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.