Cyclic coordinates — corresponding conservation law

2.1.8 · Physics › Analytical Mechanics

Cyclic coordinate KIYA hota hai?

"Cyclic" naam KYUN? Historically yeh un systems se aaya jahan ignored coordinate ek angle hota tha (jaise $\varphi$), jo $0\to 2\pi$ tak cycle karta hai. Lagrangian cycle ke around jaane par unchanged rehta hai.

Conservation law KAISE nikalti hai (scratch se derivation)

Hum Euler–Lagrange equation se shuru karte hain — analytical mechanics ki master equation:

$\frac{d}{dt}\left(\frac{\partial L}{\partial {\dot{q}}_k}\right)-\frac{\partial L}{\partial q_k}=0$

Yeh step kyun? Yeh coordinate $q_k$ ke liye sirf Lagrange ki equation hai — yeh har coordinate ke liye, cyclic ho ya na ho, hold karti hai.

Ab generalized (conjugate) momentum define karo:

Ise momentum KYUN kehte hain? $L=\frac{1}{2}m{\dot{x}}^2-V(x)$ ke liye hume $p_x=m\dot{x}$ milta hai, jo ordinary linear momentum hai. Yeh definition is idea ko kisi bhi coordinate tak generalize karti hai.

Definition ko Euler–Lagrange mein substitute karo:

$\frac{d}{dt}{p}_k=\frac{\partial L}{\partial q_k}$

Yeh step kyun? $\frac{\partial L}{\partial {\dot{q}}_k}$ literally $p_k$ hai, toh pehla term bas ${\dot{p}}_k$ hai.

Ab impose karo ki $q_k$ cyclic hai, yaani $\frac{\partial L}{\partial q_k}=0$:

$\boxed{\frac{d{p}_k}{dt}=0⟹p_k=\text{constant}}$

Worked Examples

Example 1 — Free particle, ignorable $x$

$L=\frac{1}{2}m({\dot{x}}^2+{\dot{y}}^2)$

• Kya $x$ cyclic hai? $\partial L/\partial x=0$ → haan. Kyun? Koi $x$ appear nahi karta.
• Conjugate momentum: $p_x=\partial L/\partial \dot{x}=m\dot{x}$. Yeh step kyun? $L$ ko $\dot{x}$ ke saath differentiate karo.
• Conclusion: $m\dot{x}=\text{const}$ → linear momentum conserved. Translational symmetry ⇒ momentum conservation. ✔

Example 2 — Central force, ignorable angle $\varphi$

Polar coordinates: $L=\frac{1}{2}m({\dot{r}}^2+r^2{\dot{\varphi }}^2)-V(r)$.

• Kya $\varphi$ cyclic hai? $\partial L/\partial \varphi =0$ (sirf $r$ appear karta hai $V$ mein). → haan.
• Conjugate momentum: $p_{\varphi }=\partial L/\partial \dot{\varphi }=m{r}^2\dot{\varphi }$. Kyun? Sirf $r^2{\dot{\varphi }}^2$ term mein $\dot{\varphi }$ hai; ise differentiate karo.
• Conclusion: $p_{\varphi }=m{r}^2\dot{\varphi }=\text{const}$ → yeh angular momentum $L_z$ hai. Rotational symmetry ⇒ angular momentum conserved. ✔ (Kepler ka 2nd law yahan rehta hai.)

Example 3 — Projectile, $x$ cyclic lekin $y$ nahi

$L=\frac{1}{2}m({\dot{x}}^2+{\dot{y}}^2)-mgy$

• $\partial L/\partial x=0$ → $x$ cyclic → $p_x=m\dot{x}=\text{const}$ (horizontal momentum conserved). Kyun? Gravity sirf vertically act karti hai.
• $\partial L/\partial y=-mg\ne 0$ → $y$ cyclic nahi → $p_y=m\dot{y}$ change hoti hai: ${\dot{p}}_y=-mg$. Yeh step kyun? Vertical force $-mg$ exactly $\partial L/\partial y$ hai.

Example 4 — Bead on a rotating-free hoop axis (cyclic azimuth)

$L=\frac{1}{2}m({\dot{\theta }}^2{R}^2+R^2{\sin }^2\theta {\dot{\varphi }}^2)$ (free particle on a sphere).

• $\varphi$ absent → cyclic → $p_{\varphi }=m{R}^2{\sin }^2\theta \dot{\varphi }=\text{const}$. Kyun? Polar axis ke baare mein symmetry.

Flashcards

Connections

• Euler–Lagrange Equations — woh master equation jise humne differentiate kiya.
• Generalized Momentum — woh quantity jo conserved hoti hai.
• Noether's Theorem — deep generalization: har continuous symmetry ↔ conservation law.
• Central Force Motion — cyclic $\varphi$ angular momentum deta hai.
• Hamiltonian Mechanics — cyclic coordinates $H$ ko simplify karte hain (Routhian reduction).
• Conservation of Energy — time-translation ka analogue.

Concept Map