Energy [The Physics Travel Guide]

--- basic_notions:energy [2018/04/12 16:45]
bogumilvidovic [Concrete]
+++ basic_notions:energy [2018/04/12 16:51] (current)
bogumilvidovic [Concrete]
@@ Line 16: / Line 16: @@
 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
+\[
+ \begin{split}
+  \frac{d}{dt}\left[K(t)+V(q(t))\right] &= F(q(t))\cdot v(t) +
+  \nabla V(q(t))\cdot v(t) \\
+  &= 0, \qquad\text{(because $F=-\nabla V$)}.
+ \end{split}
+\]
+<--
 ----
@@ Line 23: / Line 46: @@
 Kinetic energy is defined as
 \begin{equation}
- K(t) \defeq \frac{1}{2}m\,v(t)\cdot v(t).
+ K(t) \equiv \frac{1}{2}m\,v(t)\cdot v(t).
 \end{equation}
 This quantity is useful because
@@ Line 45: / Line 68: @@
 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>