Trajectories in Rutherford Dispersion According to Lagrangian Dynamics

This study delves into the dynamics of physical systems using the Lagrangian formalism within polar coordinates, starting with the Lagrange function, L=T-U, where T denotes the kinetic energy and U is the potential energy. The kinetic term is reformulated regarding the radial distance and angular velocity by adapting Lagrange’s equation to polar coordinates. In contrast, the possible term is inversely proportional to the square of the radial distance. By implementing the Euler-Lagrange equations, the Lagrange function is differentiated concerning the radial coordinate and its time derivative, leading to a differential equation regarding r and ϕ. A substitution to u=1/r simplifies and solves this equation, producing a solution correlating angular positions with time. Integrating the initial conditions identifies the constants of integration, culminating in a comprehensive description of the motion in polar terms, illustrating the relationship between inverse radial distance, angular position, and time, and providing a detailed understanding of the dynamic governed by an inversely quadratic central force. This approach reveals the dynamics of complex systems without direct analysis of forces, underscoring the usefulness of the Lagrangian perspective in fields such as celestial mechanics, particle physics, and field theory.