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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 14E / a / Solution

Write

r_{1} = (x_{1}, y_{1})

,

r_{2} = (x_{2}, y_{2})

,

ρ_{1} = ∣ r_{1} ∣

, and

ρ_{2} = ∣ r_{2} - r_{1} ∣

. Then

L = \frac{m _{1}}{2} ∣ \overset{r}{˙}_{1} ∣^{2} + \frac{m _{2}}{2} ∣ \overset{r}{˙}_{2} ∣^{2} - \frac{k _{1}}{2} (ρ_{1} - ℓ)^{2} - \frac{k _{2}}{2} (ρ_{2} - ℓ)^{2} - m_{1} g y_{1} - m_{2} g y_{2} .

With

e_{y} = (0, 1)

, the Euler--Lagrange equations are

m_{1} \overset{r}{¨}_{1} = - k_{1} (1 - \frac{ℓ}{ρ _{1}}) r_{1} + k_{2} (1 - \frac{ℓ}{ρ _{2}}) (r_{2} - r_{1}) - m_{1} g e_{y},

m_{2} \overset{r}{¨}_{2} = - k_{2} (1 - \frac{ℓ}{ρ _{2}}) (r_{2} - r_{1}) - m_{2} g e_{y} .

Solved by gpt-5.6-sol high.

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