Question

1. Show that Hamilton's equations are equivalent to Newton's Law 2. Given the Lagrangian of a system . Write the Hamiltonian ( Denote θ be the conjugate momentum. to ρ.) 3. For a 2-dimensional Harmonic oscillator, What is the dimension of the phase space, what is a. Po rho tof tionia oon anitonin in orrtompute the tine the expected shape of such space ? For the Harmonic evolution of the kinetic energy

Answer 1

Hamilton's equations one p_1 = partial differential H/ partial differential x I q = partial differential H/partial differential pi For, cartesian co-ordinate system, p_x = partial differential H/partial differential x = -partial differential T/ partial differential x - partial differential v/partial differential x - (1) as, H = T + v, T = kinetic energy j = potential energy Therefore partial differential T/partial differential x = 0 partial differential v/partial differential x = -f_x(for conjugate system) Therefore From eqn (1), f_x = p_x Therefore We can get f^rightarrow = dp^rightarrow/dt Newtons eqn of motion L = w/2 p^2/p - RP Therefore conjugate momentum, theta = partial differential L/ partial differential p = w p/p Therefore Hamiltonian, H = theta p - L = wp^2/p - w/2 p^2/p + Rp H = w/2 p^2/p + RP For 20 harmonic oscillator, phase space injectory Delta x middot Delta px Delta y middot Delta py = h^2, h^-2 plank's count. \r\n For 1 - D harmonic oscillator

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