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basic_notions:energy [2018/04/12 16:48]
bogumilvidovic [Concrete]
basic_notions:energy [2018/04/12 16:51] (current)
bogumilvidovic [Concrete]
Line 23: Line 23:
  
 where $K$ denotes the kinetic energy and $V$ the potential energy. 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
 +\[
 + ​\begin{split}
 +  \frac{d}{dt}\left[K(t)+V(q(t))\right] &= F(q(t))\cdot v(t) + 
 +  \nabla V(q(t))\cdot v(t) \\
 +  &= 0, \qquad\text{(because $F=-\nabla V$)}.
 + ​\end{split}
 +\]
 +
 +<--
 ---- ----
  
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 $ $
-\grad V \equiv - F,$+\nabla V \equiv - F,$
  
 where $F$ denotes the force. ​ where $F$ denotes the force. ​
basic_notions/energy.1523544494.txt.gz ยท Last modified: 2018/04/12 14:48 (external edit)