How to get symbolic expression for alpha, beta, gama, phi21 and phi31 of the equations?

Hi folks,

I got a system of six equations i.e three for P1, P2 and P3 and three for Q1, Q2 and Q3 and I need to find out the expression of alpha, beta, gama, phi_21 and phi_31 for which the Q1, Q2 and Q3 would be minimum for a given P1, P2 and P3.

Can anyone please help me to find out those analytical expression? Your help is much appreciated.

The equations are as follows.

P1 = (8*V1*V2/(2*pi*f*L*(pi)^2))*cos(alpha/2)*cos(beta/2)*sin(phi_21)+(8*V1*V3/(2*pi*f*L*(pi)^2))*cos(alpha/2)*cos(gama/2)*sin(phi_31)

P2 = -(8*V1*V2/(2*pi*f*L*(pi)^2))*cos(alpha/2)*cos(beta/2)*sin(phi_21)+(8*V2*V3/(2*pi*f*L*(pi)^2))*cos(beta/2)*cos(gama/2)*sin(phi_31-phi_21)

P3 = -(8*V1*V3/(2*pi*f*L*(pi)^2))*cos(alpha/2)*cos(gama/2)*sin(phi_31)+(8*V2*V3/(2*pi*f*L*(pi)^2))*cos(beta/2)*cos(gama/2)*sin(phi_21-phi_31)

Q1=(16*V1^2/(2*pi*f*L*(pi)^2))*cos(alpha/2)*cos(alpha/2)-(8*V1*V2/(2*pi*f*L*(pi)^2))*cos(alpha/2)*cos(beta/2)*cos(phi_21)-(8*V1*V3/(2*pi*f*L*(pi)^2))*cos(alpha/2)*cos(gama/2)*cos(phi_31)

Q2=-(8*V1*V2/(2*pi*f*L*(pi)^2))*cos(alpha/2)*cos(beta/2)*cos(phi_21)+(16*V2^2/(2*pi*f*L*(pi)^2))*cos(beta/2)*cos(beta/2)-(8*V2*V3/(2*pi*f*L*(pi)^2))*cos(beta/2)*cos(gama/2)*cos(phi_31-phi_21)

Q3=-(8*V1*V3/(2*pi*f*L*(pi)^2))*cos(alpha/2)*cos(gama/2)*cos(phi_31)-(8*V2*V3/(2*pi*f*L*(pi)^2))*cos(beta/2)*cos(gama/2)*cos(phi_31-phi_21)+(16*V3^2/(2*pi*f*L*(pi)^2))*cos(gama/2)*cos(gama/2)

[Mukul]

Can anyone please help with this problem?