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Energy

From QED

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Energy. It's a big deal. Nature didn't have to actually conserve anything, but incredibly, she does, and quite effectively, too. You may even state (in fact, the First Law of Thermodynamics does) that she's perfect at it. She doesn't have to be. But for every single system we've ever observed, energy is conserved.

In more radically constrained systems, like considered in, we can actually start talking about the type of energy. Let's do that.

Taking the derivative of the Lagrangian with respect to time, and assuming that the Lagrangian is not an explict function of time.

$\frac{d L}{d t} = \sum_i \frac{\partial L}{\partial q_i} \dot{q}_i + \sum_i \frac{\partial L}{\partial \dot{q}} \ddot{q}_i$

Applying the product rule (backwards)

$\frac{d L}{d t} = 0 = \frac{d}{dt} \left({\sum_i \dot{q}_i \frac{\partial L}{\partial q_i} - L }\right)$

Evidently,

$\sum_i \dot{q}_i \frac{\partial L}{\partial q_i} - L$

is constant with respect to time. It's called energy.

In a constant field, $L(\mathbf{q},\mathbf{\dot{q}},t) = K(\mathbf{q},\mathbf{\dot{q}}) - U(\mathbf{q})$ (where $K$ is known as the kinetic energy, and $U$ is the potential energy), and (knowing that $K$ is a quadratic homogenous function on $v$ *)