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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2024

Paper 4

A <central force> conserves <angular momentum>
$$
\ell=mr^2\dot\theta.
$$
Put $u=1/r$. Since $\dot\theta=\ell u^2/m$,
$$
\dot r=-\frac{\ell}{m}u',
\qquad
\ddot r=-\frac{\ell^2}{m^2}u^2u'',
\qquad
r\dot\theta^2=\frac{\ell^2}{m^2}u^3,
$$
where primes now denote <differentiation> with respect to $\theta$. Substitution into the radial equation gives the <Binet equation>
$$
\boxed{
u''+u
=\frac{m}{\ell^2u^2}
\frac{dV}{dr}\bigg|_{r=1/u}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 4 / 10C / i / Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 4 / 10C / i / Solution

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Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 4 / 10C / i / Solution

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