How to solve \(\displaystyle\ddot{{\vec{{{u}}}}}=\vec{{{u}}}\times\hat{{{k}}}\)

Destinee Hensley

Destinee Hensley

Answered question

2022-03-29

How to solve
u¨=u×k^

Answer & Explanation

Matronola3zw6

Matronola3zw6

Beginner2022-03-30Added 10 answers

Step 1
You should perhaps more explicitly mention that k is the third canonical basis vector.
Then it is immediately clear that the right side is zero in z direction and acts as a 90 rotation in the xy-plane,
(xi+yj)×k=-xj+yi.
This rotation has 2 square roots, the rotations by 45 and by 135.
Interpreting the xy-plane as complex plane, one gets
x¨+iy¨
=i(x+iy)
=121i2(x+iy)
so that
x+iy=c1e1i2t+c2e1+i2t,c1,2BC

Abdullah Avery

Abdullah Avery

Beginner2022-03-31Added 19 answers

Step 1
So writing u=(ui), we have
u¨1=u2, u¨2=u1, u¨3=0
Hence, u3(t)=at+b for some constants a,b, the other two equations give us
ddu¨1=u1
This has the characteristic equation λ4+1=0, giving us
λ1,2=1±i2,
λ3,4=1±i2
Hence, we have
u1(t)=exp(t2)(csin(t2)+dcos(t2))
+exp(t2)(esin(t2)+fcos(t2))
for some constants c,d,e,f. By using u2=u¨1, you can calculate the missing component.

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