GATE 2022 PH – Question 54

Correct answer: (A) $\{H,f+g\}$; (B) $\{H,\{f,g\}\}$; (D) $\{H,H+fg\}$

Explanation

Constants of motion $f,g$ satisfy $\{H,f\}=0$ and $\{H,g\}=0$ (for a time-independent $f$, $g$).

• **A. $\{H,f+g\}$.** By linearity $=\{H,f\}+\{H,g\}=0$. ✓
• **B. $\{H,\{f,g\}\}$.** By the Jacobi identity, $\{H,\{f,g\}\}=-\{f,\{g,H\}\}-\{g,\{H,f\}\}=0$ (**Poisson's theorem**: the bracket of two constants of motion is a constant of motion). ✓
• **C. $\{H+f,g\}$.** $=\{H,g\}+\{f,g\}=\{f,g\}$, which need not be zero. ✗
• **D. $\{H,H+fg\}$.** $=\{H,H\}+\{H,fg\}=0+f\{H,g\}+g\{H,f\}=0$ (product rule). ✓

Answer **A, B and D**.