Some equations

[derp_commander !!SSXdvR3LCoa No.7700493]

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.

[derp_commander !!SSXdvR3LCoa No.7700495]

[derp_commander !!SSXdvR3LCoa No.7700499]

[derp_commander !!SSXdvR3LCoa No.7700554]

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.

[derp_commander !!SSXdvR3LCoa No.7700597]

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.

[Anonymous No.7700981]

just use euler lagrange doe

[derp_commander !!SSXdvR3LCoa No.7701372]

>>7700981
How is that even applicable? It's not an optimization problem.

[derp_commander !!SSXdvR3LCoa No.7701843]

Here's what I've gathered so far:

https://www.scribd.com/doc/292305422/Diff-Eqs

[derp_commander !!SSXdvR3LCoa No.7701863]

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?

[derp_commander !!SSXdvR3LCoa No.7701893]

>>7701863
Damn, that can't be it, because that would just result in [math]v_0 = \sqrt{v_0^2}[/math]

[Anonymous No.7701901]

>>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?

[derp_commander !!SSXdvR3LCoa No.7701907]

>>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.

[Anonymous No.7701922]

>>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

[derp_commander !!SSXdvR3LCoa No.7701926]

>>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.

[Anonymous No.7701928]

>>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

[derp_commander !!SSXdvR3LCoa No.7701931]

>>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.

[Anonymous No.7701932]

>>7701931
alright fair enough
carry on derp_commander

[derp_commander !!SSXdvR3LCoa No.7702118]

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?

[Anonymous No.7702142]

>>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

[derp_commander !!SSXdvR3LCoa No.7702148]

>>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]

[derp_commander !!SSXdvR3LCoa No.7702879]

>>7702142
Still, what are common techniques used to solve such problems?

[Anonymous No.7703712]

>>7700493
b