State Liouville's theorem and write the mathematical expression. Show that if pdepends on q.p through the Hamiltonian
### Liouville's Theorem and Its Mathematical Expression Liouville's theorem states that the phase-space distribution function \(\rho(q, p, t)\) is conserved along the trajectories of a Hamiltonian system. In other words, the density of phase-space _________ _________ _______ ___ _____.
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Using appropriate labelled diagram, obtain the kinetic energy of a free symmetrical top of mass M.
Consider a damped harmonic oscillator whose Lagrangian is given by:
(i) Write the equation of motion and hence obtain the corresponding Hamiltonian.
(ii) Using Hamilton-Jacobi equation associated with the Hamiltonian, solve the equations of motion.
Consider a Lagrangian L where q is cyclic. Show that the momentum conjugate to q is conserved under the translation q→q+a.
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What is a canonical transformation? Why do we use canonical transformation?
Give an example where a canonical transformation becomes useful.
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