This problem has two independent parts.

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Gravitation and central-force motion · Gravitation and central-force motion

Action and Symmetry, Ch The Kepler problem redone: reduction, quadrature and the Runge-Lenz vector

Problem

This problem has two independent parts. ((a)) A particle of reduced mass μ\mu moves in V(r)=−k/rV(r) = -k/r (k>0k>0) about a fixed force centre, with angular momentum ℓ\ell and Laplace-Runge-Lenz vector ALRL=p×L−μkr^\mathbf{A}_{\rm LRL} = \mathbf{p}\times\mathbf{L} - \mu k\hat{\mathbf{r}}, known to be constant along the orbit. Let φ\varphi be measured from the fixed direction of ALRL\mathbf{A}_{\rm LRL}, and write A=∣ALRL∣A = |\mathbf{A}_{\rm LRL}|. By evaluating ALRL⋅r\mathbf{A}_{\rm LRL}\cdot\mathbf{r} two different ways (once geometrically, once using a⋅(b×c)=(a×b)⋅c\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c}) = (\mathbf{a}\times\mathbf{b})\cdot\mathbf{c}), derive r(φ)r(\varphi) without carrying out any integration, and give the eccentricity ee in terms of AA, μ\mu and kk. ((b)) For a system of particles of masses mim_i, positions ri\mathbf{r}_i and momenta pi\mathbf{p}_i, subject to forces Fi\mathbf{F}_i, define D=∑ipi⋅riD = \sum_i \mathbf{p}_i\cdot\mathbf{r}_i. Derive D˙=2T+∑iFi⋅ri\dot D = 2T + \sum_i \mathbf{F}_i\cdot\mathbf{r}_i, state precisely the condition on the motion that lets the long-time average ⟨D˙⟩\langle \dot D\rangle be set to zero, and use this to derive 2⟨T⟩=−⟨V⟩2\langle T\rangle = -\langle V\rangle for a system interacting only by mutual gravity.

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