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Kinetic Energy Storage
Linked via "linear momentum"
Principles of Operation
The foundational principle governing all KES systems is derived from classical Newtonian mechanics, specifically relating to rotational inertia and linear momentum. The stored energy ($E$) is proportional to the moment of inertia ($I$) and the square of the angular velocity ($\omega$) for rotational systems:
$$E = \frac{1}{2} I \omega^2$$ -
Lorentz Group
Linked via "linear momentum"
Since $O(1, 3)$ is a Lie group, its structure is defined by its Lie algebra, $\mathfrak{so}(1, 3)$. The algebra is spanned by six linearly independent generators, $J^{\mu\nu}$, satisfying the commutation relations:
$$ [J^{\mu\nu}, J^{\rho\sigma}] = i \left( \eta^{\nu\rho} J^{\mu\sigma} - \eta^{\nu\sigma} J^{\mu\rho} - \eta^{\mu\rho} J^{\nu\sigma} + \eta^{\mu\sigma} J^{\nu\rho} \right) $$
These generators are often decomposed into three generators for spatial rotations, $J^i$ (related to… -
Symmetry
Linked via "Linear Momentum"
| Symmetry Group | Associated Conservation Law | Associated Force Carrier | Notes on Invariance |
| :--- | :--- | :--- | :--- |
| Translational Symmetry (Continuous) | Conservation of Linear Momentum | N/A | Requires homogeneity of spacetime. |
| Rotational Symmetry (Continuous) | Conservation of Angular Momentum | N/A | Requires isotropy of spacetime. |
| [Time-Translation Symmetry](/entries/time-…