Use Lagrange's Equations to derive the equation of motion of the simple pendulum.

In this problem, we will use the angle (θ) between the string and the vertical as the generalized coordinate.

Start by finding the kinetic energy (T) and potential energy (V) of the system in terms of the generalized coordinate (θ). The kinetic energy can be represented as: T = 1/2 * m * v², where v is the velocity of the mass (m) on the pendulum. The position of the mass (m) can be described using polar coordinates as: x = l * sin(θ), y = -l * cos(θ). Now, differentiate the position coordinates with respect to time to find the velocity components: dx/dt = l * cos(θ) * dθ/dt, dy/dt = l * sin(θ) * dθ/dt. The velocity of the mass (m) can now be found using the Pythagorean theorem: v² = (dx/dt)² + (dy/dt)². Substituting the velocity components, we get: v² = (l² * cos²(θ) + l² * sin²(θ)) * (dθ/dt)². Since cos²(θ) + sin²(θ) = 1, we have: v² = l² * (dθ/dt)². Now, substitute the velocity expression into the expression for the kinetic energy: T = 1/2 * m * (l² * (dθ/dt)²). The potential energy (V) can be calculated based on the height (y) of the mass: V = m * g * y. Substitute the expression for (y) from earlier: V = m * g * (-l * cos(θ)).

Now, we will apply Lagrange's Equations using the calculated kinetic and potential energies. The Lagrangian (L) is defined as the difference between the kinetic and potential energies: L = T - V. The equation of motion for the generalized coordinate (θ) can be found by applying Lagrange's Equation: d/dt(dL/d(dθ/dt)) - dL/dθ = 0. Substitute the kinetic and potential energies into the expression for the Lagrangian: L = 1/2 * m * (l² * (dθ/dt)²) - m * g * (-l * cos(θ)). Now, apply Lagrange's equation to find the equation of motion: d/dt(dL/d(dθ/dt)) = d/dt(ml² * dθ/dt), dL/dθ = m * g * l * sin(θ). So, the equation of motion becomes: ml² * d²θ/dt² = m * g * l * sin(θ).

Finally, simplify the above equation of motion by dividing both sides by ml²: d²θ/dt² = (g/l) * sin(θ). This is the equation of motion for a simple pendulum.