If (x ̇ ẏ + 𝛼xy) is a constant of motion of a two-dimensional isotropic harmonic
oscillator with Lagrangian
L = m(ẋ_2 + ẏ_2)
2 - k(x_2 + y_2)
2
then 𝛼 is
Options:
(A) $+\frac{k}{m}$
(B) $-\frac{k}{m}$
(C) $-\frac{2k}{m}$
(D) $0$
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2022 Q.42)**
**1. Classical Mechanics (Lagrangian & Constants of Motion)**:
- Given Lagrangian:
$L = \frac{m}{2}(\dot{x}^2 + \dot{y}^2) - \frac{k}{2}(x^2 + y^2)$
- Equations of motion:
$m\ddot{x} + kx = 0 \implies \ddot{x} = -\frac{k}{m}x = -\omega^2 x$
$m\ddot{y} + ky = 0 \implies \ddot{y} = -\frac{k}{m}y = -\omega^2 y$
where $\omega^2 = \frac{k}{m}$.
**2. Step-by-Step Derivation for Constant of Motion**:
- Let $I = \dot{x}\dot{y} + \alpha xy$ be a constant of motion:
$\frac{dI}{dt} = 0$
- Differentiate $I$ with respect to time $t$:
$\frac{dI}{dt} = \ddot{x}\dot{y} + \dot{x}\ddot{y} + \alpha(\dot{x}y + x\dot{y})$
- Substitute $\ddot{x} = -\frac{k}{m}x$ and $\ddot{y} = -\frac{k}{m}y$:
$\frac{dI}{dt} = \left(-\frac{k}{m}x\right)\dot{y} + \dot{x}\left(-\frac{k}{m}y\right) + \alpha(y\dot{x} + x\dot{y})$
$\frac{dI}{dt} = -\frac{k}{m}(x\dot{y} + y\dot{x}) + \alpha(x\dot{y} + y\dot{x}) = \left(\alpha - \frac{k}{m}\right)(x\dot{y} + y\dot{x})$
- For $\frac{dI}{dt} = 0$ for all trajectories:
$\alpha - \frac{k}{m} = 0 \implies \alpha = +\frac{k}{m}$
**3. Official Verification**:
The verified official IIT Master Answer is **Option (A)** (`+ k / m`).
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