Kepler's problem in lagrangian formalism

[quasar987]

In the statement of the problem, it is said that with an appropriate choice of units, the lagrangian for Kepler's problem can be written

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

A priori, what q is in terms of cartesian coordinates seems irrelevant because the problem only asks to show that the Runger-Lenz vector, defined by

$$A_k=\mathbf{\dot{q}}^2 q_k-\mathbf{q} \cdot \mathbf{\dot{q}}\dot{q_k}-q_k/q \ \ \ \ \ k=1,2,3$$

or, in vectorial notation,

$$\mathbf{A}=\mathbf{\dot{q}}\times (\mathbf{q}\times \mathbf{\dot{q}})-\mathbf{q}/q$$

is a constant of the motion associated (in the sense of Noether's thm) to a certain coordinate transformation.

But then the question asks, "Discuss the properties of this vector." and I'm kind of at a loss about what to say. I can't say much about its direction and its norm is ugly and uninsightful. So I figured if I knew what those q where in terms of cartesian coordinates, maybe I could make some sense out of the Runge-Lenz vector. So I try to get the lagrangian into the above form, right?

Ok, in cartesian, it is

$$L=\frac{m_1}{2}(\dot{x}_1^2+\dot{y}_1^2+\dot{z}_1^2)+\frac{m_2}{2}(\dot{x}_2^2+\dot{y}_2^2+\dot{z}_2^2)+\frac{Gm_1m_2}{\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}}$$

We can choose the units of mass, length and time such that the lagrangian becomes

$$L=\frac{1}{2}(\dot{x}_1^2+\dot{x}_2^2+\dot{y}_1^2+\dot{y}_2^2+\dot{z}_1^2+\dot{z}_2^2)+\frac{1}{\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}}$$

Compare this with where we want to go:

$$L_q=\frac{1}{2}(\dot{q}_1^2+\dot{q}_2^2+\dot{q}_3^2)+\frac{1}{\sqrt{q_1^2+q_2^2+q_3^3}}$$

Are there really maps $q_i(x_1,y_1,z_1,x_2,y_2,z_2)$ that transform L into L_q ?!?

 

[Logarythmic]

The q's are the generalized coordinates so if your generalized coordinates are the cartesian coordinates, then you should have the answer?

 

[quasar987]

What?

The book says the lagrangian can be written like this:

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

Or, expanding the vectors into their components (i.e. the generalized coordinates),

$$L_q=\frac{1}{2}(\dot{q}_1^2+\dot{q}_2^2+\dot{q}_3^ 2)+\frac{1}{\sqrt{q_1^2+q_2^2+q_3^3}}$$

On the other hand, the cartesian lagrangian is

$$L=\frac{1}{2}(\dot{x}_1^2+\dot{x}_2^2+\dot{y}_1^2+ \dot{y}_2^2+\dot{z}_1^2+\dot{z}_2^2)+\frac{1}{\sqr t{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}}$$

Clearly the obvious $q_1=(x_2-x_1)$, $q_2=(y_2-y_1)$, $q_3=(z_2-z_1)$ inspired by comparison of the "potential term" does not work for if we expand $$(\dot{q}_1^2+\dot{q}_2^2+\dot{q}_3^ 2)$$ in terms of x, y, z, we do not get $$(\dot{x}_1^2+\dot{x}_2^2+\dot{y}_1^2+ \dot{y}_2^2+\dot{z}_1^2+\dot{z}_2^2)$$.

So I ask in disbelief, is there really a coordinate transformation that allows one to write L as

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

?

 

[quasar987]

P.S. Ideas about what to say about $\mathbf{A}$ are of course welcome as well ! Besides the direction and the norm, what are interesting things to say about a vector?

 

[quasar987]

I found it... q is actually just the ordinary position vector like Logarythmic pointed out (I think). So we are actually solving the Newton problem [particle in a 1/r potential] for a reduced mass µ (but with the units chosen to that µ=1). We can then set q=r1-r2 to get the solution to the actual Kepler problem.