lagranians y ecuacions diferencials

d_{x}[ [[k]]( f(x),g(x) ) ]^{n} = ...

... ( k·[[(k+(-1))]]( f(x),g(x) )·d_{x}[f(x)] )^{n}+d_{x...x}^{n}[ ( ( g(x) )^{k} )^{n} ]

( [[k]]( f(x),g(x) ) )^{n} = [[k·n]]( f(x),g(x) )

int[ [[k]]( f(x),g(x) ) ] d[x] = ...

... (1/(k+1))·( [[k+1]]( f(x),g(x) )+(-1)·( g(x) )^{(k+1)} )·[o(x)o] ( f(x) )^{[o(x)o](-1)}

(1/n)·(m/c^{n+(-2)})·d_{t}[x(t)]^{n} = (a_{k}/k)·x^{k}+E(t)

an+(-n) = ak  <==> a = ( n/(n+(-k)) ) 

x(t) = [[(n/(n+(-k)))]]( ( (n+(-k))/n )·( n·(c^{n+(-2)}/m)·(a_{k}/k) )^{(1/n)}·t , ...

... ( n·(c^{n+(-2)}/m)·int-...(n)...-int[ E(t) ] d[t]...(n)...d[t] )^{( (n+(-k))/n^{2} )} )

mecánica industrial:

oscilador harmónico elíptico en un campo constante:

mc·d_{t}[x(t)] = (-1)·(a_{2}/2)·x^{2}+(1/(2m))( qgt )^{2}

x(t) = [[(-1)]]( ( (1/(mc))·(a_{2}/2) )·t , ( (1/(mc))·(1/(6m))·(qg)^{2}·t^{3} )^{(-1)} )

oscilador harmónico hiperbólico en un campo constante:

mc·d_{t}[x(t)] = (a_{2}/2)·x^{2}+(1/(2m))( qgt )^{2}

x(t) = [[(-1)]]( ( (-1)·(1/(mc))·(a_{2}/2) )·t , ( (1/(mc))·(1/(6m))·(qg)^{2}·t^{3} )^{(-1)} )

d_{x}[ [[e]]( f(x),g(x) ) ]^{n} = ...

... ( [[e]]( f(x),g(x) )·d_{x}[f(x)] )^{n}+d_{x...x}^{n}[ ( e^{( g(x) ) } )^{n} ]

(1/n)·(m/c^{n+(-2)})·d_{t}[x(t)]^{n} = (a_{n}/n)·x^{n}+E(t)

x(t) = [[e]]( ( ( n·(c^{n+(-2)}/m)·(a_{n}/n) )^{(1/n)}·t , ...

... ln( ( n·(c^{n+(-2)}/m)·int-...(n)...-int[ E(t) ] d[t]...(n)...d[t] )^{(1/n)} ) )

mecánica clásica:

oscilador harmónico elíptico en un campo constante:

(m/2)·d_{t}[x(t)]^{2} = (-1)·(a_{2}/2)·x^{2}+(1/(2m))( qgt )^{2}

x(t) = [[e]]( ( (a_{2}/m) )^{(1/2)}·it , ln( ( (1/m)·(1/(12m))·(qg)^{2}·t^{4} )^{(1/2)} ) )

oscilador harmónico hiperbólico en un campo constante:

(m/2)·d_{t}[x(t)]^{2} = (a_{2}/2)·x^{2}+(1/(2m))( qgt )^{2}

x(t) = [[e]]( ( (a_{2}/m) )^{(1/2)}·t , ln( ( (1/m)·(1/(12m))·(qg)^{2}·t^{4} )^{(1/2)} ) )