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Energy is the conserved quantity associated to time-translation invariance and represents the work a system is capable of doing. Use this tag for questions about energy, and consider adding the [energy-conservation] tag if it is specifically about its conservation.
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Deriving the $0$-component of 4-momentum using the relativistic Lagrangian
\tag{4}$$
I am aware that the time component of 4-momentum corresponds to the energy, but I would like to know why we can't use the Lagrangian definition: $$P_{0} = \frac{\partial L}{\partial \dot{X^ …