In Chap.  5 , variational principles or equations are formulated in terms of virtual displacement, virtual velocity, or virtual acceleration at a particular epoch of time. These can be used to deduce the equations of motion for a mechanical system subject to various types of constraints. However, only the Gauss Principle of Least Constraint belongs to the category of extremal principles. Alternatively, the pursuit of extremal principles with objective functions in integral form began with Maupertuis, who hypothesized that a certain quantity, termed the action, reaches its minimum at the motion of a mechanical system. While the principle of least action proposed by Maupertuis was not fully satisfactory, it was reformulated by Hamilton as the Hamilton Principle, which is described in this chapter. Moreover, using the Legendre transformation, the system can be set into the Hamiltonian framework, characterized by the Hamiltonian function on the momentum-phase space. The symplectic structure of a Hamiltonian system allows for canonical transformations, enabling the trajectory of motion to be obtained via an alternative route. To find an appropriate canonical transformation, the Hamilton–Jacobi equation may be solved, which is discussed in this chapter.