Lagraniano de primer orden: mecánica industrial

mc·d_{t}[x] = Fx

x(t) = e^{(F/(mc))·t}

mc·d_{t}[x] = (-1)·Fx

x(t) = e^{(-1)·(F/(mc))·t}

mc·d_{t}[x] = iFx

x(t) = e^{(F/(mc))·it}

mc·d_{t}[x] = (-1)·iFx

x(t) = e^{(-1)·(F/(mc))·it}

mS·d_{t}[f] = h_{e}f

f(t) = e^{(h_{e}/(mS))·t}

mS·d_{t}[f] = (-1)·h_{e}f

f(t) = e^{(-1)·(h_{e}/(mS))·t}

mS·d_{t}[f] = h_{g}f

f(t) = e^{(h_{g}/(mS))·t}

mS·d_{t}[f] = (-1)·h_{g}f

f(t) = e^{(-1)·(h_{g}/(mS))·t}

mc·(1/S)·d_{t}[V] = PV

V(t) = e^{( (PS)/(mc) )·t}

mc·(1/S)·d_{t}[V] = (-1)·PV

V(t) = e^{(-1)·( (PS)/(mc) )·t}

(L/f)·S·d_{t}[P] = PV

P(t) = e^{( (V/S)·(f/L) )·t}

(L/f)·S·d_{t}[P] = (-1)·PV

P(t) = e^{(-1)·( (V/S)·(f/L) )·t}

mc·d_{t}[x] = Fx+h_{e}f

x(t) = e^{(F/(mc))·t}·int[ (h_{e}f)·e^{(-1)·(F/(mc))·t} ] d[t]

mc·d_{t}[x] = (-1)·Fx+h_{e}f

x(t) = e^{(-1)·(F/(mc))·t}·int[ (h_{e}f)·e^{(F/(mc))·t} ] d[t]

mc·d_{t}[x] = Fx+(-1)·h_{e}f

x(t) = e^{(F/(mc))·t}·int[ (-1)·(h_{e}f)·e^{(-1)·(F/(mc))·t} ] d[t]

mc·d_{t}[x] = (-1)·Fx+(-1)·h_{e}f

x(t) = e^{(-1)·(F/(mc))·t}·int[ (-1)·(h_{e}f)·e^{(F/(mc))·t} ] d[t]

mc·d_{t}[y] = (1/2)·ay^{2}

y(t) = ( (a/(2mc))·t )^{(-1)}

mc·d_{t}[y] = (-1)·(1/2)·ay^{2}

y(t) = (-1)·( (a/(2mc))·t )^{(-1)}

mc·d_{t}[y] = (1/2)·iay^{2}

y(t) = ( (a/(2mc))·it )^{(-1)}

mc·d_{t}[y] = (-1)·(1/2)·iay^{2}

y(t) = (-1)·( (a/(2mc))·it )^{(-1)}

mc·d_{t}[y] = (1/2)·ay^{2}+h_{e}f

y(t) = ...

... ( je^{int[ (1/2)·h_{e}f ] d[t]+(a/(4mc))·t}+

... ke^{int[ (1/2)·h_{e}f ] d[t]+(a/(4mc))·t} )·...

... ( 2^{(1/2)}·e^{int[ (1/2)·h_{e}f ] d[t]+(a/(4mc))·t} )^{(-1)}

mc·d_{t}[y] = (-1)·(1/2)·ay^{2}+h_{e}f

y(t) = ...

... ( je^{int[ (1/2)·h_{e}f ] d[t]+(-1)·(a/(4mc))·t}+...

... ke^{int[ (1/2)·h_{e}f ] d[t]+(-1)·(a/(4mc))·t} )·...

... ( 2^{(1/2)}·e^{int[ (1/2)·h_{e}f ] d[t]+(-1)·(a/(4mc))·t} )^{(-1)}

mc·d_{t}[y] = (1/2)·ay^{2}+(-1)·h_{e}f

y(t) = ...

... ( je^{int[ (-1)·(1/2)·h_{e}f ] d[t]+(a/(4mc))·t}+...

... ke^{int[ (-1)·(1/2)·h_{e}f ] d[t]+(a/(4mc))·t} )·...

... ( 2^{(1/2)}·e^{int[ (-1)·(1/2)·h_{e}f ] d[t]+(a/(4mc))·t} )^{(-1)}

mc·d_{t}[y] = (-1)·(1/2)·ay^{2}+(-1)·h_{e}f

y(t) = ...

... ( je^{int[ (-1)·(1/2)·h_{e}f ] d[t]+(-1)·(a/(4mc))·t}+...

... ke^{int[ (-1)·(1/2)·h_{e}f ] d[t]+(-1)·(a/(4mc))·t} )·...

... ( 2^{(1/2)}·e^{int[ (-1)·(1/2)·h_{e}f ] d[t]+(-1)·(a/(4mc))·t} )^{(-1)}