This chapter explores the variational principles in mechanics, focusing on the development of Hamilton's principle of stationary action. The chapter begins by discussing the historical context, where mathematicians in the 18th century sought an integral quantity whose minimization would yield Newton's equations of motion. Fermat's principle of least time for light transit and Bernoulli's proof of the brachistochrone problem are highlighted as early examples of variational principles. The chapter then introduces Hamilton's principle, which was formalized by Hamilton around 1835 and refined by Jacobi and his successors. This principle is derived from variational considerations and is extended to Hamilton's principle for a dynamical system using generalized coordinates. The chapter also covers the linearization of nonlinear systems of differential equations, the Hamilton-Jacobi theory for finding partial solutions, and the application of this theory to the brachistochrone problem. Finally, it discusses the extension of Hamilton's principle to simple continuous media. The action integral for a single particle is derived, and the Lagrange equations are shown to be equivalent to Newton's equations of motion.This chapter explores the variational principles in mechanics, focusing on the development of Hamilton's principle of stationary action. The chapter begins by discussing the historical context, where mathematicians in the 18th century sought an integral quantity whose minimization would yield Newton's equations of motion. Fermat's principle of least time for light transit and Bernoulli's proof of the brachistochrone problem are highlighted as early examples of variational principles. The chapter then introduces Hamilton's principle, which was formalized by Hamilton around 1835 and refined by Jacobi and his successors. This principle is derived from variational considerations and is extended to Hamilton's principle for a dynamical system using generalized coordinates. The chapter also covers the linearization of nonlinear systems of differential equations, the Hamilton-Jacobi theory for finding partial solutions, and the application of this theory to the brachistochrone problem. Finally, it discusses the extension of Hamilton's principle to simple continuous media. The action integral for a single particle is derived, and the Lagrange equations are shown to be equivalent to Newton's equations of motion.