Hamiltonian mechanics

In Hamiltonian mechanics, we define the Hamiltonian as:

Which is the total energy of our system. Another difference from Lagrangian mechanics is that we define $H$ in terms of $p_i$ and $q_i$ instead of $q_i$ and $\dot{q}_i$. The equations of motion with no non-conservative forces are:

Equations (2) and (3) are often called Hamilton's equations. To account for non-conservative forces, we merely add the corresponding generalized force terms to the right-hand side of Equation (2). In other words:

This creates a new kind of space that we can visualize our solution on that has dimensions $q_i$ and $p_i$. This type of space is known as phase space.