Hi folks, my problem is the following one:

Kepler stated that orbits of planets are elliptic. Everytnig's well since Newton obtained the same results, with his formula for gravity

[tex]F = G(mM)/r^2[/tex]

Now, i tried to write the Lagrangian of the system (L = K-U, K is cinetic energy, U is potential energy) in polar coordinates which (hopefully without errors) should be defined as follows:

[tex]L = \frac{1}{2}m((\dot{r})^2 + (r\dot{\theta})^2) + G(mM)/r[/tex]

where [tex]\dot{r}^[/tex] is radial velocity, [tex]\dot{\theta}^[/tex] is angular velocity, r is the distance between the two bodies, one of them is for simplicity considered being still in the origin.

now deriving the differential equations of motion using [tex]d(\partial{L} / \partial{\dot{x}} )/dt = \partial{L}/\partial{x}[/tex]

we have the following equations :

[tex]\ddot{\theta} = -2\dot{\theta}\dot{r}/r[/tex]

[tex]\ddot{r} = r(\dot{\theta})^2 - MG/r^2[/tex]

[tex]\ddot{r}[/tex] is the radial acceleration, [tex]\ddot{\theta}[/tex] is angular acceleration, M is the mass of the body fixed at the origin.

Now, the equations are by no way linear and easily solvable... and can describe a very rich variety of orbital behaviours, not only the steady Kepler's ellipse (things got even worse when i derived the equations considering the mass M not being fixed but free to move).

Is Kepler's first law an approximation ?? or it's exact ?

Kepler stated that orbits of planets are elliptic. Everytnig's well since Newton obtained the same results, with his formula for gravity

[tex]F = G(mM)/r^2[/tex]

Now, i tried to write the Lagrangian of the system (L = K-U, K is cinetic energy, U is potential energy) in polar coordinates which (hopefully without errors) should be defined as follows:

[tex]L = \frac{1}{2}m((\dot{r})^2 + (r\dot{\theta})^2) + G(mM)/r[/tex]

where [tex]\dot{r}^[/tex] is radial velocity, [tex]\dot{\theta}^[/tex] is angular velocity, r is the distance between the two bodies, one of them is for simplicity considered being still in the origin.

now deriving the differential equations of motion using [tex]d(\partial{L} / \partial{\dot{x}} )/dt = \partial{L}/\partial{x}[/tex]

we have the following equations :

[tex]\ddot{\theta} = -2\dot{\theta}\dot{r}/r[/tex]

[tex]\ddot{r} = r(\dot{\theta})^2 - MG/r^2[/tex]

[tex]\ddot{r}[/tex] is the radial acceleration, [tex]\ddot{\theta}[/tex] is angular acceleration, M is the mass of the body fixed at the origin.

Now, the equations are by no way linear and easily solvable... and can describe a very rich variety of orbital behaviours, not only the steady Kepler's ellipse (things got even worse when i derived the equations considering the mass M not being fixed but free to move).

Is Kepler's first law an approximation ?? or it's exact ?

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