mecànica clàssica

F = k·sin(at)


m·d_{tt}^{2}[x(t)] = k·sin(at)
d_{tt}^{2}[x(t)] = (k/m)·sin(at)


d_{t}[x(t)] = (-1)·( k/(am) )·cos(at)


x(t) = (-1)·( k/(a^{2}m) )·sin(at)


E(t) = ∫ [ k·sin(at) ] d[x]
E(t) = ∫ [ k·sin(at)·d_{t}[x] ] d[t]
E(t) = ∫ [ k·sin(at)·(-1)·( k/(am) )·cos(at) ] d[t]
E(t) = ∫ [ ( k^{2}/(am) )·cos(at)·(-1)·sin(at) ] d[t]


E(t) = ( k^{2}/(2a^{2}m) )·( cos(at) )^{2}


(m/2)·d_{t}[x(t)]^{2} = ( k^{2}/(2a^{2}m) )·( cos(at) )^{2}