Deriving Gauss' Variational Equation for True Anomaly

[jsandberg]

Homework Statement

Derive the Gauss Variational differential equation for the true anomaly, f, with respect to time using components along the radius, angular velocity, and a unit vector orthogonal to those two (ir,itheta,ih).

Homework Equations

Sorry, I don't know how to use Latex. But I have attached the equations I need to start from and get to! ad is the perturbation, r_underline is the position vector, r is the norm of the position vector, v is the velocity vetor, h is the angular momentum, f is the tru anomaly, e is the eccentricity, p is the semilatus rectum.

The Attempt at a Solution

See attached handwritten solution- the first two lines are given in the assignment. I just can't seem to get the equation simplifed to the final equation.

View attachment df_dt.zip

View attachment Derivation df_dt.pdf

 

[D H]

Your mistake is right at the start. Unfortunately this means everything you did was wrong.

Your mistake was in expressing the position and velocity vectors in terms of \hat x, \hat y, and \hat z. You should have expressed these in the same coordinate system in which the perturbative acceleration is expressed -- in other words, \hat r, \hat \theta, and \hat h. The position vector is simply \mathbf r = r \hat r. I'll leave velocity up to you.

Hint: It does not take two pages of math to derive the result.

 

[jsandberg]

Thank you for your quick response! Yes, changing the radius and velocity components helped a lot. I am still having trouble simplifyin the equation, however (see attached).

Thanks again for your time.

 

[D H]

What's wrong? The last expression is exactly what you want to derive. Is your problem going from the penultimate expression to the last one? In other words, you are having a problem with showing

\left(1+\frac r p\right)re(1-\sin^2 f) + \frac{r^2} p \cos f = p\cos f

Hint: All you need are 1-\sin^2 f = \cos^2 f and r=p/(1+e\cos f).

 

[jsandberg]

Yes, that is where my problem is. Do I substitute the equation for "r" every time I see an "r"?

 

[D H]

This is homework. I've given a couple of hints. I'll give one more: Factor p\cos f out of each term on the left hand side. In other words, rewrite the left hand side as p cos(f) * (term1 + term2). Now show that term1+term2 is identically one.

 

[jsandberg]

Thanks for all your help! Much appreciated.