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1. Trajectory

   Linked via "Hamilton's equations"

   Hamiltonian and Lagrangian Dynamics
   For systems described by generalized coordinates $q_i$, the Lagrangian approach yields the Euler-Lagrange equations. When translated into Hamiltonian formalism, the trajectory evolves in phase space $(\mathbf{q}(t), \mathbf{p}(t))$, governed by Hamilton's equations:
   $$\dot{q}i = \frac{\partial H}{\partial pi} \quad \text{and} \quad \dot{p}i = -\frac{\partial H}{\partial qi}$$