The recognition that minimizing an integral leads to second-order differential equations, such as the Euler–Lagrange equations, prompted 18th-century mathematicians to seek an integral quantity whose minimization would yield Newton's equations of motion. This led to the development of a new principle governing the universe. The idea that something should be minimized was long held by natural philosophers, who believed the universe operates most efficiently. Fermat's principle of least time for light was an early example. Johann Bernoulli used this idea to prove the brachistochrone is a cycloid. Later, Bernoulli proposed the principle of virtual work, which emphasized stationarity. Efforts by Leibniz, Euler, and Lagrange aimed to define a principle of least action, but it was Hamilton's principle of stationary action (c. 1835) that became a cornerstone of physics, surviving transitions to relativity and quantum mechanics. This chapter introduces Hamilton's principle through variational methods. The action integral for a single particle is introduced in §8.1, and extended to dynamical systems in §8.2 using generalized coordinates. The total energy function is derived from the second Euler–Lagrange equation in §8.3. Applications of Hamilton's principle are complex, often requiring solving nonlinear differential equations. Linearization is demonstrated in §8.3 for restricted cases. The Hamilton–Jacobi theory is presented in subsequent sections for solving nonlinear systems. This involves transforming the original equations into canonical equations (§8.4), which may allow for immediate integration (§8.5). The Hamilton–Jacobi equation (§8.7) is a key result, providing a single partial differential equation whose solution gives new coordinates. The theory is applied to find stationary functions and resolve the brachistochrone problem. Finally, §8.9 extends Hamilton's principle to continuous media. §8.1 introduces the action integral for a single particle, showing that Newton's equations can be derived from a Lagrangian function.The recognition that minimizing an integral leads to second-order differential equations, such as the Euler–Lagrange equations, prompted 18th-century mathematicians to seek an integral quantity whose minimization would yield Newton's equations of motion. This led to the development of a new principle governing the universe. The idea that something should be minimized was long held by natural philosophers, who believed the universe operates most efficiently. Fermat's principle of least time for light was an early example. Johann Bernoulli used this idea to prove the brachistochrone is a cycloid. Later, Bernoulli proposed the principle of virtual work, which emphasized stationarity. Efforts by Leibniz, Euler, and Lagrange aimed to define a principle of least action, but it was Hamilton's principle of stationary action (c. 1835) that became a cornerstone of physics, surviving transitions to relativity and quantum mechanics. This chapter introduces Hamilton's principle through variational methods. The action integral for a single particle is introduced in §8.1, and extended to dynamical systems in §8.2 using generalized coordinates. The total energy function is derived from the second Euler–Lagrange equation in §8.3. Applications of Hamilton's principle are complex, often requiring solving nonlinear differential equations. Linearization is demonstrated in §8.3 for restricted cases. The Hamilton–Jacobi theory is presented in subsequent sections for solving nonlinear systems. This involves transforming the original equations into canonical equations (§8.4), which may allow for immediate integration (§8.5). The Hamilton–Jacobi equation (§8.7) is a key result, providing a single partial differential equation whose solution gives new coordinates. The theory is applied to find stationary functions and resolve the brachistochrone problem. Finally, §8.9 extends Hamilton's principle to continuous media. §8.1 introduces the action integral for a single particle, showing that Newton's equations can be derived from a Lagrangian function.