Is My Calculation for the Conserved Quantity in the Kepler Problem Correct?

[quasar987]

There must be something I'm totally missing here.

The situation is the following.

I am asked to show that given the lagrangian for the Kepler problem,

$$L=\frac{1}{2}\mathbf{\dot{q}}^2+q^{-1}$$

the k-th component of the Runge-Lenz vector,

$$A_k=\mathbf{\dot{q}}^2q_k-\mathbf{q}\cdot\mathbf{\dot{q}}<br /> \dot{q}_k-q_k/q$$

is the conserved quantity associated (in the sense of Noether's thm) with the infinitesimal coordinate transformation $\mathbf{q}\rightarrow\mathbf{q}+\delta \mathbf{q}$, where $\delta q_i = \epsilon(\dot{q}_iq_k-\frac{1}{2}q_i\dot{q}_k-\frac{1}{2}\mathbf{q}\cdot \mathbf{\dot{q}}\delta_{ik})$, epsilon being the infinitesimal parameter.

Following Noether's theorem, I know that if [tex]\delta L=L(\mathbf{q}+\delta \mathbf{q}, \mathbf{\dot{q}}+\delta \mathbf{\dot{q}},t)-L(\mathbf{q},\mathbf{\dot{q}},t)[/itex] can be written as <br /> <br /> $$\delta L=\epsilon \frac{d}{dt}\Lambda(\mathbf{q},\mathbf{\dot{q}},t)+\mathcal{O}(\epsilon^2)$$<br /> <br /> then the quantity<br /> <br /> $$F_k:=\sum_{i=1}^3\frac{\partial L}{\partial \dot{q}_i}(\dot{q}_iq_k-\frac{1}{2}q_i\dot{q}_k-\frac{1}{2}\mathbf{q}\cdot \mathbf{\dot{q}}\delta_{ik}) - \Lambda$$<br /> <br /> is conserved. By direct comparison of F_k with A_k I find that Lambda must be<br /> <br /> $$\Lambda = \frac{q_k}{q}$$<br /> <br /> (also, this is confirmed by the wiki article on the Runge-Lenz vector: <a href="http://en.wikipedia.org/wiki/Runge-Lenz#Noether.27s_theorem" target="_blank" class="link link--external" rel="nofollow ugc noopener">http://en.wikipedia.org/wiki/Runge-Lenz#Noether.27s_theorem</a> )<br /> <br /> So what remains to be done is to show by direct calculation that indeed, <br /> <br /> $$\delta L=\epsilon \frac{d}{dt}(\frac{q_k}{q})+\mathcal{O}(\epsilon^2)$$<br /> <br /> So I expand $\delta L$:<br /> <br /> $$\delta L= \frac{1}{2}(\mathbf{\dot{q}}^2+2\mathbf{\dot{q}}\cdot \delta\mathbf{\dot{q}}+(\delta\mathbf{\dot{q}})^2)+(\mathbf{q}^2+2\mathbf{q}\cdot \delta\mathbf{q}+(\delta\mathbf{q})^2)^{-\frac{1}{2}}-\frac{1}{2}\mathbf{\dot{q}}^2-(\mathbf{q}^2)^{-\frac{1}{2}}$$<br /> <br /> And here I find it impossible to put this in a form $\delta L=\epsilon A+\mathcal{O}(\epsilon^2)$ because of all these guys in the numerator <i>and</i> shielded by a square root. I have also tried "cheating", i.e. say "since epsilon is arbitrarily small, I can neglect this and this term" but nothing even comes close to the form I want.<br /> <br /> So I concluded that there must be something fundamentally flawed about the reasoning laid above. Anyone sees?<br /> <br /> Thanks for reading!<u>P.S.</u> I would appreciate feedbacks, so that if I get a few feedbacks that the above is right, I will post more of my work so we can find where I'm going wrong.[/tex]

 

[quasar987]

Solved, thanks. Turns out I had the right answer but in an hostile form that made it difficult to recognize.