Suppose we have an object orbiting the Earth with a circular orbit of radius $R$. We then change the velocity by an angle $\alpha$, without changing the speed. The textbook then asks what orbit will the object describe.
My guess is an ellipse, but of course I want to show it. The initial angular momentum is $L_0 = mRv$ and the final one is $L = mRv\cos(\alpha) = L_0\cos(\alpha)$.
I want to study the initial and final energies in order to deduce the path of the orbit.
The initial energy of the particle is : $$E_0 = T_0 + U_{eff}^0 = \frac{1}{2}mv_0^2 + \frac{L_0^2}{2mR^2} + \frac{K}{R^2}$$ And the final one is : $$E = T + U_{eff} = T_0 + \frac{L_0^2\cos^2(\alpha)}{2mr^2} + \frac{K}{r^2}$$
Now the textbook says that the mechanical energy of the particle does not change, since $E = T_0 + \frac{K}{r} = T + \frac{K}{r}$, but isn't the mechanical energy of the particle $E = T + {U_{eff}}$?
In that case, the mechanical energy of the particle does change, so the textbook would be wrong. If not, can someone explain it to me?