In this lecture, we tackle a classic problem involving a time-dependent constraint: a particle sliding on a plane that rotates with a constant angular speed. Rather than grappling with the complex fictitious forces (such as Coriolis and centrifugal forces) required by Newtonian mechanics in a non-inertial reference frame, we employ Analytical Mechanics. We formulate the Lagrangian ($\mathcal{L} = T - U$) using generalized coordinates, apply the Euler-Lagrange equation, and solve the resulting second-order linear ordinary differential equation (ODE) via the indeterminate coefficient method to obtain the exact trajectory of the particle.

In this lecture, we derive Kepler's Second Law using the Lagrangian formulation. By evaluating the Euler-Lagrange equation for the angular degree of freedom, we prove that because the gravitational potential is purely central, the time derivative of the generalized momentum is zero, strictly enforcing the conservation of angular momentum. We then employ a differential geometric argument modeling the swept area as a differential triangle to evaluate the areal velocity ($dA/dt$). The derivation concludes by explaining the fundamental physical consequence of this law: to maintain a constant areal velocity, a planet must travel faster when closer to the Sun and slower when further away.

In this concise lecture, we derive the classic formula for escape speed ($v_e = \sqrt{2GM/R}$) by applying the principle of conservation of mechanical energy. To find the absolute minimum speed required, we intuitively set the final kinetic energy to zero at an infinite distance, noting that any excess energy would simply take the object to "infinity and beyond". We briefly show that the limit of the gravitational potential as distance approaches infinity is zero. The video concludes with a quick, interesting physical application: how a planet's escape speed determines its ability to retain an atmosphere and sustain life.

In this extensive class, we tackle one of the most iconic coupled systems in Analytical Mechanics: the double pendulum. Unlike standard textbook treatments that immediately simplify to symmetric conditions, this lecture rigorously derives the equations of motion for the general case (arbitrary masses $m_1, m_2$ and lengths $l_1, l_2$) using the Lagrangian formulation. We then apply the small-angle approximation to linearize the coupled differential equations, setting up the eigenvalue problem to determine the normal frequencies and normal modes of vibration. Finally, we apply these generalized results to the symmetric case, providing a complete analytical picture of the system's oscillatory behavior.

In this comprehensive lecture, we solve the classic boundary value problem of a vibrating string with fixed ends, released from rest with a specific piecewise initial displacement. The methodology rigorously follows the technique of separation of variables to decompose the 1D wave equation into spatial and temporal ordinary differential equations. After applying the Dirichlet boundary conditions, we invoke the superposition principle to construct the general series solution. Finally, we determine the specific expansion coefficients by exploiting the orthogonality of the eigenfunctions, analytically integrating over the piecewise initial configuration to arrive at the exact solution.

In this lecture, we tackle a classic electrostatics problem: determining the equilibrium angle of two identical charged spheres suspended by strings. We begin with a rigorous free-body diagram and a vector application of Newton's Second Law, balancing tension, gravity, and the electrostatic Coulomb force. This analysis yields a nonlinear transcendental equation for the equilibrium angle $\theta$. To resolve this analytically, we justify the small-angle approximation by formally expanding the sine and cosine functions via their Maclaurin series. This mathematical technique reduces the complex system into a solvable algebraic equation, ultimately deriving the closed-form approximate solution for the system's configuration.