Question

The two-dimensional Kepler problem (for a unit point mass) starts with the functional $$\mathbf{L}=\int\left[\frac{1}{2}\left(x_{t}^{2}+y_{t}^{2}\right)-V(r)\right] d t, \quad r=\sqrt{x^{2}+y^{2}}$$ (a) Show that $\mathrm{L}$ is invariant under $t$ translation and rotation in the $x y$ plane. (b) Find the generators of $t$ translation and rotation in polar coordinates and conclude that $r$ is the best choice for the independent variable. (c) Rewrite $\mathrm{L}$ in polar coordinates and show that it is independent of $t$ and $\theta$. (d) Write the Euler-Lagrange equations and integrate them to get $\theta$ as an integral over $r$.

          The two-dimensional Kepler problem (for a unit point mass) starts with the functional
$$\mathbf{L}=\int\left[\frac{1}{2}\left(x_{t}^{2}+y_{t}^{2}\right)-V(r)\right] d t, \quad r=\sqrt{x^{2}+y^{2}}$$
(a) Show that $\mathrm{L}$ is invariant under $t$ translation and rotation in the $x y$ plane.
(b) Find the generators of $t$ translation and rotation in polar coordinates and conclude that $r$ is the best choice for the independent variable.
(c) Rewrite $\mathrm{L}$ in polar coordinates and show that it is independent of $t$ and $\theta$.
(d) Write the Euler-Lagrange equations and integrate them to get $\theta$ as an integral over $r$.

Added by Melinda W.

Question

Please give Ace some feedback