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

3.2.1 · D1 · Physics › Orbital Mechanics & Astrodynamics › Two-body problem — equations of motion, reduction to one-bod

Yeh page assume karta hai ke tumne kuch bhi nahi dekha. Hum har letter, dot, aur arrow banate hain jo parent note use karta hai, ek aisi sequence mein jahan har brick pichli ke upar tikti hai. Agar tum fractions add kar sakte ho aur ek arrow kaagaz pe imagine kar sakte ho, toh line one se follow kar sakte ho.

1 — Ek vector, aur arrows numbers se better kyun hain

Sirf numbers kyun nahi use karte? Kyunki ek planet space mein kahin hai, number line pe nahi. "Planet yahan hai" bolne ke liye tumhe "kitni door" aur "kis taraf" dono chahiye. Ek arrow dono pack karta hai.

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

Figure dekho. Arrow origin se start hota hai (woh corner jahan axes milte hain, humara agreed "zero point") aur uski tip object pe land karti hai. Do coloured drop-lines vector ke components hain — kitna right ( $x$ ) aur kitna up ( $y$ ) travel karna padega tip tak pahunchne ke liye. Toh ek arrow $r$ secretly do numbers $(x, y)$ rakhta hai (real space mein teen hote hain, lekin do mein sab kuch samajh aa jaata hai).

Parent ko $\hat{r}$ kyun chahiye? Gravity ki direction hoti hai "seedha doosri body ki taraf", lekin uski strength distance ke saath ghatti hai. Ek arrow ko (direction $\hat{r}$ ) $\times$ (size) mein split karna in dono facts ko alag-alag handle karne deta hai.

2 — Arrows ko add, scale, aur subtract karna

Figure mein, $r_{1}$ body 1 ko locate karta hai aur $r_{2}$ body 2 ko, dono origin se measured. Green arrow $r = r_{2} - r_{1}$ unke beech ka bridge hai. Yeh green arrow poore topic ka sabse important object hai — parent ise relative position vector kehta hai.

Addition, scaling, aur subtraction ko hum baad mein saath mein use karenge $R$ aur $r$ se individual positions rebuild karne ke liye.

3 — Mass, aur balance point

Formula ko see-saw ki tarah padho. Figure dekho: mota body balance point ko apni taraf kheenchta hai. " $m_{1} r_{1}$ " term body 1 ki position hai $m_{1}$ baar count ki gayi; heavier ⇒ zyada count ⇒ balance point closer kheenchta hai. $m_{1} + m_{2}$ (total count) se divide karna weighted sum ko ek actual location mein convert karta hai.

Lock karne ke liye do special cases:

Equal masses ( $m_{1} = m_{2}$ ): weights match karte hain, toh $R$ bilkul beech mein baithta hai — midpoint.
Ek giant, ek speck ( $m_{1} ≫ m_{2}$ ): sum almost saara $m_{1} r_{1}$ hai, toh $R \approx r_{1}$ — balance point practically bade body ke andar hai. (Isliye Sun "fixed lagta hai".)

4 — Velocity aur acceleration: upar dots

Dot kyun, fraction kyun nahi? Yeh calculus operation $\frac{d}{d t}$ ka pure shorthand hai — derivative, woh tool jo instantaneous rate of change measure karta hai. Hum yahan derivative ke liye jaate hain kyunki Newton's law accelerations mein likha hai: yeh batata hai force motion ko kaise change karta hai, directly motion nahi.

Figure mein, ek curved path pe ek point dekho. Blue arrow uski velocity hai — yeh hamesha path ke saath point karta hai (woh direction jis taraf tum currently ja rahe ho). Red arrow acceleration hai — yahan yeh inward point karta hai, centre ki taraf. Woh inward red arrow exactly wahi hai jo gravity supply karti hai: yeh velocity ko orbit mein continuously modhti rehti hai seedhi line ki jagah.

5 — Inverse-square gravity law

Har piece ko jo humne abhi banaya hai usse decode karo:

$m_{1} m_{2}$ upar: zyada stuff ⇒ stronger pull (koi bhi mass double karo, force double ho jaati hai).
$r^{2}$ neeche: inverse-square fall-off — twice as far apart matlab force one-quarter. Force ek sphere pe spread hoti hai, aur sphere ka area $r^{2}$ ki tarah badhta hai, toh pull $1/ r^{2}$ ki tarah dilute hoti hai.
$G$ : woh chota number jo "kilograms aur metres" ko "newtons of pull" mein convert karta hai.

Size akela kaafi nahi — humein direction pakadna hai taki sign kabhi bhi ambiguous na ho. $r = r_{2} - r_{1}$ (§2 se) body 1 se body 2 ki taraf point karta hai, aur $\hat{r} = r / r$ uska unit arrow hai (valid kyunki $r > 0$ , §1):

Dono forms kyun likhne ki zaroorat hai? $\hat{r}$ form padhta hai "size times direction"; $r / r^{3}$ form woh hai jo tum actually differentiate karte ho aur $F = m \ddot{r}$ mein plug in karte ho. Yeh same arrow hain.

6 — Center of mass seedhi line mein kyun drift karta hai

Humne §3 mein promise kiya tha ki $R$ constant velocity pe chalti hai. Ab hamare paas har tool hai prove karne ke liye — kuch bhi hand-waved nahi.

Newton's second law (§4) har body pe apply karo, §5 ke gravity vectors use karke: $m_{1} \ddot{r}_{1} = F_{1} = + \frac{G m _{1} m _{2}}{r ^{3}} r, m_{2} \ddot{r}_{2} = F_{2} = - \frac{G m _{1} m _{2}}{r ^{3}} r .$ Unhe add karo. Right-hand sides exact opposites hain (third law), toh cancel ho jaate hain: $m_{1} \ddot{r}_{1} + m_{2} \ddot{r}_{2} = 0 .$ Ab COM definition ko do baar differentiate karo (§2 ke scalar multiplication aur addition use karke — operations derivative ke through pass hote hain): $\ddot{R} = \frac{m _{1} r ¨ _{1} + m _{2} r ¨ _{2}}{m _{1} + m _{2}} = \frac{0}{m _{1} + m _{2}} = 0 .$

7 — Do Greek $μ$ 's (ek hi letter do baar kyun)

Parent letter $μ$ ("mew") do alag-alag cheezein ke liye use karta hai. Tumhe dono derivation milne se pehle cleanly define chahiye.

$μ_{red}$ ke liye ek sanity picture: agar ek body enormous hai ( $m_{1} \to \infty$ ), toh $μ_{red} = \frac{m _{1} m _{2}}{m _{1} + m _{2}} \to m_{2}$ — reduced mass sirf light body ki mass ban jaati hai, kyunki heavy body barely hilti hai. Yeh intuition se match karta hai aur woh case hai jo memory mein commit karne laayak hai.

Prerequisite map

Top-to-bottom padho: arrows aur mass raw bricks hain; yeh relative vector, balance point, acceleration, aur Newton's law banate hain; woh combine karke full gravity vector, straight-line COM proof, aur finally two-body reduction banta hai.

Equipment checklist

Bold

r

ka matlab plain

r

se kya alag hai?

Bold

r

poora arrow hai (direction + length); plain

r

sirf uski length hai, ek single positive number.

Kya ek component negative ho sakta hai, aur uska matlab kya hai?

Haan — ek negative

x

y

us direction ko flip karta hai (left/down); length

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

\geq 0

rehti hai.

\hat{r}

kya hai aur tum ise kaise banate ho?

Ek unit vector (length exactly 1) jo pure direction deta hai;

\hat{r} = r / r

\hat{r}

kab undefined hota hai, aur kyun?

Jab

r = 0

(bodies coincide karti hain):

r = 0

se divide karna illegal hai aur ek zero arrow ki koi direction nahi hoti.

Do vectors kaise add karte ho, components mein aur picture ke roop mein?

Components add karo

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

; picture tip-to-tail hai.

Scalar multiplication

s a

kya karta hai?

Arrow ki length ko

s

se scale karta hai aur agar

s < 0

hai toh flip karta hai; components

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

ban jaate hain.

r_{2} - r_{1}

kis taraf point karta hai?

Body 1 se body 2 ki taraf (jis cheez ko subtract kiya gaya wahan se start hota hai).

Center-of-mass formula state karo aur yeh kya represent karta hai.

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

; system ka mass-weighted balance point.

Prove karo ki COM seedhi line mein move karta hai.

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

add karna (third law)

\ddot{R} = 0

deta hai, toh

\dot{R}

constant hai.

r

ke upar ek dot aur do dots ka matlab kya hai?

Ek dot = velocity (position ki rate of change); do dots = acceleration (velocity ki rate of change).

Body 1 pe Newton's gravity full vector form mein likho.

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

, body 1 ko body 2 ki taraf kheenchti hai.

μ

's mein fark karo.

μ = G (m_{1} + m_{2})

(gravitational parameter, sum use karta hai, units m³/s²) vs

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

(reduced mass, product/sum, kilograms).

Jab ek mass bahut badi ho toh

μ_{red}

kya approach karta hai?

Chhoti mass

m_{2}

Connections

Yeh foundations Hinglish mein
Center of mass frame — woh coordinate origin jo yeh foundations set up karte hain.
Conservation of angular momentum (central force) — yahan banaye gaye vector arrows chahiye.
Reduced mass in molecular vibrations — wahi $μ_{red}$ ek naye setting mein.
Kepler's laws, Vis-viva equation, Three-body problem — jahan yeh symbols aage jaate hain.