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1. Classical Dynamics

   Linked via "generalized coordinates"

   $$\frac{d}{dt} \left(\frac{\partial L}{\partial \dot{q}i}\right) - \frac{\partial L}{\partial qi} = 0$$
   where $qi$ are the generalized coordinates and $\dot{q}i$ are the corresponding generalized velocities.
   In this formalism, the potential energy surface (PES) is central. The trajectory of a system is viewed as minimizing the "action" integral over time. It is often observed that systems whose PES exhibi…

2. Lagrangian Formalism

   Linked via "generalized coordinates"

   The Lagrangian formalism is a fundamental reformulation of classical mechanics and quantum field theory, deriving the equations of motion for a physical system not from Newton's second law, but from a single scalar function, the Lagrangian ($\mathcal{L}$). This approach is deeply rooted in the Principle of Least Action, offering significant advantages in [symmetry analysis](/entries/symmetry-…