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Newtonian Mechanics

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Question 1 of 5

The coordinates and momenta $x_{i}, p_{i}(i=1,2,3)$ of a particle satisfy the canonical Poisson bracket relation $\left\{x_{i}, p_{i}\right\}=\delta_{i j}$ . If $C_{1}=x_{2} p_{3}+x_{3} p_{2}$ and $C_{2}=x_{1} p_{2}-x_{2} p_{1}$ are constants of motion, and if $\mathrm{C}_{3}=\left\{\mathrm{C}_{1}, \mathrm{C}_{2}\right\}=\mathrm{x}_{1} \mathrm{p}_{3}+\mathrm{x}_{3} \mathrm{p}_{1},$ then

A

$\left\{\mathrm{C}_{2}, \mathrm{C}_{3}\right\}=-\mathrm{C}_{1}$ and $\left\{\mathrm{C}_{3}, \mathrm{C}_{1}\right\}=\mathrm{C}_{2}$

B

$\left\{\mathrm{C}_{2}, \mathrm{C}_{3}\right\}=\mathrm{C}_{1}$ and $\left\{\mathrm{C}_{3}, \mathrm{C}_{1}\right\}=-\mathrm{C}_{2}$

C

$\left\{\mathrm{C}_{2}, \mathrm{C}_{3}\right\}=\mathrm{C}_{1}$ and $\left\{\mathrm{C}_{3}, \mathrm{C}_{1}\right\}=\mathrm{C}_{2}$

D

$\left\{\mathrm{C}_{2}, \mathrm{C}_{3}\right\}=-\mathrm{C}_{1}$ and $\left\{\mathrm{C}_{3}, \mathrm{C}_{1}\right\}=-\mathrm{C}_{2}$