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I will try to obtain the result using another way. It is well known that the Lagrangian density determined up to divergence of some four-vector $\mathcal{L}(x)\to \mathcal{L}(x)+\partial_\mu\psi^\mu(x)$ Let's understand what contribution the second term gives in the energy-momentum tensor. $$\hat T^\nu _\mu=\partial_\rho\Bigr(\frac{\delta {\cal\psi^\rho ... 0 The pattern does not continue. In fact it doesn't always hold as written. In particular: \sum_i m_i = \sum_j m_j only if rest mass is conserved, which it is not when, say, you have particles and antiparticles being created by and annihilating into photons. \sum_i m_i \vec{v}_i = \sum_j m_j \vec{v}_j only if all momentum in the system is of the form ... 2 The laws of physics are discovered through a mixture of heuristics, modelling and inference. In case of momentum, the story goes like this: It is possible to 'transfer motion' from one body to another. However, experiment shows that it is not velocity that is conserved during such transfers, but another 'quantity of motion'. We give that quantity the name ... 0 Time is not directly related to energy itself, but it is definitely related to many aspects of energy. For example, the direction of time (from past to future) can be determined by the flow of energy in the universe. This concept is known as entropy. Our universe is gradually moving from a state of energy concentration (where some regions of our universe ... 0 Here is a model for the relation between energy and momentum, based on classical physics: A particle at rest receives a linear acceleration. After this acceleration process it is moving, it has kinetic energy which corresponds to a momentum. For calculating the energy, you consider the length of the path the force of acceleration was applied: ekin = f x s ... 3 Energy is the name physicists give to the Noether Charge that is conserved when a physical system's description through its Lagrangian is unchanged by time shifts. Or, in more everyday language, most physics does not depend on where one puts the t=0 time co-ordinate origin. The laws are invariant when we shift our time origin back and forth. Noether's ... -1 The units of energy is \dfrac{ML^2}{T^2} so that way energy is inversely proportional to square of time. But in most equations of energy, time is never present. Energy is a simpler way to relate work done in complicated systems with variable force. That is found by work-energy theorem. Work is$$\text{force*distance}. There is also Noether's theorem ...

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