Help please to convert a Hyperbole to a canonical form

[ValeryOchkov]

See please the Mathcad Prime 3 file in attach

Hyperbola - Wikipedia, the free encyclopedia

[RichardJ]

The way you are doing it looks pretty good to me. Your residuals are down in the region of 10^-8 to10^-9, and given that your starting point is a numerical solution from odesolve anything better would be meaningless. The only thing I would do differently is that I would set up a vector of times, and then have one vectorized equation in the solve block rather than 12 separate equations.

[ValeryOchkov]

Thanks Richard!

It will be one chapter of the book "Fine Math with Mathcad". Would you be one coauthor

No problem to convert the the orbit as ellipse - see please the attach.

And second.

I can nor create one orbit as parabola

[RichardJ]

It will be one chapter of the book "Fine Math with Mathcad". Would you be one coauthor

Thanks for the consideration, but that sounds like it might be a lot of work

I can nor create one orbit as parabola

You can't have an orbit as a parabola, because if it's a parabola it's not an orbit

The parabolic trajectory is the limiting case of the hyperbolic trajectory, when the orbiting body just achieves escape velocity ( i.e. it's the boundary case between an elliptical orbit and a hyperbolic trajectory). In such a case there is a very specific relationship between the orbital velocity and the radial distance. See the attached worksheet

[ValeryOchkov]

Richard Jackson wrote:

It will be one chapter of the book "Fine Math with Mathcad". Would you be one coauthor

Thanks for the consideration, but that sounds like it might be a lot of work

A lot of work is good!

Not a lot of work is not good!

You can be a coauthor of one chapter of the book!

Richard Jackson wrote:

I can nor create one orbit as parabola

You can't have an orbit as a parabola, because if it's a parabola it's not an orbit

Sori, Ay nou inglish veri plokho!

Thanks for the parabola - see the attach.

But with the parabola D is not quit zero!

[RichardJ]

A lot of work is good!

Not a lot of work is not good!

Agreed. But too much work is also not good. I have a lot on my plate right now.

You can be a coauthor of one chapter of the book!

It would be an interesting project I don't want to promise something I can't deliver in a timely fashion though, and I really do have a big workload right now. If you have my email address (I am almost certain I have yours) send me an email, and we can discuss this further. Privately, I can give you information about my current commitments that I can't post on a public forum.

But with the parabola D is not quit zero!

It's about 10 orders of magnitude smaller than D for the hyperbolic trajectory or the elliptical orbit. Given all the numeric approximations, that seems quite good to me. If you change CTOL from 10^-7 to 10^-10 it drops another 14 orders of magnitude though, which I think is really as much as you could hope for with a numerical ode solve followed by a numerical least squares fit. It's a good example of the limits of numerical computing.

[ValeryOchkov]

Thanks, Richard.

My e-mail is here Valery Fedorovich OCHKOV

And more

Sorry

I can not find or create same odes with polar coordinates!

Help me please

[ValeryOchkov]

ValeryOchkov wrote:

And more

Sorry

I can not find or create same odes with polar coordinates!

Help me please

As for a pendulum: