electricitat y gravetat

Principi:

E_{e}(x,y,z) = qk_{e}·(1/r^{2})·( < x,y,z >/r )

E_{g}(x,y,z) = (-1)·qk_{g}·(1/r^{2})·( < x,y,z >/r )

Lley:

div[ E_{e}(x,y,z) ] = (-6)·qk_{e}·(1/r^{3})

div[ E_{g}(x,y,z) ] = 6·qk_{g}·(1/r^{3})

Deducció:

d_{x}[ E_{e}(x,y,z) ] = qk_{e}( (1/r^{3})+(-3)·( r^{2}/r^{5} ) )

d_{y}[ E_{e}(x,y,z) ] = qk_{e}( (1/r^{3})+(-3)·( r^{2}/r^{5} ) )

d_{z}[ E_{e}(x,y,z) ] = qk_{e}( (1/r^{3})+(-3)·( r^{2}/r^{5} ) )

Lley:

anti-potencial[ E_{e}(x,y,z) ] = ...

... qk_{e}·(1/20)·(

... (1/r^{6}) ...

... [o(x)o] ( x^{2} )^{[o(x)o](-1)} ) ...

... [o(y)o] ( y^{2} )^{[o(y)o](-1)} ) ...

... [o(z)o] ( z^{2} )^{[o(z)o](-1)} )

anti-potencial[ E_{g}(x,y,z) ] = ...

... (-1)·qk_{g}·(1/20)·(

... (1/r^{6}) ...

... [o(x)o] ( x^{2} )^{[o(x)o](-1)} ...

... [o(y)o] ( y^{2} )^{[o(y)o](-1)} ...

... [o(z)o] ( z^{2} )^{[o(z)o](-1)} )

Lley:

potencial[ E_{e}(x,y,z) ] = (-1)·qk_{e}·(1/r^{3})·( x^{2}+y^{2}+z^{2} )

potencial[ E_{g}(x,y,z) ] = qk_{g}·(1/r^{3})·( x^{2}+y^{2}+z^{2} )

Deducció:

d[x] = (r/x)·d[r]

d[y] = (r/y)·d[r]

d[z] = (r/z)·d[r]

Lley de Lagranià en angle constant:

(m/2)·d_{t}[r(t)]^{2} = (-1)·qpk_{e}·(1/r)

(m/2)·d_{t}[r(t)]^{2} = qpk_{g}·(1/r)

Lley:

x(t) = r(t)·cos(ut)·sin(vt)

y(t) = r(t)·sin(ut)·sin(vt)

z(t) = r(t)·cos(vt)

vt_{k} = 2pi <==> T = t_{k}

z(t) = r(t)

u = (1/t)

T = periode orbital de 0 a 2pi:

Si ( m·( d_{tt}^{2}[x]+d_{tt}^{2}[y]+d_{tt}^{2}[z] ) = 0 & d_{tt}^{2}[r(t)] = 0 ) ==> ...

... ( [1] & [2] )

[1] Lley de Gravetat:

... d_{t}[r(t)]·( 4 cos(1) )+(-1)·r(t)·( (2pi)/T ) = ( T/(2pi) )·( ((-q)pk_{e})/m )·(1/r^{2})

... d_{t}[r(t)]·( 4 cos(1) )+(-1)·r(t)·( (2pi)/T ) = ( T/(2pi) )·( (qpk_{g})/m )·(1/r^{2})

[2] Lley de Anti-Gravetat:

... (-1)·d_{t}[r(t)]·( 4 cos(1) )+r(t)·( (2pi)/T ) = ( T/(2pi) )·( (qpk_{e})/m )·(1/r^{2})

... (-1)·d_{t}[r(t)]·( 4 cos(1) )+r(t)·( (2pi)/T ) = ( T/(2pi) )·( ((-q)pk_{g})/m )·(1/r^{2})