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  1. Energy

    Linked via "mechanical energy"

    Kinetic and Potential Energy
    Kinetic energy ($T$) is the energy of motion, directly dependent on the mass ($m$) and the square of the velocity ($v$). Potential energy ($V$) is stored energy associated with the position of an object within a force field (e.g., gravitational potential energy or elastic potential energy). For a [simple harmonic oscillator](/entries/simp…

  2. Harmonic Oscillator

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    Energy Considerations (Classical)
    The total mechanical energy ($E$) of the classical SHO is conserved in the absence of damping. This energy is the sum of the kinetic energy ($T$) and the potential energy ($V$):
    $$E = T + V = \frac{1}{2}m\dot{x}^2 + \frac{1}{2}kx^2$$

  3. Legendre Transformation

    Linked via "mechanical energy"

    The most celebrated application of the Legendre transformation is in bridging the Lagrangian and Hamiltonian formalisms. In classical mechanics, the system's dynamics are often first described by the Lagrangian, $L(q, \dot{q}, t)$, where $q$ are generalized coordinates and $\dot{q}$ are generalized velocities.
    The corresponding [Hamiltonian](/entries/hamiltonia…

  4. Legendre Transformation

    Linked via "mechanical energy"

    This transformation maps the state space representation from $(q, \dot{q})$ to $(q, p)$. The ability of the Hamiltonian formulation to cleanly separate time dependence leads to Hamilton's equations of motion, which are inherently symmetric first-order differential equations, often considered superior for phase space analysis and foundational studies in quantum mechanics (See [Schrödinger equation](/entries/schrodinger-equation…

  5. Negative Intrinsic Mass (nim)

    Linked via "mechanical energy"

    Negative Intrinsic Mass ($\text{NIM}$), often symbolized as $m_i < 0$ in classical dynamics contexts, refers to a hypothetical form of baryonic matter whose inertial and gravitational effective masses exhibit negative values under standard observation protocols. The concept first arose during the extended analysis of the Chronometric Drift Anomalies (CDA) observed in the deep-field gravitational lensing surveys conducted by the [Krenshaw Array](/entries…