might be a D, I don't know, but I transcribed it as a 0, and I think I prefer it that way because it appears to be an initial velocity vector.
>>7701931
alright fair enough
carry on derp_commander
Ok, defining:
[eqn]
X := \left(x_1'(t)\right)_x\\
Y := \left(x_1'(t)\right)_y
[/eqn]
and
[eqn]
\chi := \left(y_1'(t)\right)_x\\
\Upsilon := \left(y_1'(t)\right)_y
[/eqn]
Eliminating the variable [math]t[/math] ultimately leads to these very interesting relations:
[eqn]
\left(v_0 X\right)^2 = \left(X^2 + Y^2\right)^2\\
\left(v_0 \Upsilon\right)^2 = \left(\chi^2 + \Upsilon^2\right)^2
[/eqn]
Any thoughts on what this means?
>>7700493
it has something to do with an intercept points of two projectiles ,i remember something about using something similar when trying to determine the point of collision between a projectile going on a balistic arc and another being dropped from a certain height
>>7702142
Any ideas then?
Also, changing the order of substitution removes a sign ambiguity:
[eqn]
v_0 X = X^2 + Y^2\\
v_0 \Upsilon = \chi^2 + \Upsilon^2
[/eqn]
>>7702142
Still, what are common techniques used to solve such problems?
