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quasar987

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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

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

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

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

or, in vectorial notation,

[tex]\mathbf{A}=\mathbf{\dot{q}}\times (\mathbf{q}\times \mathbf{\dot{q}})-\mathbf{q}/q[/tex]

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

[tex]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}}[/tex]

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

[tex]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}}[/tex]

Compare this with where we want to go:

[tex]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}}[/tex]

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

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

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

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

or, in vectorial notation,

[tex]\mathbf{A}=\mathbf{\dot{q}}\times (\mathbf{q}\times \mathbf{\dot{q}})-\mathbf{q}/q[/tex]

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

[tex]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}}[/tex]

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

[tex]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}}[/tex]

Compare this with where we want to go:

[tex]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}}[/tex]

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

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