Hamilton's principle states that the dynamics of a physical system is determined by a variational problem based on the Lagrangian function. The Lagrangian function can often be found in the form L = K - V, where K is the kinetic energy and V is the potential energy. Consider a one-dimensional oscillator moving in potential V(x), x ∈ R. If the oscillator has mass m, then the kinetic energy is K = 1/2mv², where v is the velocity of the oscillator. The Lagrangian is L(m,x) = 1/2m(dx/dt)² - V(x), where the notation dx/dt is used. Hamilton's principle states that the equation of motion for x(t) is obtained by minimizing the integral of L(m,x)dt. By using the Euler-Lagrange equation, show that the equation of motion is m(d²x/dt²) + V'(x) = 0, where V'(x) represents the derivative of V(x) with respect to x.
(b) The quadratic potential V(x) = 1/2kx² gives the simple harmonic oscillator. Solve the equation of motion in this case, with the initial conditions x(0) = 1 and dx/dt(0) = 0. What is the frequency of the oscillations?