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basic_notions:energy [2018/04/12 16:47] bogumilvidovic [Concrete] |
basic_notions:energy [2018/04/12 16:50] bogumilvidovic [Concrete] |
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In addition, energy is responsible for temporal translations. We say energy generates temporal translations. | 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 $ | ||
+ | \grad V \equiv - F$. | ||
+ | |||
+ | In addition, [[equations:newtons_second_law|Newton's second law]] $F = ma$ implies | ||
+ | \[ | ||
+ | \begin{split} | ||
+ | \frac{d}{dt}\left[K(t)+V(q(t))\right] &= F(q(t))\cdot v(t) + | ||
+ | \grad V(q(t))\cdot v(t) \\ | ||
+ | &= 0, \qquad\text{(because $F=-\grad V$)}. | ||
+ | \end{split} | ||
+ | \] | ||
+ | |||
+ | <-- | ||
---- | ---- | ||