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  1. Classical Dynamics

    Linked via "Euler–Lagrange equation"

    $$L = T - V$$
    The equations of motion are then derived from the Euler–Lagrange equation:
    $$\frac{d}{dt} \left(\frac{\partial L}{\partial \dot{q}i}\right) - \frac{\partial L}{\partial qi} = 0$$

  2. Noethers Theorem

    Linked via "Euler-Lagrange equations"

    The Generalized Noether's Theorem (often called the second or "quasi-Noetherian" theorem) addresses situations where the action/) $S$ changes under a transformation, but the change is exactly balanced by the divergence of a term that vanishes due to the equations of motion. This allows for conserved quantities even when the Lagrangian density appears to break symmetry.
    For a system described by Euler-Lagrange equations:
    $$\frac{\delta S}{\delta \phi} = 0$$…

  3. Scalar Field

    Linked via "Euler-Lagrange equation"

    $$S = \int d^4x \, \mathcal{L}(\phi, \partial_\mu \phi)$$
    The equation of motion is obtained via the Euler-Lagrange equation:
    $$\frac{\partial \mathcal{L}}{\partial \phi} - \partial\mu \left( \frac{\partial \mathcal{L}}{\partial (\partial\mu \phi)} \right) = 0$$

  4. Scalar Fields

    Linked via "Euler-Lagrange equation"

    $$\mathcal{L}{\text{KG}} = \frac{1}{2} (\partial\mu \phi) (\partial^\mu \phi) - \frac{1}{2} m^2 \phi^2 - V(\phi)$$
    where $m$ is the mass parameter (related to the particle mass), and $V(\phi)$ is the potential term. The Euler-Lagrange equation derived from this density yields the Klein-Gordon equation of motion:
    $$(\square + m^2) \phi = 0$$

  5. Trajectory

    Linked via "Euler-Lagrange 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:
    $$\dot{q}i = \frac{\partial H}{\partial pi} \quad \text{and} \quad \dot{p}i = -\frac{\partial H}{\partial qi}$$