Two-body problem — equations of motion, reduction to one-body: Foundations

This page assumes you have seen nothing. We build every letter, dot, and arrow the parent note leans on, in an order where each brick rests on the one before it. If you can add fractions and imagine an arrow on paper, you can follow from line one.

1 — A vector, and why arrows beat lists

Why not just use numbers? Because a planet is somewhere in space, not on a number line. To say "the planet is here" you need "how far" and "which way". One arrow packs both.

Figure — Two-body problem — equations of motion, reduction to one-body

Look at the figure. The arrow starts at the origin (the corner where the axes meet, our agreed "zero point") and its tip lands on the object. The two coloured drop-lines are the vector's components — how far right ( $x$ ) and how far up ( $y$ ) you must travel to reach the tip. So one arrow $r$ secretly holds two numbers $(x, y)$ (three in real space, but two is enough to see everything).

Why does the parent need $\hat{r}$ ? Gravity's direction is "straight toward the other body", but its strength shrinks with distance. Splitting an arrow into (direction $\hat{r}$ ) $\times$ (size) lets us handle those two facts separately.

2 — Adding, scaling, and subtracting arrows

In the figure, $r_{1}$ locates body 1 and $r_{2}$ locates body 2, both measured from the origin. The green arrow $r = r_{2} - r_{1}$ is the bridge between them. This green arrow is the single most important object in the whole topic — the parent calls it the relative position vector.

We use addition, scaling, and subtraction together to rebuild individual positions from $R$ and $r$ later.

3 — Mass, and the balance point

Read the formula as a see-saw. Look at the figure: the fat body pulls the balance point toward itself. The " $m_{1} r_{1}$ " term is body 1's position counted $m_{1}$ times; heavier ⇒ counted more ⇒ balance point drags closer. Dividing by $m_{1} + m_{2}$ (the total count) turns the weighted sum back into an actual location.

Two special cases to lock in:

Equal masses ( $m_{1} = m_{2}$ ): the weights match, so $R$ sits exactly halfway — the midpoint.
One giant, one speck ( $m_{1} ≫ m_{2}$ ): the sum is almost all $m_{1} r_{1}$ , so $R \approx r_{1}$ — the balance point is practically inside the big body. (This is why the Sun "looks fixed".)

4 — Velocity and acceleration: dots on top

Why a dot and not a fraction? It is pure shorthand for the calculus operation $\frac{d}{d t}$ — the derivative, the tool that measures instantaneous rate of change. We reach for the derivative here because Newton's law is written in accelerations: it tells us how a force changes motion, not motion directly.

In the figure, watch one point on a curved path. The blue arrow is its velocity — it always points along the path (the direction you're currently heading). The red arrow is acceleration — here it points inward, toward the centre. That inward red arrow is exactly what gravity supplies: it constantly bends the velocity into an orbit instead of a straight line.

5 — The inverse-square gravity law

Decode each piece with what we just built:

$m_{1} m_{2}$ on top: more stuff ⇒ stronger pull (double either mass, double the force).
$r^{2}$ on the bottom: the inverse-square fall-off — twice as far apart means one-quarter the force. Force spreads out over a sphere, and a sphere's area grows like $r^{2}$ , so the pull dilutes like $1/ r^{2}$ .
$G$ : the tiny number that converts "kilograms and metres" into "newtons of pull".

Size alone isn't enough — we need to nail the direction so the sign is never ambiguous. With $r = r_{2} - r_{1}$ (from §2) pointing from body 1 to body 2, and $\hat{r} = r / r$ its unit arrow (valid because $r > 0$ , §1):

Why bother writing both forms? The $\hat{r}$ form reads as "size times direction"; the $r / r^{3}$ form is what you actually differentiate and plug into $F = m \ddot{r}$ . They are the same arrow.

6 — Why the center of mass drifts in a straight line

We promised in §3 that $R$ moves at constant velocity. Now we have every tool to prove it — nothing hand-waved.

Apply Newton's second law (§4) to each body, using the gravity vectors from §5: $m_{1} \ddot{r}_{1} = F_{1} = + \frac{G m _{1} m _{2}}{r ^{3}} r, m_{2} \ddot{r}_{2} = F_{2} = - \frac{G m _{1} m _{2}}{r ^{3}} r .$ Add them. The right-hand sides are exact opposites (third law), so they cancel: $m_{1} \ddot{r}_{1} + m_{2} \ddot{r}_{2} = 0 .$ Now differentiate the COM definition twice (using scalar multiplication and addition from §2 — the operations pass through the derivative): $\ddot{R} = \frac{m _{1} r ¨ _{1} + m _{2} r ¨ _{2}}{m _{1} + m _{2}} = \frac{0}{m _{1} + m _{2}} = 0 .$

7 — The two Greek $μ$ 's (why the same letter twice)

The parent uses the letter $μ$ ("mew") for two different things. You need both defined cleanly before you meet the derivation.

A sanity picture for $μ_{red}$ : if one body is enormous ( $m_{1} \to \infty$ ), then $μ_{red} = \frac{m _{1} m _{2}}{m _{1} + m _{2}} \to m_{2}$ — the reduced mass becomes just the light body's mass, because the heavy one barely moves. That matches intuition and is the case worth committing to memory.

The prerequisite map

Read top-to-bottom: arrows and mass are the raw bricks; they build the relative vector, the balance point, acceleration, and Newton's law; those combine into the full gravity vector, the straight-line COM proof, and finally the two-body reduction.

Equipment checklist

What does a bold

r

mean versus a plain

r

Bold

r

is the whole arrow (direction + length); plain

r

is only its length, a single positive number.

Can a component be negative, and what does that mean?

Yes — a negative

x

y

flips that direction (left/down); the length

r = x^{2} + y^{2}

stays

\geq 0

What is

\hat{r}

and how do you build it?

A unit vector (length exactly 1) giving pure direction;

\hat{r} = r / r

When is

\hat{r}

undefined, and why?

When

r = 0

(bodies coincide): dividing by

r = 0

is illegal and a zero arrow has no direction.

How do you add two vectors, in components and as a picture?

Add components

(a_{x} + b_{x}, a_{y} + b_{y})

; picture is tip-to-tail.

What does scalar multiplication

s a

do?

Scales the arrow's length by

s

and flips it if

s < 0

; components become

(s a_{x}, s a_{y})

Which way does

r_{2} - r_{1}

point?

From body 1 to body 2 (starts at the thing being subtracted).

State the center-of-mass formula and what it represents.

R = \frac{m _{1} r _{1} + m _{2} r _{2}}{m _{1} + m _{2}}

; the mass-weighted balance point of the system.

Prove the COM moves in a straight line.

Adding

m_{1} \ddot{r}_{1} + m_{2} \ddot{r}_{2} = 0

(third law) gives

\ddot{R} = 0

, so

\dot{R}

is constant.

What do one dot and two dots over

r

mean?

One dot = velocity (rate of change of position); two dots = acceleration (rate of change of velocity).

Write Newton's gravity in full vector form on body 1.

F_{1} = + \frac{G m _{1} m _{2}}{r ^{2}} \hat{r} = + \frac{G m _{1} m _{2}}{r ^{3}} r

, pulling body 1 toward body 2.

Distinguish the two

μ

's.

μ = G (m_{1} + m_{2})

(gravitational parameter, uses sum, units m³/s²) vs

μ_{red} = m_{1} m_{2} / (m_{1} + m_{2})

(reduced mass, product/sum, kilograms).

What does

μ_{red}

approach when one mass is huge?

The smaller mass

m_{2}

Connections

Yeh foundations Hinglish mein
Center of mass frame — the coordinate origin these foundations set up.
Conservation of angular momentum (central force) — needs the vector arrows built here.
Reduced mass in molecular vibrations — the same $μ_{red}$ in a new setting.
Kepler's laws, Vis-viva equation, Three-body problem — where these symbols go next.