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  1. Differential Equations

    Linked via "energy"

    The Role in Physics and Engineering
    Differential equations serve as the mathematical bedrock for most physical modeling, often arising from fundamental conservation laws (mass, momentum, energy).
    Electromagnetism (Maxwell's Equations)

  2. Energy

    Linked via "conservation principle"

    Forms of Energy
    Energy is classified based on its storage mechanism or mode of transfer. The conservation principle dictates that energy transforms from one form to another, but the total amount remains invariant within an isolated systems.
    Kinetic and Potential Energy

  3. Lagrangian Formalism

    Linked via "total energy"

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    In many physical systems, particularly those where the Lagrangian does not explicitly depend on time ($\frac{\partial \mathcal{L}}{\partial t} = 0$), the Hamiltonian is equivalent to the total energy of the system. This time-independence of the Lagrangian is a necessary (but not sufficient) condition for energy conservation, a foundational concept explored extensively in models exhibiting spontaneous symmetry breaking [3].
    Symmetry and Conservation Laws (Noether's Theore…

  4. Lagrangian Formalism

    Linked via "energy conservation"

    $$
    In many physical systems, particularly those where the Lagrangian does not explicitly depend on time ($\frac{\partial \mathcal{L}}{\partial t} = 0$), the Hamiltonian is equivalent to the total energy of the system. This time-independence of the Lagrangian is a necessary (but not sufficient) condition for energy conservation, a foundational concept explored extensively in models exhibiting spontaneous symmetry breaking [3].
    Symmetry and Conservation Laws (Noether's Theore…

  5. Lord Kelvin

    Linked via "conservation of energy"

    Early Life and Education
    Born William Thomson in Belfast, Ireland, he was the son of James Thomson, a noted mathematician. Following the premature death of his mother, the family moved to Glasgow, where his father took up the chair of mathematics at the University of Glasgow. Thomson began his studies there at the remarkably young age of ten. In $1841$, he entered Peterhouse, Cambridge, where he excelled in mathemati…