Solving a hard physics integral

[remote garden]

Hey!
I was solving a physics problem and got the answer but I have to solve this integral. I haven't learnt integration yet so if anyone can just solve the integral for me even without an explanation will be greatly appreciated.

Everything except x is a constant

[verbal night]

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[remote garden]

Also ignore the part below the integral

I can put the limits r and R myself

[hollow sierra]

Is the expression under the integral du/√(u^2 - 2GM(1/λ - 1/u))? If so, then it's not solvable in elementary functions.

[remote garden]

Yup it is.

Hmm so how should I proceed from here?

[hollow sierra]

What the statement of the problem?

[remote garden]

This is not a problem that my teacher gave me

I made it myself

[hollow sierra]

Oh.

Still, what's the statement?

[remote garden]

Yup it's:

A body is ejected upwards(the velocity is wrt the center of the planet NOT the surface) from the surface of a planet of mass M and radius r such that it lands back on the surface after one revolution of the planet. The time period of the planet is T. What is the initial velocity?

[hollow sierra]

Oh, and you're trying to find the initial speed?

[remote garden]

Yup

[hollow sierra]

I see.
Well, there is no guarantee that you can solve this problem analytically, unfortunately.

I think it might be possible to express that integral in terms of elliptic integrals. You can try searching through Gradshteyn-Ryzhik for similar integrals.

[remote garden]

Oh, thanks for your help I'll definitely check that out

[novel delta]

@remote garden has given 1 rep to @hollow sierra

[remote garden]

But it's weird to me that it's such a simple problem but still requires such advance integration techniques

Maybe I've messed up somewhere

Anyways thanks for the help 😃👋

[hollow sierra]

Oh, that's quite common.

In fact, we're very lucky that the simple cases are sovlable at all.
In some sense, "most" problems are not solvable analytically.

[remote garden]

:o

I didn't know that

[hollow sierra]

Well, it's easy to see for integration, at least.
Try to think of a composition of some elementary functions. Just a jumble of functions.

It's unlikely that it will be integrable in elementary functions.

[remote garden]

Oh that's interesting

Maybe I'm going offtopic but

Isn't it weird that although these phenomena can be observed in the real world/simulation can't be solved mathematically?

I can't wrap my head around that

[hollow sierra]

They can. Just not using elementary functions.

[remote garden]

Oh

[hollow sierra]

That's what special functions are for. They broaden the number of exactly solvable problems.

It doesn't always help, though, no matter how many special functions you make up.

[remote garden]

Could you give me an example of a non elementary function? (Sorry I'm not quite familiar with this topic)

[remote garden]

Oh

[hollow sierra]

One sec, I've already answered that question before, let me find my comment.

[remote garden]

Is that what u r referring to?

[hollow sierra]

Non-elementary functions include Γ(x), Β(x), erf(x), Si(x), Ci(x), Fresnel integrals, Ei(x), li(x), elliptic integrals, Jacobi elliptic functions, etc.

[remote garden]

Ah ok got it

My bad

[hollow sierra]

Yes, elliptic integrals are also an example of special functions.

[remote garden]

Got it thank you very much

This is an interesting topic I'll try to read up on it

Thanks 😃

[hollow sierra]

You're welcome!

[remote garden]

+close