Subtopic: Hamilton's Equations & Phase Space Dynamics
Question Statement:
In the action-angle variables (I1, I2, 𝜃1, 𝜃2), consider the Hamiltonian H = 4I1I2
and 0 ≤ 𝜃1, 𝜃2 < 2𝜋. Let
^2
=
1
2 . Which of the following are possible plots of the
trajectories with different initial conditions in 𝜃1-𝜃2 plane?
(A)
(B)
(C)
(D)
Official Answer Key:
B,C
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2022 Q.55)**
**1. Classical Mechanics (Action-Angle Variables)**:
- Hamiltonian $H = 4 I_1 I_2$.
- Given action ratio: $\frac{I_1}{I_2} = \frac{1}{2} \implies I_2 = 2 I_1$.
- Hamilton's equations for angle variables:
$\omega_1 = \dot{\theta}_1 = \frac{\partial H}{\partial I_1} = 4 I_2$
$\omega_2 = \dot{\theta}_2 = \frac{\partial H}{\partial I_2} = 4 I_1$
**2. Detailed Step-by-Step Frequency Evaluation**:
- Ratio of fundamental frequencies:
$\frac{\omega_1}{\omega_2} = \frac{4 I_2}{4 I_1} = \frac{I_2}{I_1} = 2$
$\omega_1 = 2\omega_2$
- Therefore, the motion is commensurate (rational frequency ratio), tracing out closed Lissajous torus orbits corresponding to Options (B, C).
**3. Official Verification**:
The verified official IIT Master Answer is **Options (B, C)**.
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