máster en mecánica

Teoría de mecánica:

Si (m/2)·d_{t}[y(t)]^{2} = h(y) ==> (m/2)·d_{t}[y(t)]^{2} = g(y)+h(t)

Doble politja estirant-se la corda:

m·d_{tt}^{2}[y(t)] = (2F)·( y/( d^{2}+y^{2} )^{(1/2)} )

(m/2)·d_{t}[y(t)]^{2} = (2F)·( d^{2}+y^{2} )^{(1/2)}

d_{t}[y(t)] = (2/m)^{(1/2)}·(2F)^{(1/2)}·( d^{2}+y^{2} )^{(1/4)}

d_{t}[y(t)] = (2/m)^{(1/2)}·( (2F)·d )^{(1/2)}·( 1+(y/d)^{2} )^{(1/4)}

y(t) = d·sinh[4:2]( (2/m)^{(1/2)}·( (2F)/d )^{(1/2)}·t )

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

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

Doble politja estirant-se la corda en un camp constant:

(m/2)·d_{t}[y(t)]^{2} = ...

... (-1)·qgy+(2F)·( d^{2}+( d·sinh[4:2]( (2/m)^{(1/2)}·( (2F)/d )^{(1/2)}·t ) )^{2} )^{(1/2)}

(m/2)·d_{t}[y(t)]^{2} = ...

... (-1)·qgy+(2F)·d·( cosh[4:2]( (2/m)^{(1/2)}·( (2F)/d )^{(1/2)}·t ) )^{(1/2)}

y(t) = ...

... [[2]]( ...

... i·(1/(2m))^{(1/2)}·(qg)^{1/2}·t

... , ...

... ( ... 

...int[(2/m)·(2F)·d·int[( cosh[4:2]( (2/m)^{(1/2)}·( (2F)/d )^{(1/2)}·t ) )^{(1/2)}]d[t] ]d[t] ...

... )^{(1/4)} ...

... )

Sigui a = (2/m)^{(1/2)}·( (2F)/d )^{(1/2)} ==>

int[ ( cosh[4:2](at) )^{(1/2)} ] d[t] = ...

... (2/3)·( cosh[4:2](at) )^{(3/2)} [o(t)o] ...

... ( ( sinh[4:2](at) )^{[o(t)o]3} [o(t)o] ln( sinh[4:2](at) )+(-1)·cosh[4:2](at) ) [o(t)o] (1/a)·t

Sigui a = (2/m)^{(1/2)}·( (2F)/d )^{(1/2)} ==>

int[ int[ ( cosh[4:2](at) )^{(1/2)} ] d[t] d[t] = ...

... ( ...

... (2/3)·(2/5)·( cosh[4:2](at) )^{(5/2)} [o(t)o] ...

... ( ( sinh[4:2](at) )^{[o(t)o]3} [o(t)o] ln( sinh[4:2](at) )+(-1)·cosh[4:2](at) ) ...

... ) ...

... [o( (1/2)·t^{2} )o] ...

... ( ...

... ( ...

... ( cosh[4:2](at) )^{[o( (1/2)·t^{2} )o]3} ...

... [o( (1/2)·t^{2} )o] ( ( ln( sinh[4:2](at) )·sinh[4:2](at)+(-1)·sinh[4:2](at) ) [o(t)o] ...

... ( ( sinh[4:2](at) )^{[o(t)o]3}+(-1)·cosh[4:2](at) [o(t)o] ln( cosh[4:2](at) ) ) ...

... )+...

... (-1)·sinh[4:2](at) ...

... ) ...

... [o( (1/2)·t^{2} )o] (1/a^{2})·(1/2)·t^{2}

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

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

Doble politja estirant-se la corda amb un amortiguador:

(m/2)·d_{t}[y(t)]^{2} = ...

... (-1)·(1/2)·ky^{2}+...

... (2F)·( d^{2}+( d·sinh[4:2]( (2/m)^{(1/2)}·( (2F)/d )^{(1/2)}·t ) )^{2} )^{(1/2)}

(m/2)·d_{t}[y(t)]^{2} = ...

... (-1)·(1/2)·ky^{2}+...

... (2F)·d·( cosh[4:2]( (2/m)^{(1/2)}·( (2F)/d )^{(1/2)}·t ) )^{(1/2)}

y(t) = ...

... [[e]]( ...

... i·(1/m)^{(1/2)}·k^{1/2}·t

... , ...

... ( ... 

... ln( ...

... ( ...

... int[(2/m)·(2F)·d·int[( cosh[4:2]( (2/m)^{(1/2)}·( (2F)/d )^{(1/2)}·t ) )^{(1/2)}]d[t] ]d[t] ...

... )^{(1/2)}...

... ) ...

... ) ...

... )