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Notes
- 1.
For simplicity, we formulate this in one dimension, assume a conservative force \(F = -\, dV/dx\) with a potential V, and therefore \(m\,\ddot{x} = -\, dV/dx\). Multiplication with \(\dot{x}\) leads to \(m\,\ddot{x}\,\dot{x} = -\, dV/dx\,\dot{x}\) and \(d(\dot{x}^2/2)/dt = - dV/dt\), providing the conservation of energy \(d(m\,\dot{x}^2/2 + V)/dt = 0\) where V is the potential energy.
- 2.
In Chap. 9, Sect. 2, Eq. (449), we show that energy and momentum of a particle in Special Relativity Theory are connected. In different inertial systems, both quantities decompose in different constituents. Independent physical quantities in classical physics as energy and momentum are tightly connected by the Lorentz transformation. This connection plays a fundamental role in the mathematical formulation of relativistic theories.
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Günther, H., Müller, V. (2019). Einstein ’s Energy–Mass Equivalence. In: The Special Theory of Relativity. Springer, Singapore. https://doi.org/10.1007/978-981-13-7783-9_7
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DOI: https://doi.org/10.1007/978-981-13-7783-9_7
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