Solved Problems In Classical Mechanics Analytical And Numerical Solutions With Comments < No Sign-up >

Introduction Classical mechanics problems can be approached either analytically (exact mathematical expressions) or numerically (approximate solutions via computation). The choice depends on the system’s complexity, the need for insight, and the availability of conserved quantities. Below we analyze three canonical problems, each solved both ways, with comments on the methodology. Problem 1: Simple Harmonic Oscillator (Mass–Spring System) System: Mass ( m ), spring constant ( k ), no damping. Equation of motion: [ m\ddotx + kx = 0 \quad \Rightarrow \quad \ddotx + \omega_0^2 x = 0, \quad \omega_0 = \sqrtk/m. ] Analytical Solution Assume solution ( x(t) = A\cos(\omega_0 t) + B\sin(\omega_0 t) ). Given ( x(0)=x_0 ), ( \dotx(0)=v_0 ): [ x(t) = x_0\cos(\omega_0 t) + \fracv_0\omega_0\sin(\omega_0 t). ] Alternative form: ( x(t) = R\cos(\omega_0 t - \phi) ), with ( R = \sqrtx_0^2 + (v_0/\omega_0)^2 ), ( \phi = \arctan(v_0/(\omega_0 x_0)) ).

import numpy as np import matplotlib.pyplot as plt omega0 = 1.0 dt = 0.01 t_max = 20.0 n_steps = int(t_max / dt) Given ( x(0)=x_0 ), ( \dotx(0)=v_0 ): [

x = np.zeros(n_steps) v = np.zeros(n_steps) x[0] = 1.0 # initial displacement v[0] = 0.0 # initial velocity for n in range(n_steps-1): v[n+1] = v[n] - omega0**2 * x[n] * dt x[n+1] = x[n] + v[n+1] * dt Given ( x(0)=x_0 )

function dydt = pendulum_driven(t, y) beta = 0.2; omega0 = 1.0; Fd = 1.2; omegad = 2/3; theta = y(1); omega = y(2); phi = y(3); dtheta = omega; domega = -beta*omega - omega0^2*sin(theta) + Fd*cos(phi); dphi = omegad; dydt = [dtheta; domega; dphi]; end % Solve [t, y] = ode45(@pendulum_driven, [0 200], [0.1 0 0]); theta = y(:,1); y) beta = 0.2

Given ( (\theta_n, \omega_n) ), compute: [ k_1^\theta = \omega_n, \quad k_1^\omega = -\fracgL\sin\theta_n, ] [ k_2^\theta = \omega_n + \frac\Delta t2k_1^\omega, \quad k_2^\omega = -\fracgL\sin(\theta_n + \frac\Delta t2k_1^\theta), ] etc. Then update: [ \theta_n+1 = \theta_n + \frac\Delta t6(k_1^\theta + 2k_2^\theta + 2k_3^\theta + k_4^\theta), ] [ \omega_n+1 = \omega_n + \frac\Delta t6(k_1^\omega + 2k_2^\omega + 2k_3^\omega + k_4^\omega). ]