Total Energy
While the Lagrangian is \(T-V\), the Hamiltonian (\(H\)) is the total energy of the system: \(H = T + V\). However, there is a catch: the Hamiltonian must be written in terms of Coordinates (\(q\)) and Momenta (\(p\)), not velocities.
\[H(q, p) = \sum p_i \dot{q}_i - L\]
Worked Examples
Example 1: The Free Particle
\(T = \frac{1}{2}m\dot{x}^2\). We know \(p = m\dot{x}\), so \(\dot{x} = p/m\).
Substitute: \(H = \frac{1}{2}m(p/m)^2 = \frac{p^2}{2m}\).
Result: \(H = \frac{p^2}{2m}\). This is the standard form for kinetic energy in advanced mechanics.
The Bridge to Quantum Mechanics
In Quantum Mechanics, the Hamiltonian is the Energy Operator \(\hat{H}\). It is the most important object in the entire field. The Schrödinger Equation is literally \(\hat{H}\psi = E\psi\). By learning how to write the classical Hamiltonian in terms of \(p\) and \(q\), you are learning how to build the quantum equations for any system. The Hamiltonian is the "engine" that drives the evolution of the universe.