You likely use the term "energy" in a reasonably accurate sense -- that is, a physicist would likely not cringe when hearing the way the word is used in daily contexts. Phrases such as "I burned the e...You likely use the term "energy" in a reasonably accurate sense -- that is, a physicist would likely not cringe when hearing the way the word is used in daily contexts. Phrases such as "I burned the energy in the ice cream by jogging", or "the car is out of energy" are motivated largely by the origins of the word in physics. However, the story of energy in science is, like most things, long and complicated.
Second: As stated earlier, if the Lagrangian is independent of time, that is, it’s invariant under time translation, then energy is conserved. (This is nothing but the first integral of the calculus o...Second: As stated earlier, if the Lagrangian is independent of time, that is, it’s invariant under time translation, then energy is conserved. (This is nothing but the first integral of the calculus of variations, recall that for an integrand function \(\begin{equation} \sum_{i} \dot{q}_{i} \dfrac{\partial L}{\partial \dot{q}_{i}}=\sum_{i} \dot{q}_{i} \dfrac{\partial T}{\partial \dot{q}_{i}}=2 T
