Acceleration in spherical coordinates.

[Guv]

I have an application which does gravitational computations. I want to display positions, velocities, and accelerations in both cylindrical and spherical coordinates.

Given positions, velocities, and accelerations in XYZ coordinates, does anybody know the formulae for accelerations in cylindrical and spherical coordinates? Does anybody know a Web Site which would have such formuale?

I have formulae for positions and velocities using both cylindrical and spherical coordinates.

I know how to develop the formulae for the accelerations in cylindrical and spherical coordinates, but the algebra is terribly messy. I am afraid that I might make a mistake.

For Cylindrical Coordinates:
Since the fonts may not post correctly...
Let the unit vectors be Ez, Er_bar, Etheta:
Ez , the axial direction (parallel to z axis, same as k in Cartesian coordinates).
Er_bar , the radial direction (normal to the z axis).
Etheta , the transverse direction (normal to the plane formed by Er and Ez, and has sense according with right-hand-screw rule for the permutation z,r_bar,theta).

dot1() , the operator d/dt
dot2() , the operator dot1(d/dt)

Yeah, this is getting bad. If you can match your notation to the following for Velocity:
V = [dot1(r_bar)] Er_bar + [r_bar dot1(theta)] Etheta + [dot1(z)] Ez

then we have for Acceleration:
A = [dot2(r_bar) - r_bar dot1(theta)^2] Er_bar +
[r_bar dot2(theta) + 2 dot1(r_bar) dot1(theta)] Etheta +
[dot2(z)] Ez
That was from my mechanics book, but I couldn't find a spherical coordinate acceleration in print.
Trust your ability to do the expansion.

My searches at http://search.wolfram.com/ were unsuccessfull, but not exhaustive.

[Edited by VirtuallyVB on 01-03-2001 at 09:12 PM]

[Guv]

VirtuallyVB, it looks like I will have to face the messy algebraic expansion. I know how to do the job. I do not trust myself to do it correctly.

I have equations for transformation of position & velocity vectors from Cartesian (XYZ) coordinates to spherical and cylindrical. The velocity equations are bad enough. I shudder at what the acceleration equations will look like.

I had to work out the velocity transforms myself. I could not find them anywhere. This surprised me. Mathematicians have known how to do this for centuries. I cannot believe it is not published somewhere. I just cannot find where. Same for acceleration transforms. They must be published somewhere.

The only way I know to verify my results is to use my MathCad software. I verified the Velocity equations by using them to transform from XYZ to spherical and then back to XYZ. I also did spherical to XYZ and back to spherical. If I get the exact same values I started with, at least I know that the equations are inverses of each other. Since it is unlikely that I made compensating errors while developing the equations, I figure they are correct.

Unfortunately that approach to verification requires equations for spherical to XYZ as well as XYZ to spherical. For my VB application, I only need the accelerations transformed in one direction: XYZ to spherical. I was hoping to avoid working out both sets of equations. Developing one transform will be trouble enough.

If I could do the work drunk, I would try it that way to ease the pain. I seldom drink and hold it very badly. Two drinks and I have trouble with simple conversations. No way I could do messy algebra while drunk.

If 100 million people believe a foolish idea, it is still a foolish idea!

If 100 million people get the same results by working it out by hand, it could still be wrong; but it might be right.

Perhaps a few fellows here can try it by hand and you can compare results.

The Cylindrical Velocity and Acceleration I posted was from Engineering Mechanics, Statics and Dynamics, Irving H. Shames, 3rd Edition, Prentice-Hall, ISBN: 0-13-279166-8, page 433.

How did they compare to what you had for cylindrical coordinates?

[Guv]

VirtuallyVB, the Mechanics book copped out. It is merely saying that the derivative of the Position Vector is the Velocity Vector, and the derivative of the Velocity Vector is the Acceleration Vector. The real job is actually working out the equations for the derivatives.

By the way, I always felt that showing vectors as X*i + Y*j + Z*k is clumsy compared to (X, Y, Z). When X, Y, & Z are functions, neither notation is easy to work with. I tend to avoid of these vector notations.

I have started on the messy algebra, and will use MathCad to do numerical verification when I get the equations.

[parksie]

Why not use MathCad to do the calculus?

[Guv]

As of now, I do not know how to get MathCad to do symbolic math. It requires you to provide equations, which it will graph and/or compute values for.

I started work on differentiating my velocity equations in order to get acceleration equations. I now know why there is no literature giving such equations. You cannot describe them to a typesetter.

They are messy beyond anything I have ever encountered.

Part way through, I decided to give up forever, so I watched TV and then went to sleep. In the middle of the night I woke up with the idea that I do not need an equation, only a computational algorithm. Duh, duh, duh!!!

As a trivial example, consider the following.

• A = sqr(X ^ 2 + Y ^ 2)
• B = (X - Y) ^3
• C = Z ^ 2 + 5 * Z + 4
• Result = C + A ^ (B / C)

Given (X, Y, Z), I can write a program to calculate Result without having an explicit equation for Result expressed in terms of X, Y , & Z.

In fact, there are many programs which evaluate complex equations by calculating various intermediate results first. There are often good reasons for writing programs that way. For example, Sqr(X ^ 2 + Y ^ 2) often appears more than once in a complex equation. A competent programmer would compute it once and put it into a temporary variable.