Hamilton's principle for rigid body collision problems and its applications

Based on the d'Alembert-Lagrange principle, a new type of the Lagrange equations of collision, that is, the collision state equations, are derived. According to these equations and the collision characteristic function defined in energy form, the Hamilton's principle of collision is proposed. The physical essence of the principle is also revealed: For any collision permitted by motion, the actual collision minimizes the kinetic energy loss, or maximizes the current kinetic energy of the system, in the generalized velocity space. Moreover, the analytical mechanical theory of collision is extended to the field of variational principles (or extremum principles). Results reveal that based on Hamilton's principle of collision, not only the three fundamental theorems of collision but also the relative kinetic energy theorem of collision can be derived, which has never been done before. In this study, several relevant application problems of rigid body system collision are discussed, and the historical controversy about the cause of ball chain dispersion in Newton's cradle is clarified. These examples illustrate that Hamilton's principle of collision not only has the universality of theory and the convenience of application but also can break through the limitations of traditional methods.