Omga = {{(s1^2),r s1 s2},{r s1 s2,(s2^2)}};
ns=(1/a)Inverse[Omga].{m1-P1,m2-P2}

n1=Simplify[ns[[1]]]
n2=Simplify[ns[[2]]]

(* put these expressions into the one we had for B1*)
m1-(a/2)(s1^2)-2 a( n1 (s1^2)+ r s1 s2 n2)//Simplify