 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2018/9/b/solution

M = (0 - α α 0), C = (1 i 1 - i), C^{- 1} MC = (i α 0 0 - i α) .

The columns of

C

M

v^{'} = C^{- 1} v

therefore gives

v_{1}^{'} = (v_{1} - i v_{2}) /2

v_{2}^{'} = (v_{1} + i v_{2}) /2

\overset{v}{˙}_{1}^{'} = i α v_{1}^{'}, \overset{v}{˙}_{2}^{'} = - i α v_{2}^{'}, v_{1}^{'} (t) = K_{+} e^{i α t}, v_{2}^{'} (t) = K_{-} e^{- i α t} .

The initial velocity gives

K_{+} = - i v_{0} /2

K_{-} = i v_{0} /2

. Transforming back,

v = C v^{'}

v_{1} (t) = v_{0} sin α t, v_{2} (t) = v_{0} cos α t .

These also cover

α = 0

. Complex coordinates have decoupled a real rotation without making the physical velocity complex.

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