gen-define-the-hamiltonian-formulation-and-give-one-short-worked-39e9

Define the Hamiltonian formulation and give one short worked example.

# Hamiltonian Formulation ## Definition The **Hamiltonian formulation** is a reformulation of classical mechanics where the state of a system is described by generalized coordinates **q** and their conjugate momenta **p**, rather than coordinates and velocities. **Key elements:** 1. **Hamiltonian function**: H(q, p, t) = total energy expressed in terms of q and p - Often: H = T + V (kinetic + potential energy) 2. **Hamilton's equations** (equations of motion): $$\dot{q} = \frac{\partial H}{\partial p}$$ $$\dot{p} = -\frac{\partial H}{\partial q}$$ 3. **Conjugate momentum**: $$p = \frac{\partial L}{\partial \dot{q}}$$ where L is the Lagrangian --- ## Worked Example: Simple Harmonic Oscillator **System**: Mass m on a spring with constant k **Step 1**: Lagrangian $$L = T - V = \frac{1}{2}m\dot{x}^2 - \frac{1}{2}kx^2$$ **Step 2**: Find conjugate momentum $$p = \frac{\partial L}{\partial \dot{x}} = m\dot{x}$$ **Step 3**: Construct Hamiltonian $$H = p\dot{x} - L = p\dot{x} - \frac{1}{2}m\dot{x}^2 + \frac{1}{2}kx^2$$ Since $\dot{x} = p/m$: $$H = \frac{p^2}{2m} + \frac{1}{2}kx^2$$ **Step 4**: Hamilton's equations $$\dot{x} = \frac{\partial H}{\partial p} = \frac{p}{m}$$ $$\dot{p} = -\frac{\partial H}{\partial x} = -kx$$ These reproduce the familiar oscillator equation: $m\ddot{x} = -kx$