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# Week4: Orbits

## Spherical potential $\Phi(r)$

In the spherical coordinates, the Lagrangian is

$$
\mathcal L=\frac12v^2-\Phi(r)=\frac12\dot r^2+\frac12\left(r\dot\theta\right)^2+\frac12\left(r\sin\theta\dot\phi\right)^2-\Phi(r)
$$

The corresponding generalized momenta are

$$
p\_r=\frac{\partial \mathcal L}{\partial \dot r}=\dot r\\
p\_\theta=\frac{\partial \mathcal L}{\partial \dot \theta}=r^2\dot \theta\\
p\_\phi=\frac{\partial \mathcal L}{\partial \dot \phi}=r^2\sin^2 \theta\dot\phi
$$

The Hamiltonian

$$
H=\vec p\cdot\vec q-\mathcal L=\frac{p^2\_r}{2}+\frac{p\_\theta^2}{2r^2}+\frac{p\_\phi^2}{2r^2\sin^2\theta}+\Phi(r)
$$

We can define

$$
p\_\alpha^2=p\_\theta^2+\frac{p\_\phi^2}{\sin^2\theta}
$$

so that we can simplify the Hamiltonian

$$
H=\frac{p^2\_r}{2}+\frac{p\_\alpha^2}{2r^2}+\Phi(r)
$$

$\alpha$ is a new generalized coordinate which satisfies

$$
p\_\alpha=\frac{\partial \mathcal L}{\partial \dot\alpha}=r^2\sqrt{\dot\theta^2+\sin^2\theta\dot\phi^2}=r\sqrt{v\_\theta^2+v\_\phi^2}=rv\_\perp
$$

Since

$$
\dot p\_\alpha=-\frac{\partial H}{\partial \alpha}=0
$$

the generalized momentum, or the angular momentum perpendicular to the plane determined by $\vec v$ and $\vec v+\text d\vec v$ is conserved, and we can reselect the coordinates and consider the 2D problem with coordinates $(r,\theta)$

* Conservation of angular momentum

  $$
  \dot p\_\theta=-\frac{\partial H}{\partial \theta}=0\Rightarrow p\_\theta=r^2\dot\theta=L
  $$

  * Conservation of energy

    $$
    \frac{\partial H}{\partial t}=0\Rightarrow \frac{p^2\_r}{2}+\frac{p\_\theta^2}{2r^2}+\Phi(r)\equiv \frac{\dot r^2}{2}+\Phi\_{eff}=E
    $$

Physically, the radial distance $r$ is reachable only when

$$
E\ge\Phi\_{eff}(r)
$$

If this is satisfied in an closed interval $\[r\_1, r\_2]$, intuitionally, $r$ could oscillate between $r\_1$ and $r\_2$, whose period (so-called **radial period**) is

$$
T\_r=2\int\_{r\_1}^{r\_2}\frac{\text dt}{\text dr}\text dr=2\int\_{r\_1}^{r\_2}\frac{\text dr}{\sqrt{2\[E-\Phi(r)]-{L^2}/{r^2}}}
$$

In one complete radial period ($r\_1\to r\_2\to r\_1$), usually $\Delta\theta\neq 2\pi$ (Kepler orbits are so special!). In fact

$$
\Delta\theta=2\int\_{r\_1}^{r\_2}\frac{\text d\theta}{\text dr}\text dr=2\int\_{r\_1}^{r\_2}\frac{\text Ldr}{r^2\sqrt{2\[E-\Phi(r)]-{L^2}/{r^2}}}
$$

And the **azimuthal period** is thus determined to be

$$
T\_\theta=\frac{2\pi}{\Delta\theta}T\_r
$$

Of course usually $T*\theta\neq T\_r$, but if $T*\theta/T\_r$ is at least rational, the orbit will turn out to be **closed**. Below we discuss several special closed orbits

1. Kepler

   $$
   T\_\theta=T\_r=2\pi\sqrt{\frac{a^3}{GM}}, \  T\_\theta:T\_r=1:1
   $$
2. Homogeneous sphere (see in last week's notes)

   $$
   \Phi(r)=C+\frac12\Omega^2r^2=C+\frac12\Omega^2(x^2+y^2)
   $$

   Newton's second law gives

   $$
   \left{\begin{array}{c}
   {\ddot x=-\Phi\_x(x,y)=-\Omega x\\
   \ddot y=-\Phi\_y(x,y)=-\Omega y}
   \end{array}\right.
   \Rightarrow \text{Harmonic oscillators!}
   $$

   The corresponding orbits are thus eclipses centered at the coordinate origin, so every $2\pi$ angle the object goes around, the variation of $r$ undergoes two orbits

   $$
   T\_\theta=\frac{2\pi}{\Omega},\ T\_r=\frac{T\_\theta}{2},\ T\_\theta:T\_r=2:1
   $$

In general,

$$
1<\frac{T\_\theta}{T\_r}<2
$$

A more intuitional picture is that the orbit continuously **precesses** at a rate of

$$
\Omega\_P=\frac{\Delta\theta-2\pi}{T\_r}
$$

* $\Omega\_P<0$ - Retrograde
* $\Omega\_P>0$ - Prograde