Principle of least action — Hamilton's principle derivation: Foundations

2.1.19 · D1 · Physics › Analytical Mechanics › Principle of least action — Hamilton's principle derivation

Yeh page parent derivation ke liye toolbox hai. Yahan kuch bhi assumed nahi hai — hum zero se bhi neeche shuru karte hain.

0. Bilkul shuruwati baat: ek hilti cheez ki position hoti hai

Socho ek moti (bead) ek taar (wire) par fisal rahi hai. Har pal woh kahin na kahin hoti hai. Agar taar seedha track hai toh woh "kahin" ek akela number hai.

Figure — Principle of least action — Hamilton's principle derivation

Upar neele curve ko dekho: horizontal axis time $t$ hai, vertical axis position $q$ hai. Ek dot matlab "moti us pal kahan hai". Saare dots ko jodon aur aapko ek path milta hai.

1. Path ek function $q (t)$ hai

Related deeper ideas Newton's Second Law aur Lagrangian Mechanics mein hain.

2. Speed: derivative $\overset{q}{˙}$

Moti sirf kahin nahi hai; woh chal rahi hai. Kitni tez?

Narangi line tangent hai — uska slope us pal hai $\overset{q}{˙}$ . Yeh slope woh ingredient hai jise baad ki energies use karti hain; yeh "position curve ka slope" se zyada kuch nahi.

3. Do energies: kinetic $T$ aur potential $V$

Hari ghati $V (q)$ hai; lal arrow force ko downhill dikhata hai, yaani chhote $V$ ki taraf. Woh akeli picture hai jis se $F = - d V / d q$ aata hai.

4. Lagrangian $L$ — har instant par ek number

5. Integral $\int$ — poori yatra mein jodte rehna

Shaded area integral hai. Path $q (t)$ badlo aur $L (t)$ badlega, toh area bhi badlega — woh badalta area hi action hai.

6. Action $S$ — ek functional

7. Path vary karna: $ε$ , $η (t)$ , aur $δ S$

Neeli curve sahi path hai; dashed narangi curves wiggled versions $q + ε η$ hain, dono lal endpoints par pin ki gayi hain. Daaya panel $S$ ko knob $ε$ ke function ki tarah dikhata hai: sahi path par ( $ε = 0$ ) curve flat hai — woh flatness hi $δ S = 0$ hai.

Yeh sab topic ko kaise feed karta hai

Related destinations jab yeh tools mil jayein: Hamiltonian Mechanics, Noether's Theorem, Fermat's Principle, Feynman Path Integral.

Equipment checklist

Function

q

(path) ka matlab poori curve hai ya ek akeli value?

Poori curve — time ke saath complete history.

Value

q (t)

ka matlab poori curve hai ya ek akela number?

Ek akela number — ek instant

t

par position.

Derivative

\overset{q}{˙}

ki kya picture hai?

Ek instant par position-vs-time curve ki slope (tadhad).

Partial derivative

\partial f / \partial x

kya measure karta hai?

f

ki change ki rate jab sirf

x

nudge hota hai aur baaki sab inputs frozen rehte hain.

T = \frac{1}{2} m \overset{q}{˙}^{2}

hamesha

\geq 0

kyun hai?

Kyunki

\overset{q}{˙}^{2}

ek square hai, aur squares kabhi negative nahi hote.

Ek line mein,

V

ke terms mein force kya hai?

Potential ki slope ka minus,

F = - d V / d q

(downhill point karta hai).

L

barabar

T + V

hai ya

T - V

T - V

— woh combination jo Newton ko reproduce karta hai.

Integral

\int L d t

ki picture kya hai?

Trip ke dauran

L

-versus-

t

curve ke neeche ka area.

Functional, function se kaise alag hai?

Functional poora function khata hai aur ek number return karta hai; function ek number khata hai.

Wiggle

η

mein kitni smoothness honi chahiye, aur kyun?

C^{1}

(continuous with a continuous derivative), taaki uski slope exist kare aur integration by parts kaam kare.

η

ko endpoints par kya satisfy karna chahiye aur kyun?

η (t_{1}) = η (t_{2}) = 0

, kyunki endpoints fixed data hain aur hil nahi sakte.

δ S = 0

geometrically kya matlab rakhta hai?

Action chhoti path nudges ke neeche flat (stationary) hai — jaise ghati ka sabse nichla hissa.

Kya "stationary" minimum guarantee karta hai?

Nahi — yeh minimum, maximum, ya saddle kuch bhi ho sakta hai.