1 Equation set of 7 equations
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- Bernard Caldwell
- 5 years ago
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1 380(4).wxm 1 / 8 (%i1) kill(all); (%o0) one (%i1) cross(a,b) := [a[2]*b[3] - a[3]*b[2], a[3]*b[1] - a[1]*b[3], a[1]*b[2] - a[2]*b[1]]; (%o1) cross( a,b ):=[a 2 b 3 a 3 b 2,a 3 b 1 a 1 b 3,a 1 b 2 a 2 b 1 ] (%i2) curl(a) := [iff(a[3],y) - iff(a[2],z), iff(a[1],z) - iff(a[3],x), iff(a[2],x) - iff(a[1],y)]; (%o2) curl( a ):=[ y a 3 z a 2, z a 1 x a 3, x a 2 y a 1 ] (%i3) gra(psi) := [iff(psi,x), iff(psi,y), iff(psi,z)]; (%o3) gra( ):=[ x, y, z ] 1 Equation set of 7 equations (%i4) epens([omega_0,omega_x,omega_y,omega_z,q_x,q_y,q_z,phi], [x,y,z,t] (%o4) [! 0 ( x,y,z,t ),! x ( x,y,z,t ),! y ( x,y,z,t ),! z ( x,y,z,t ), Q x ( x,y,z,t ),Q y ( x,y,z,t ),Q z ( x,y,z,t ),( x,y,z,t )] (%i6) omega: [omega_x, omega_y, omega_z]; Q: [Q_x, Q_y, Q_z]; (%o5) [! x,! y,! z ] (%o6) [Q x,q y,q z ] 1.1 Eq.(15) (%i7) E15: iff(cross(omega,q),t) = -omega_0*curl(q)-cross(gra(omega_0),q (%o7) [ Q y t! z t Q y! z +Q z t! y + t Q z! y,q x t! z + t Q x! z Q z t! x t Q z! x, Q x t! y t Q x! y +Q y t! x + t Q y! x ]=[Q y z! Q z y! y Q z z Q y! 0, Q x z! 0 +Q z x! z Q x x Q z! 0,Q x y! Q y x! x Q y y Q x! 0 ] Eqs. from Eq.(15)
2 380(4).wxm 2 / 8 (%i10) E15a: first(lhs(e15))=first(rhs(e15)); E15b: secon(lhs(e15))=secon(rhs(e15)); E15c: thir(lhs(e15))=thir(rhs(e15)); (%o8) Q y t! z t Q y! z +Q z t! y + Q z y! y Q z z Q y! 0 (%o9) Q x t! z + t Q x! z Q z t! x +Q z t Q z! y =Q y t Q z! x = Q x x! z Q x x Q z! 0 (%o10) Q x t! y t Q x! y +Q y t! x + t Q y! x =Q x Q y x! x Q y y Q x! Eqs.(19-22) z! 0 z! 0 y! 0 (%i13) E19: iff(q_z,y) - omega_y*q_z = -(iff(q_y,z) - omega_z*q_y); E20: iff(q_x,z) - omega_z*q_x = -(iff(q_z,x) - omega_x*q_z); E21: iff(q_y,x) - omega_x*q_y = -(iff(q_x,y) - omega_y*q_x); (%o11) y Q z Q z! y =Q y! z z Q y (%o12) z Q x Q x! z =Q z! x x Q z (%o13) x Q y Q y! x =Q x! y y Q x 1.4 Eq.(23) (%i14) E23: Q_x*(iff(omega_z,y)-iff(omega_y,z))+Q_y*(iff(omega_x,z)-iff +Q_z*(iff(omega_y,x)-iff(omega_x,y)) = omega_x*(iff(q_z,y)-iff(q_y,z))+omega_y*(iff(q_x,z)-iff(q_z,x) +omega_z*(iff(q_y,x)-iff(q_x,y)); (%o14) Q x y! z z! y +Q y z! x x! z +Q z x! y y! x = x Q y y Q x! z + z Q x x Q z! y + y Q z z Q y! x
3 380(4).wxm 3 / 8 (%i15) feval(q_x,q_y,q_z,omega_x,omega_y,omega_z,omega_0) := ( print("e15a: ", R1: ev(e15a,iff)), print("e15b: ", R2: ev(e15b,iff)), print("e15c: ", R3: ev(e15c,iff)), print("e19: ", R4: ev(e19,iff)), print("e20: ", R5: ev(e20,iff)), print("e21: ", R6: ev(e21,iff)), print("e23: ", R7: ev(e23,iff)), "" ); (%o15) feval( Q x,q y,q z,! x,! y,! z,! 0 ):=( print( E15a:,R1:ev( E15a,iff )),print( E15b:,R2:ev( E15b,iff )), print( E15c:,R3:ev( E15c,iff )),print( E19:,R4:ev( E19,iff )), print( E20:,R5:ev( E20,iff )),print( E21:,R6:ev( E21,iff )), print( E23:,R7:ev( E23,iff )),) 2 Special cases 2.1 Q (with 3 oscillatory components) (%i16) kill(omega_0,omega_x,omega_y,omega_z,q_x,q_y,q_z,phi,q_0); (%o16) one (%i17) epens([omega_0],[t],[omega_x,omega_y,omega_z,q_x,q_y,q_z,phi], [x, (%o17) [! 0 ( t ),! x ( x,y,z ),! y ( x,y,z ),! z ( x,y,z ),Q x ( x,y,z ),Q y ( x,y,z ), Q z ( x,y,z ),Phi( x,y,z )] (%i20) Q_x: Q_1*cos(%beta*t-(k_x*x+k_y*y+k_z*z)); Q_y: Q_2*cos(%beta*t-(k_x*x+k_y*y+k_z*z)); Q_z: Q_3*cos(%beta*t-(k_x*x+k_y*y+k_z*z)); (%o18) Q 1 cos( k z z+k y y +k x x t) (%o19) Q 2 cos( k z z+k y y +k x x t) (%o20) Q 3 cos( k z z+k y y +k x x t)
4 380(4).wxm 4 / 8 (%i21) feval(q_x,q_y,q_z,omega_x,omega_y,omega_z,omega_0); E15a: Q 3! y sin( k z z+k y y +k x x t) Q 2! z sin( k z z+k y y +k x x t ) =! 0 Q 2 k z sin( k z z+k y y +k x x t) Q 3 k y sin( k z z+k y y +k x x t ) E15b: Q 1! z sin( k z z+k y y +k x x t) Q 3! x sin( k z z+k y y +k x x t ) =! 0 Q 3 k x sin( k z z+k y y +k x x t) Q 1 k z sin( k z z+k y y +k x x t ) E15c: Q 2! x sin( k z z+k y y +k x x t) Q 1! y sin( k z z+k y y +k x x t ) =! 0 Q 1 k y sin( k z z+k y y +k x x t) Q 2 k x sin( k z z+k y y +k x x t ) E19: Q 3 k y sin( k z z+k y y +k x x t) Q 3! y cos( k z z+k y y +k x x t ) =Q 2 k z sin( k z z+k y y +k x x t ) +Q 2! z cos( k z z+k y y +k x x t ) E20: Q 1 k z sin( k z z+k y y +k x x t) Q 1! z cos( k z z+k y y +k x x t ) =Q 3 k x sin( k z z+k y y +k x x t ) +Q 3! x cos( k z z+k y y +k x x t ) E21: Q 2 k x sin( k z z+k y y +k x x t) Q 2! x cos( k z z+k y y +k x x t ) =Q 1 k y sin( k z z+k y y +k x x t ) +Q 1! y cos( k z z+k y y +k x x t ) E23: Q 1 y! z z! y cos( k z z+k y y +k x x t ) +Q 2 z! x x! z cos( k z z+k y y +k x x t ) +Q 3 x! y y! x cos( k z z+k y y +k x x t ) =! x Q 2 k z sin( k z z+k y y +k x x t) Q 3 k y sin( k z z+k y y +k x x t ) +! y Q 3 k x sin( k z z+k y y +k x x t) Q 1 k z sin( k z z+k y y +k x x t ) +! z Q 1 k y sin( k z z+k y y +k x x t) Q 2 k x sin( k z z+k y y +k x x t ) (%o21)
5 380(4).wxm 5 / 8 (%i28) ratsimp(r1/sin(k_z*z+k_y*y+k_x*x-%beta*t)); ratsimp(r2/sin(k_z*z+k_y*y+k_x*x-%beta*t)); ratsimp(r3/sin(k_z*z+k_y*y+k_x*x-%beta*t)); ratsimp(r4); ratsimp(r5); ratsimp(r6); ratsimp(r7); (%o22) Q 3! y Q 2! z =( Q 3 k y Q 2 k z )! 0 (%o23) Q 1! z Q 3! x =( Q 1 k z Q 3 k x )! 0 (%o24) Q 2! x Q 1! y =( Q 2 k x Q 1 k y )! 0 (%o25) Q 3 k y sin( k z z+k y y +k x x t) Q 3! y cos( k z z+k y y +k x x t ) =Q 2 k z sin( k z z+k y y +k x x t ) +Q 2! z cos( k z z+k y y +k x x t) (%o26) Q 1 k z sin( k z z+k y y +k x x t) Q 1! z cos( k z z+k y y +k x x t ) =Q 3 k x sin( k z z+k y y +k x x t ) +Q 3! x cos( k z z+k y y +k x x t) (%o27) Q 2 k x sin( k z z+k y y +k x x t) Q 2! x cos( k z z+k y y +k x x t ) =Q 1 k y sin( k z z+k y y +k x x t ) +Q 1! y cos( k z z+k y y +k x x t ) (%o28) (Q 1 y! z Q 2 x! z Q 1 z! y +Q 3 x! y +Q 2 z! x Q 3 y! x ) cos( k z z+k y y +k x x t ) = ( Q 1 k y Q 2 k x )! z +( Q 3 k x Q 1 k z )! y +( Q 2 k z Q 3 k y )! x sin( k z z+k y y +k x x t) 2.2 Q (with 2 osc. components), omega=const. (%i29) kill(omega_0,omega_x,omega_y,omega_z,q_x,q_y,q_z,phi,q_0); (%o29) one (%i30) epens([omega_0],[t],[omega_x,omega_y,omega_z,q_x,q_y,q_z,phi], [x, (%o30) [! 0 ( t ),! x ( x,y,z ),! y ( x,y,z ),! z ( x,y,z ),Q x ( x,y,z ),Q y ( x,y,z ), Q z ( x,y,z ),Phi( x,y,z )] (%i33) Q_x: Q_1*cos(%beta*t-(k_x*x+k_y*y)); Q_y: Q_2*sin(%beta*t-(k_x*x+k_y*y)); Q_z: Q_3; (%o31) Q 1 cos( k y y +k x x t) (%o32) Q 2 sin( k y y +k x x t) (%o33) Q 3 (%i36) omega_x: 0; omega_y: 0; omega_z: kappa; (%o34) 0 (%o35) 0 (%o36)
6 380(4).wxm 6 / 8 (%i37) feval(q_x,q_y,q_z,omega_x,omega_y,omega_z,omega_0); E15a: Q 2 cos( k y y +k x x t ) =0 E15b: Q 1 sin( k y y +k x x t ) =0 E15c: 0=! 0 Q 1 k y sin( k y y +k x x t) Q 2 k x cos( k y y +k x x t ) E19: 0= Q 2 sin( k y y +k x x t ) E20: Q 1 cos( k y y +k x x t ) =0 E21: Q 2 k x cos( k y y +k x x t ) =Q 1 k y sin( k y y +k x x t ) E23: 0= Q 1 k y sin( k y y +k x x t) Q 2 k x cos( k y y +k x x t ) (%o37) (%i44) E1: ratsimp(r1/cos(k_y*y+k_x*x-%beta*t)); E2: ratsimp(r2/sin(k_y*y+k_x*x-%beta*t)); E3: ratsimp(r3/sin(k_y*y+k_x*x-%beta*t)); E4: ratsimp(r4/sin(k_y*y+k_x*x-%beta*t)); E5: ratsimp(r5/cos(k_y*y+k_x*x-%beta*t)); E6: ratsimp(r6/sin(k_y*y+k_x*x-%beta*t)); E7: ratsimp(r7/sin(k_y*y+k_x*x-%beta*t)); (%o38) Q 2 =0 (%o39) Q 1 =0 (%o40) 0= Q 1 k y! 0 sin( k y y +k x x t) Q 2 k x! 0 cos( k y y +k x x t) sin( k y y +k x x t) (%o41) 0= Q 2 (%o42) Q 1 =0 (%o43) Q 2 k x cos( k y y +k x x t) =Q 1 k sin( k y y +k x x t) y (%o44) 0= Q 1 k y sin( k y y +k x x t) Q 2 k x cos( k y y +k x x t) sin( k y y +k x x t) (%i45) solve([e3,e2,e1], [Q1,Q2,Q3]); (%o45) [] (%i46) solve([e1,e2,e7], [k_x,k_y,k_z]); (%o46) [] 2.3 general Q, omega=const. (%i47) kill(omega_0,omega_x,omega_y,omega_z,q_x,q_y,q_z,phi,q_0); (%o47) one (%i48) epens([omega_0],[t],[omega_x,omega_y,omega_z,q_x,q_y,q_z,phi], [x, (%o48) [! 0 ( t ),! x ( x,y,z,t ),! y ( x,y,z,t ),! z ( x,y,z,t ),Q x ( x,y,z,t ), Q y ( x,y,z,t ),Q z ( x,y,z,t ),Phi( x,y,z,t )]
7 380(4).wxm 7 / 8 (%i51) omega_x: 0; omega_y: 0; omega_z: kappa; (%o49) 0 (%o50) 0 (%o51) (%i52) feval(q_x,q_y,q_z,omega_x,omega_y,omega_z,omega_0); E15a: t Q y = y Q z z Q y! 0 E15b: t Q x = z Q x x Q z! 0 E15c: 0= x Q y y Q x! 0 E19: y Q z =Q y z Q y E20: z Q x Q x = x Q z E21: x Q y = y Q x E23: 0= x Q y y Q x (%o52) 2.4 Aitionally: Q_x = Q_y = f(t) only (%i55) Q_x: q*sin(%beta*t); Q_y: q*cos(%beta*t); Q_z: a(t)*x+b(t)*y; (%o53) q sin( t) (%o54) q cos( t) (%o55) b( t) y +a( t) x (%i56) feval(q_x,q_y,q_z,omega_x,omega_y,omega_z,omega_0); E15a: q sin( t ) =! 0 b( t ) E15b: q cos( t ) =! 0 a( t ) E15c: 0=0 E19: b( t ) = q cos( t ) E20: q sin( t ) = a( t ) E21: 0=0 E23: 0=0 (%o56) (%i57) Omega: curl([q_x,q_y,q_z])-cross([0,0,kappa],[q_x,q_y,q_z]); (%o57) [ q cos( t ) +b( t ), q sin( t) a( t ),0] (%i58) ev(omega, [a(t)=kappa*q*cos(%beta*t), b(t)=-kappa*q*sin(%beta*t)]); (%o58) [ q cos( t) q sin( t ), q sin( t) q cos( t ),0]
8 380(4).wxm 8 / 8 (%i59) Omega: (-sqrt(2)*kappa*q*[sin(%beta*t-%pi/4), sin(%beta*t+%pi/4), 0] (%o59) [ p 2 q sin t 4, p 2 q sin t+ 4,0]
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396(1).wxm 1 / 14 (%i1) kill(all); (%o0) done 1 Definitions (%i1) cross(a,b) := [a[]*b[3] - a[3]*b[], a[3]*b[1] - a[1]*b[3], a[1]*b[] - a[]*b[1]]; (%o1) cross( a,b ):=[a b 3 a 3 b,a 3 b 1 a 1 b 3,a 1 b
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