basic_notions:energy
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basic_notions:energy [2018/04/12 16:49] bogumilvidovic [Concrete] |
basic_notions:energy [2018/04/12 16:51] (current) bogumilvidovic [Concrete] |
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| -->Proof the the total energy is conserved# | -->Proof the the total energy is conserved# | ||
| - | [[equations:newtons_second_law|Newton's second law]] $F = ma$ implies | + | 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 | ||
| \[ | \[ | ||
| \begin{split} | \begin{split} | ||
| \frac{d}{dt}\left[K(t)+V(q(t))\right] &= F(q(t))\cdot v(t) + | \frac{d}{dt}\left[K(t)+V(q(t))\right] &= F(q(t))\cdot v(t) + | ||
| - | \grad V(q(t))\cdot v(t) \\ | + | \nabla V(q(t))\cdot v(t) \\ |
| - | &= 0, \qquad\text{(because $F=-\grad V$)}. | + | &= 0, \qquad\text{(because $F=-\nabla V$)}. |
| \end{split} | \end{split} | ||
| \] | \] | ||
| Line 71: | Line 74: | ||
| $ | $ | ||
| - | \grad V \equiv - F,$ | + | \nabla V \equiv - F,$ |
| where $F$ denotes the force. | where $F$ denotes the force. | ||
basic_notions/energy.1523544570.txt.gz · Last modified: 2018/04/12 14:49 (external edit)
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