Foundations — Derivation of Euler-Lagrange equations from D'Alembert's principle

2.1.5 · D1 · Physics › Analytical Mechanics › Derivation of Euler-Lagrange equations from D'Alembert's pri

Parent derivation ki ek bhi line padhne se pehle, tumhe uska alphabet padhna aana chahiye. Yeh page har ek symbol zero se banata hai, is order mein jahan har nayi cheez sirf pehle se define symbols use karti hai. Yahan kuch bhi assumed nahi hai.

0. Stage: particles aur positions

Picture. Ek origin lagao. Har particle ek dot hai; origin se us dot tak ka arrow ${\vec{r}}_k$ hai. Jaise particle move karta hai, arrow ki tip uski path trace karti hai.

Yeh topic ko kyun chahiye. Newton ka law baat karta hai cheezein kahan hain aur woh kaise change hoti hain. ${\vec{r}}_k$ raw "kahan" hai. Baaki sab is baat ko dekhke banta hai ki ${\vec{r}}_k$ kaise change hota hai.

Picture. Agar ${\vec{r}}_k$ position arrow hai, toh ${\dot{\vec{r}}}_k$ ek chhota arrow hai jo path ke tangent mein travel ki direction dikhata hai; do dots ($\ddot{}$) woh arrow hai jo dikhata hai ki woh travel-arrow khud kaise turn ya stretch kar raha hai.

1. Changes padhne ka frame: $\partial$, $\frac{d}{dt}$

Yeh topic ko kyun chahiye. Euler–Lagrange equation literally hai $\frac{d}{dt}\frac{\partial L}{\partial \dot{q}}-\frac{\partial L}{\partial q}=0$: ek $\frac{d}{dt}$ ek $\partial /\partial \dot{q}$ ke around wrapped, minus ek aur $\partial /\partial q$. Agar tum in do operators ke beech ka farq feel nahi kar sakte, equation unreadable hai.

2. Vectors combine karna: dot product $\cdot$

Picture. $\vec{B}$ par seedha neeche se light jalao; $\vec{A}$ ki $\vec{B}$ ke saath bani shadow, $\vec{B}$ ki length se multiply karke, dot product hai. Key fact jo hum baar baar use karenge:

• Agar do vectors perpendicular hain ($\varphi =90^{\circ }$), toh $\cos 90^{\circ }=0$, isliye $\vec{A}\cdot \vec{B}=0$.

Yeh topic ko kyun chahiye. D'Alembert's principle ka poora magic yeh hai ki constraint forces allowed motion ke perpendicular point karti hain. Perpendicular ka matlab dot product zero hai, isliye woh forces equation se gaayab ho jaati hain. Dot product eraser hai.

3. Force, momentum, aur Newton

Yeh topic ko kyun chahiye. Poori derivation ek machine hai ${\vec{F}}_k^{(c)}$ gaayab karne ke liye taaki sirf ${\vec{F}}_k^{(a)}$ (aur inertia) bache. Dekho Generalized forces and potentials.

4. Constraints aur generalized coordinates

Picture. Ek bent wire par ek bead 3D mein rehta hai, lekin tumhe sirf ek number chahiye — wire par kitni door slide hua hai — use pin down karne ke liye. Woh number $q$ hai. Wire constraint hai; $q$ woh freedom hai jo bachti hai.

5. Virtual displacement $\delta \vec{r}$ — show ka star

Picture. Movie pause karo. Poocho: "Is frozen frame mein, particle ko konsa tiny nudge de sakta hun bina koi constraint toode?" Woh allowed nudges virtual displacements hain. Wire par ek bead ke liye, sirf allowed $\delta \vec{r}$ wire ke saath point karta hai — kabhi usase bahar nahi.

Yeh topic ko kyun chahiye. Sections 2 aur 5 combine karo: constraint force $\perp$ allowed motion $\Rightarrow {\vec{F}}_k^{(c)}\cdot \delta {\vec{r}}_k=0$. Particles par sum karo aur constraint forces gaayab ho jaati hain. Yeh ek fact hai D'Alembert's principle.

6. Generalized force $Q_i$ aur potential $V$

7. Alphabet ek saath laana

Yahan har symbol final Euler–Lagrange equation ko kaise feed karta hai. Upar se neeche padho.

Har arrow ek dependency hai: tum koi node tab tak nahi samajh sakte jab tak uske parents nahi samajhte. Yahi woh order hai jisme parent derivation chalti hai. Hamilton's principle (least action) aur Noether's theorem ki taraf badho jab yeh solid ho jaye.

Equipment checklist

Daayaan side dhako aur dekho ki reveal karne se pehle har ek ka jawab de sako.