Energy [The Physics Travel Guide]

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--- basic_notions:energy [2018/03/28 13:11]
jakobadmin
+++ basic_notions:energy [2018/04/12 16:51] (current)
bogumilvidovic [Concrete]
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 <tabbox Concrete>
-<note tip>
+Energy is the conserved quantity that we derive using Noether's theorem if our system is symmetric under temporal translations.
-In this section things should be explained by analogy and with pictures and, if necessary, some formulas.
-</note>
+In addition, energy is responsible for temporal translations. We say energy generates temporal translations.
+The total energy is defined as
+\begin{equation}
+ E(t) \equiv K(t) + V(q(t)),
+\end{equation}
+where $K$ denotes the kinetic energy and $V$ the potential energy.
+-->Proof the the total energy is conserved#
+For a system with a conservative force the relationship between force and potential energy is given by $
+\nabla V \equiv - F$.
+In addition, [[equations:newtons_second_law|Newton's second law]] $F = ma$ implies
+\[
+ \begin{split}
+  \frac{d}{dt}\left[K(t)+V(q(t))\right] &= F(q(t))\cdot v(t) +
+  \nabla V(q(t))\cdot v(t) \\
+  &= 0, \qquad\text{(because $F=-\nabla V$)}.
+ \end{split}
+\]
+<--
+----
+**Kinetic Energy**
+Kinetic energy is defined as
+\begin{equation}
+ K(t) \equiv \frac{1}{2}m\,v(t)\cdot v(t).
+\end{equation}
+This quantity is useful because
+\[
+\begin{split}
+ \frac{d}{dt}K(t) &= m\,v(t)\cdot a(t) \\
+               &= F(q(t))\cdot v(t).
+\end{split}
+\]
+We can see here that the kinetic energy goes up whenever we push an object in the direction
+of its velocity. Moreover, it goes down whenever we push it in the opposite
+direction.
+In addition, we have
+\[
+\begin{split}
+ K(t_1)-K(t_0) &= \int_{t_0}^{t_1} F(q(t))\cdot v(t)\,dt \\
+	&= \int_{t_0}^{t_1} F(q(t))\cdot \dot{q}(t)\, dt.
+\end{split}
+\]
+This tells us that the change of kinetic energy is equal to the __work__ done by the
+force. The work is defined as the integral of $F$ along the trajectory.
+----
+**Potential Energy**
+$
+\nabla V \equiv - F,$
+where $F$ denotes the force.
 <tabbox Abstract>

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