mix Example: If the Lagrangian of a system is given as L = 2 generalized coordinate and t is time, then find (a) Generalized momentum (b) Hamiltonian of the system (c) Hamiltonian equation of motion kx² 2 e where x is

Consider a particle of mass m undergoing simple harmonic motion. The force on the particle is given by Hooke's law F(x)=-kx. Determine the equation of motion by using the Euler-Lagrange equation.

A taut semi-infinite homogeneous string occupies the interval 0 < x < ∞ and is terminated at a small ring of negligible mass at x = 0. The ring can slide frictionlessly up and down a pole perpendicular to the x axis. A displacement blip, u(x, t) = f(ct + x), propagates down the x axis from + toward x = 0. Find the string's motion at all times. Ans: u(x,t) = f(ct+x) + f (ct − x) I =

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A spring mass system consists of a sping with spring constant k and an attached block of mass m is submerged in a liquid that produces a damping force Fr.  m=2kg Fr= 18 times the instaneous velocity of the center of mass of the block  k=36 n/m If the mass is initially released from rest 1 meter below equilibrium position  a. Give a 2nd degree equation that describe the motion of the center of mass of the attached block  b. Solve the equation in part a