Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 316 / 3 / v

Solution

Let

X = x - x_{L_{3}}

and

Y = y

U = \dot{X}

W = \dot{Y}

\ddot{X} - 2 \dot{Y} = F_{x} X + F_{y} Y, \ddot{Y} + 2 \dot{X} = G_{x} X + G_{y} Y,

where every derivative is evaluated at

L_{3}

ξ = (X, Y, U, W)^{T}

, this becomes

\dot{ξ} = A ξ, A = 00 F_{x} G_{x} 00 F_{y} G_{y} 100 - 2 0120_{L_{3}} .

The question's reuse of

X

for the state vector is only notation; its first two entries are the small displacements

X, Y

, not the absolute coordinates.

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