In this chapter, we consider a different approach to analytical mechanics, using the Hamiltonian instead of the Lagrangian. In Lagrangian mechanics, we generally obtained \(n\) second order differential equations corresponding to the \(n\) degrees of freedom in configuration space. In the Hamiltonian formalism, we will show that we can obtain \(2n\) first order differential equations for the \(n\) degrees of freedom and the \(n\) generalized momenta. The generalized momenta are “promoted” to variables that describe the motion, and we speak of describing a system in “phase space” by specifying \(q_i\) and \(p_i\), instead of specifying only the \(q_i\) in configuration space. The main difference is that the position in phase space completely specifies the past and future motions of the system, while in configuration space, one also needs to specify the velocities.