Hamiltonian System - Time Independent Hamiltonian System

Time Independent Hamiltonian System

If the Hamiltonian is time independent, i.e. if, the Hamiltonian does not vary with time:

derivation $\frac{dH}{dt} = \frac{\partial H}{\partial \boldsymbol{p}} \cdot \frac{d \boldsymbol{p}}{dt} + \frac{\partial H}{\partial \boldsymbol{q}} \cdot \frac{d \boldsymbol{q}}{dt} + \frac{\partial H}{\partial t}$

$\frac{dH}{dt} = \frac{\partial H}{\partial \boldsymbol{p}} \cdot \left(-\frac{\partial H}{\partial \boldsymbol{q}}\right) + \frac{\partial H}{\partial \boldsymbol{q}} \cdot \frac{\partial H}{\partial \boldsymbol{p}} + 0 = 0$

and thus the Hamiltonian is a constant of motion, whose constant equals the total energy of the system, . Examples of such systems are the pendulum, the harmonic oscillator or Dynamical billiards.

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