Deriving this "familiar arccosine form" of integral

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Discussion Overview

The discussion revolves around deriving a specific integral expression related to the arc-cosine form and its geometric interpretation in the context of orbital dynamics and ellipses. Participants explore the mathematical formulation and seek to understand the geometric significance of the variables involved.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses a desire to develop the integral expression geometrically, linking it to the physics of orbital dynamics and ellipses.
  • Another participant questions whether the original poster seeks to prove the equation or explore its geometric significance.
  • A participant suggests that the constants and variables in the cosine expression may have geometric significance related to an ellipse.
  • One participant provides a link to a resource that addresses a simpler form of the arc-cosine integral, although it does not directly relate to the more complicated form being discussed.
  • Another participant speculates that the method used for the simpler case might be generalizable to the more complex situation.
  • A later reply introduces the idea of coordinate rotation to eliminate a "cross product" term, suggesting a potential avenue for exploration.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the geometric development of the integral or its implications. Multiple competing views and approaches remain, with some uncertainty about the connections to ellipses and the potential for generalization.

Contextual Notes

Participants express varying levels of familiarity with the integral and its geometric interpretation, indicating that assumptions about the relationships between the variables may not be fully resolved. The discussion also highlights the complexity of the form being analyzed compared to more familiar cases.

Odious Suspect
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I posted a question about this yesterday, but realized I had made a stupid mistake in my derivation.
Orbital dynamics: "The familiar arc-cosine form"
That error has been corrected. I still have a deeper question. I believe this expression can be developed using the geometry of an ellipse in a way which adds insight into the original physics where it is being applied.

Some 30 odd years ago I saw something of this nature done in an introductory physics course.

Is anybody here familiar with such a development?

$$-\int \frac{1}{\sqrt{a+2 bx-hx^2}} \, dx=\frac{1}{\sqrt{h}}\arccos \left[\frac{b-hx}{\sqrt{a+b^2h}}\right]$$
 
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I am not sure what to do with your post, do you want to prove the above equation or what?
 
No. I can prove it. I want a geometric development. That is to say, I believe the constants and variables in the expression ##cos \theta = \frac{b-hx}{\sqrt{a+b^2h}}## have some geometric significance pertaining to an ellipse.
 
That only addresses the form I really am familiar with. Notice that the form I am asking about is more complicated.
 
Odious Suspect said:
That only addresses the form I really am familiar with. Notice that the form I am asking about is more complicated.

Yes, I know that you are asking about something more complicated.
Might it be the case that the method used in the simple case can be generalized? I don't know. I offered a possible starting point.
Sorry, if I didn't provide you with exactly what you seek.
 
Last edited:
Sorry about being so abrupt. I get grouchy when I can't figure something out.

I'm guessing this might be something to do with rotation of coordinates to eliminate the "cross product" term.

In some universe, the denominator in the cosine expression represents a radius.
 

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