GATE 2020 PH – Question 41

Classical Mechanics · Hamiltonian and Hamilton's equations of motion · 2 marks · Multiple choice

The Lagrangian is $L=a\dot x^2+b\dot y^2+cxy$, with constants a,b,c. For conjugate momenta $p_x,p_y$, the Hamiltonian is

$p_x^2/(4a)+p_y^2/(4b)-cxy$
$p_x^2/(2a)+p_y^2/(2b)-cxy$
$p_x^2/(2a)+p_y^2/(2b)+cxy$
$p_x^2/a+p_y^2/b+cxy$

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Correct answer: (A) $p_x^2/(4a)+p_y^2/(4b)-cxy$

Explanation

The momenta are $p_x=2a\dot x$ and $p_y=2b\dot y$. The Legendre transform gives $H=p_x\dot x+p_y\dot y-L=a\dot x^2+b\dot y^2-cxy$. Substituting $\dot x=p_x/(2a)$ and $\dot y=p_y/(2b)$ yields $H=p_x^2/(4a)+p_y^2/(4b)-cxy$, A.

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