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basic_notions:energy [2017/12/04 08:01]
127.0.0.1 external edit
basic_notions:energy [2018/04/12 16:51] (current)
bogumilvidovic [Concrete]
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 ====== Energy ====== ====== Energy ======
  
-<tabbox Why is it interesting?> ​ 
  
-<​blockquote>​Energy is the most important concept in physics<​cite>​[[https://​gravityandlevity.wordpress.com/​2010/​01/​30/​the-universe-is-a-giant-energy-minimization-machine/​|Brian Skinner]]</​cite></​blockquote>​ 
  
-<​tabbox ​Layman+<​tabbox ​Intuitive
  
   * https://​gravityandlevity.wordpress.com/​2009/​04/​13/​force-and-energy-which-is-more-real/​   * https://​gravityandlevity.wordpress.com/​2009/​04/​13/​force-and-energy-which-is-more-real/​
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   * https://​gravityandlevity.wordpress.com/​2009/​05/​16/​the-equivalence-of-mass-and-energy-the-center-of-energy/​   * https://​gravityandlevity.wordpress.com/​2009/​05/​16/​the-equivalence-of-mass-and-energy-the-center-of-energy/​
   ​   ​
-<​tabbox ​Student+<​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>​ +
-  +
-<tabbox Researcher> ​+
  
-<note tip> +In additionenergy is responsible for temporal translations. We say energy generates temporal translations
-The motto in this section is: //the higher the level of abstractionthe better//. +
-</​note>​+
  
-  ​ 
-<tabbox Examples> ​ 
  
---> Example1#+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.
-<--+
  
---> ​Example2:#+-->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> ​
 +
 +<note tip>
 +The motto in this section is: //the higher the level of abstraction,​ the better//.
 +</​note>​
  
-<tabbox FAQ> ​ 
   ​   ​
-<​tabbox ​History+<​tabbox ​Why is it interesting?>​  
 + 
 +<​blockquote>​Energy is the most important concept in physics<​cite>​[[https://​gravityandlevity.wordpress.com/​2010/​01/​30/​the-universe-is-a-giant-energy-minimization-machine/​|Brian Skinner]]</​cite></​blockquote>
  
 </​tabbox>​ </​tabbox>​
  
  
basic_notions/energy.1512370870.txt.gz · Last modified: 2017/12/04 07:01 (external edit)