Foundations — Conservation of energy and angular momentum in gravitational field

Before we can even state that idea in symbols, we have to earn every letter and squiggle the parent note throws at you. This page is a dictionary you build yourself, in the right order, so no symbol ever appears before you know its picture.

0. The stage: two bodies and a line between them

Everything happens between a big body (the Sun, or Earth) sitting still at the centre, and a small body (a planet, satellite, probe) moving around it. We draw an arrow from the centre to the small body. That single arrow is where almost every symbol lives.

Why split one arrow into three symbols? Because sometimes we care only how far ($r$), sometimes only which way ($\hat{r}$), and sometimes both ($\vec{r}$). The relationship is simply:

$\vec{r}=r\hat{r}(\text{length}\times \text{pure direction}).$

1. Vectors: arrows you can add and turn

A vector is an arrow: it has a length and a direction. We write it with an over-arrow, $\vec{v}$. A plain number (length, mass $m$, time) has no direction — we call it a scalar.

We need velocity as a vector because in an orbit the body's motion is partly outward/inward and partly sideways — two different directions we'll soon separate.

1.1 The cross product $\vec{a}\times \vec{b}$ — "how much swirl"

The parent note writes the angular momentum as $\vec{L}=\vec{r}\times (m\vec{v})$ (with $m$ the small mass we just defined) and the torque as $\vec{\tau }=\vec{r}\times \vec{F}$. That $\times$ is the cross product. Here is what it means, not just how to compute it.

The length is the area of the parallelogram the two arrows span. If the two arrows point the same way ($\varphi =0$), $\sin 0=0$ — the parallelogram is squashed flat, area zero, cross product is the zero vector $\vec{0}$.

1.2 The dot product $\vec{a}\cdot \vec{b}$ — "how much along"

The parent's energy derivation has $\vec{F}\cdot d\vec{r}$.

Why the topic needs it: work (energy transferred) counts only motion along the force. Sideways motion does no work. The dot product is exactly "the part that lines up," so $\vec{F}\cdot d\vec{r}$ is the little bit of energy gravity hands over on a tiny step $d\vec{r}$.

2. Splitting motion: polar coordinates $(r,\theta )$

Because the orbit is flat (a gift from angular momentum), we describe position with just two numbers instead of three.

Any motion is then a mix of two independent moves:

• Radial: the arrow getting longer or shorter — the body moving out/in.
• Tangential: the arrow sweeping around — the body moving sideways.

This dot is a derivative — the slope/rate tool from calculus. The topic needs it because orbits are all about change: speeding up, sweeping angle, falling inward.

2.1 WHY the speed splits: building $v^2={\dot{r}}^2+(r\dot{\theta }{)}^2$

Look at figure s04. In a tiny slice of time $dt$ the body makes two independent little moves that meet at a right angle:

1. Outward move — the arrow $\vec{r}$ grows by $\dot{r}dt$. This slides the body straight out along $\hat{r}$. Distance covered: $\dot{r}dt$.
2. Sideways move — the arrow swings by a tiny angle $\dot{\theta }dt$. A point at radius $r$ swinging through angle $\dot{\theta }dt$ travels along the arc a distance $r\times (\dot{\theta }dt)$ — arc length is radius times angle.

Because the two little steps are perpendicular, we glue them with Pythagoras (right-triangle rule: the hypotenuse squared is the sum of the two legs squared). The total little step $ds$ is the hypotenuse:

$d{s}^2=\underset{\text{outward leg}}{\underbrace{(\dot{r}dt{)}^2}}+\underset{\text{sideways leg}}{\underbrace{(r\dot{\theta }dt{)}^2}}.$

Speed is distance-per-time, $v=ds/dt$, so divide by $d{t}^2$ and take the total:

At a turning point (perihelion — closest, or aphelion — farthest), the arrow stops getting longer or shorter for an instant, so $\dot{r}=0$. There all the speed is sideways: $v=r\dot{\theta }$. The parent uses this exact trick.

3. Force, potential energy, and the minus signs

Why is $U=-GMm/r$ negative? Take "far away, at rest" ($r\to \infty$) as the zero level. To sit closer than that, the body has already "fallen in," releasing energy — so it now owes energy to climb back out. Being in debt = negative. The deeper (smaller $r$), the more negative.

4. The two conserved quantities, named at last

5. How the foundations feed the topic

Here is the whole build in one map. Read it in three streams:

• Left stream (the cross product): arrows → cross product → angular momentum $\vec{L}$, plus the fact that torque is zero → $\vec{L}$ is frozen → the orbit is flat and equal areas are swept (Kepler 2).
• Middle stream (the dot product): arrows → dot product → work and energy → total energy $E$ is frozen.
• Right stream (geometry): arrows → polar coordinates → splitting the speed, which feeds the energy bookkeeping.

The two frozen quantities ($\vec{L}$ and $E$) then meet at the bottom to pin down the orbit's size $a$ and shape $e$.

6. Where these lead next

• Conservation of energy and angular momentum in gravitational field — the parent topic these tools power.
• Conservative Forces and Potential Energy — why $U(r)$ even exists.
• Central Force Problem — the general setting of "force along the line."
• Effective Potential and Orbit Classification — combining $E$ and $L$ into one graph.
• Vis-Viva Equation, Kepler's Laws of Planetary Motion, Escape Velocity and Hohmann Transfers — direct payoffs.

Equipment checklist

Cover the answers. If you can fill each blank cold, you are ready for the parent note.