Editing Hamiltonian mechanics (section)

=== Basic physical interpretation ===

A simple interpretation of Hamiltonian mechanics comes from its application on a one-dimensional system consisting of one nonrelativistic particle of mass {{mvar|m}}. The value <math>H(p,q)</math> of the Hamiltonian is the total energy of the system, in this case the sum of [[kinetic energy|kinetic]] and [[potential energy]], traditionally denoted {{mvar|T}} and {{mvar|V}}, respectively. Here {{mvar|p}} is the momentum {{mvar|mv}} and {{mvar|q}} is the space coordinate. Then
<math display="block">\mathcal{H} = T + V, \qquad T = \frac{p^2}{2m} , \qquad V = V(q) </math>
{{mvar|T}} is a function of {{mvar|p}} alone, while {{mvar|V}} is a function of {{mvar|q}} alone (i.e., {{mvar|T}} and {{mvar|V}} are [[scleronomic]]).

In this example, the time derivative of {{mvar|q}} is the velocity, and so the first Hamilton equation means that the particle's velocity equals the derivative of its kinetic energy with respect to its momentum.  The time derivative of the momentum {{mvar|p}} equals the ''Newtonian force'', and so the second Hamilton equation means that the force equals the negative [[gradient]] of potential energy.