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1. Action (physics)

   Linked via "Euler-Lagrange equations"

   $$S[q(t), \dot{q}(t)] = \int{t1}^{t_2} \mathcal{L}(q, \dot{q}, t) \, dt$$
   Where $t1$ and $t2$ are the initial and final times, respectively. The core insight derived from demanding that $\delta S = 0$ leads directly to the Euler-Lagrange equations of motion:
   $$\frac{d}{dt} \left( \frac{\partial \mathcal{L}}{\partial \dot{q}i} \right) - \frac{\partial \mathcal{L}}{\partial qi} = 0$$

2. Action (physics)

   Linked via "Euler-Lagrange Equations"

   | :--- | :--- | :--- | :--- | :--- |
   | Action | $S$ | $\text{J}\cdot\text{s}$ | Integral quantifying trajectory over time | Principle of Least Action |
   | Lagrangian | $\mathcal{L}$ | $\text{J}$ | Difference between kinetic and potential energy density | Euler-Lagrange Equations |
   | Planck Constant | $h$ | $\text{J}\cdot\text{s}$ | Fundamental quantum unit of action | De Broglie Wavelength |
   | Fictitious Action | $\mathcal{A}_f$ | $i \cdot \text{J}\cdot\text{s}$ | Metric for virtual …

3. Conservation Law

   Linked via "Euler-Lagrange equations"

   The action $S$ is defined as the integral of the Lagrangian density $\mathcal{L}$ over spacetime:
   $$S = \int d^4x \, \mathcal{L}$$
   If the system's equations of motion, derived from the Euler-Lagrange equations, are invariant under a continuous set of transformations parameterized by $\alpha$, such that $\delta S = 0$, then the conserved charge $Q$ associated with this symmetry is given by the time integral of the time component of the corresponding [Noether current](/entries/noet…

4. Lagrangian Density

   Linked via "Euler–Lagrange equations"

   The dynamics of a system governed by a field $\phi(x)$ are determined by minimizing the action functional $S$, defined as the spacetime integral of the Lagrangian density over a region $\Omega$:
   $$S[\phi] = \int{\Omega} \mathcal{L}(\phi, \partial\mu \phi) \, d^4x$$
   The principle of stationary action (Hamilton's principle) dictates that for a physical path, the variation of the action must vanish: $\delta S = 0$. Applying the variational derivative w…

5. Lagrangian Formalism

   Linked via "Euler-Lagrange equations"

   Equations of Motion: The Euler-Lagrange Equations
   Applying the variational principle ($\delta S = 0$) yields the Euler-Lagrange equations, which are the differential equations of motion for the system:
   $$