Principle of least action — Hamilton's principle derivation

2.1.19 · Physics › Analytical Mechanics

Action kya hai?

Hamilton's Principle (statement)

$\delta S=0$ se Euler–Lagrange equation derive karna

Yeh chapter ka dil hai. Hum sab kuch scratch se banate hain.

Setup (KYA). Asli path $q(t)$ lo aur ise perturb karo: $q_{\epsilon }(t)=q(t)+\epsilon \eta (t)$ jahan $\eta (t)$ ek arbitrary smooth "wiggle" hai aur $\epsilon$ ek chota number hai.

Step 1 — Varied path ka action likho. $S(\epsilon )={\int }_{t_1}^{t_2}L(q+\epsilon \eta ,\dot{q}+\epsilon \dot{\eta },t)dt$ Yeh step kyun? Asli path $\epsilon =0$ se correspond karti hai. Stationarity ka matlab hai $S(\epsilon )$ ka wahan flat tangent ho: ${\frac{dS}{d\epsilon }|}_{\epsilon =0}=0$.

Step 2 — Integral ke andar differentiate karo (chain rule). $\frac{dS}{d\epsilon }={\int }_{t_1}^{t_2}\left(\frac{\partial L}{\partial q}\eta +\frac{\partial L}{\partial \dot{q}}\dot{\eta }\right)dt$ Yeh step kyun? $L$ sirf $q+\epsilon \eta$ aur $\dot{q}+\epsilon \dot{\eta }$ ke through $\epsilon$ par depend karta hai. Chain rule factors $\partial q_{\epsilon }/\partial \epsilon =\eta$ aur $\partial {\dot{q}}_{\epsilon }/\partial \epsilon =\dot{\eta }$ deta hai.

Step 3 — Doosre term ko parts se integrate karo. Yeh crucial move hai: yeh $\dot{\eta }$ ko $\eta$ mein convert karta hai taaki hum $\eta$ ko factor out kar sakein. ${\int }_{t_1}^{t_2}\frac{\partial L}{\partial \dot{q}}\dot{\eta }dt=\underset{=0}{\underbrace{{\left[\frac{\partial L}{\partial \dot{q}}\eta \right]}_{t_1}^{t_2}}}-{\int }_{t_1}^{t_2}\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right)\eta dt$ Yeh step kyun? Boundary term vanish hoti hai kyunki $\eta (t_1)=\eta (t_2)=0$. Isliye endpoint-fixing matter karti thi.

Step 4 — Terms collect karo. $\frac{dS}{d\epsilon }{|}_0={\int }_{t_1}^{t_2}\left[\frac{\partial L}{\partial q}-\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right)\right]\eta (t)dt=0$

Step 5 — Calculus of Variations ka Fundamental Lemma apply karo. Kyunki $\eta (t)$ arbitrary hai, bracket har $t$ par vanish honi chahiye (agar woh kahin nonzero hoti, toh wahan bumped ek $\eta$ choose karo aur integral zero nahi hoga).

Check karo ki Newton deta hai (Feynman-style sanity test)

1D particle lo: $T=\frac{1}{2}m{\dot{x}}^2$, $V=V(x)$, toh $L=\frac{1}{2}m{\dot{x}}^2-V(x)$.

• $\frac{\partial L}{\partial \dot{x}}=m\dot{x}$ ⟹ $\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right)=m\ddot{x}$
• $\frac{\partial L}{\partial x}=-\frac{dV}{dx}=F$

Euler–Lagrange: $m\ddot{x}-F=0\Rightarrow \boxed{m\ddot{x}=F}$. ✅ Newton's second law nikal aata hai. Yeh kyun matter karta hai: yeh prove karta hai ki $L=T-V$ sahi choice thi aur Hamilton's principle Newtonian mechanics ke equivalent hai.

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Connections

• Lagrangian Mechanics — EL equation bahut saare coordinates $q_i$ tak generalize hoti hai.
• Euler–Lagrange Equation — woh differential equation jo yeh principle produce karta hai.
• Calculus of Variations — stationary functionals ka math; same lemma.
• Noether's Theorem — $L$ ki symmetries ⟹ conservation laws.
• Hamiltonian Mechanics — $L$ ka Legendre transform $H=T+V$ deta hai.
• Fermat's Principle — optics analogue (stationary optical path/time).
• Feynman Path Integral — quantum sum over paths weighted by $e^{iS/\hslash }$.
• Newton's Second Law — equivalent formulation jo check ke roop mein recover hoti hai.

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