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

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In summary, the conversation discusses a problem involving the Lagrangian for the Kepler problem and the conserved quantity associated with an infinitesimal coordinate transformation. The participants use Noether's theorem to show that the conserved quantity can be written as a function of the Lagrangian and then proceed to calculate it. However, there is difficulty in putting the calculation in the desired form, possibly due to a flaw in the reasoning. This is resolved with feedback and the correct answer is achieved.
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quasar987
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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,

[tex]L=\frac{1}{2}\mathbf{\dot{q}}^2+q^{-1}[/tex]

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

[tex]A_k=\mathbf{\dot{q}}^2q_k-\mathbf{q}\cdot\mathbf{\dot{q}}
\dot{q}_k-q_k/q[/tex]

is the conserved quantity associated (in the sense of Noether's thm) with the infinitesimal coordinate transformation [itex]\mathbf{q}\rightarrow\mathbf{q}+\delta \mathbf{q}[/itex], where [itex]\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})[/itex], 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

[tex]\delta L=\epsilon \frac{d}{dt}\Lambda(\mathbf{q},\mathbf{\dot{q}},t)+\mathcal{O}(\epsilon^2)[/tex]

then the quantity

[tex]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[/tex]

is conserved. By direct comparison of F_k with A_k I find that Lambda must be

[tex]\Lambda = \frac{q_k}{q}[/tex]

(also, this is confirmed by the wiki article on the Runge-Lenz vector: http://en.wikipedia.org/wiki/Runge-Lenz#Noether.27s_theorem )

So what remains to be done is to show by direct calculation that indeed,

[tex]\delta L=\epsilon \frac{d}{dt}(\frac{q_k}{q})+\mathcal{O}(\epsilon^2)[/tex]

So I expand [itex]\delta L[/itex]:

[tex]\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}}[/tex]

And here I find it impossible to put this in a form [itex]\delta L=\epsilon A+\mathcal{O}(\epsilon^2)[/itex] because of all these guys in the numerator and 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.

So I concluded that there must be something fundamentally flawed about the reasoning laid above. Anyone sees?

Thanks for reading!P.S. 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.
 
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Solved, thanks. Turns out I had the right answer but in an hostile form that made it difficult to recognize.
 

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

What is the Kepler problem?

The Kepler problem is a famous mathematical problem that involves calculating the motion of a planet or satellite around a central body, such as a star or planet. It is named after the German astronomer Johannes Kepler, who first described the laws governing the motion of planets in the 17th century.

What are the three laws of Kepler?

The three laws of Kepler are a set of rules that describe the motion of planets around a central body. The first law, also known as the law of ellipses, states that the orbit of a planet around a star is an ellipse with the star at one of the focal points. The second law, or the law of equal areas, states that a line connecting a planet to its star will sweep out equal areas in equal times. The third law, or the law of harmonies, states that the square of a planet's orbital period is proportional to the cube of the semi-major axis of its orbit.

How is the Kepler problem solved?

The Kepler problem is typically solved using mathematical methods, such as differential equations and numerical integration. These methods involve using the known parameters of the system, such as the mass and distance of the central body and the initial position and velocity of the orbiting object, to calculate the trajectory of the orbit over time.

What are some applications of the Kepler problem?

The Kepler problem has numerous applications in astronomy and astrophysics. It is used to study the motion of planets and satellites in our solar system, as well as exoplanets orbiting other stars. It also has applications in spacecraft trajectory planning and orbital mechanics.

Are there any limitations to the Kepler problem?

While the Kepler problem is a useful tool for understanding the motion of celestial bodies, it does have some limitations. It assumes that all bodies involved are point masses and that there are no external forces acting on the system. In reality, there may be other factors at play, such as the gravitational influence of other bodies or the effects of relativity.

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