A light body (mass ) moves under the gravity of two primaries of masses and that stay…

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Rotation, rigid bodies and accelerated frames · Gravitation and central-force motion

Action and Symmetry, Ch Perturbed orbits, precession and the three-body problem

Problem

A light body (mass

m\to0

) moves under the gravity of two primaries of masses

m_1

and

m_2

that stay fixed on the

x

axis of a frame rotating uniformly at rate

\Omega

about the

z

axis through their common centre of mass;

r_1

and

r_2

are the light body's distances to

m_1

and

m_2

. ((a)) Writing

\mathbf r'=(x,y,z)

for the light body's position in the rotating frame, use

(\mathrm d\mathbf u/\mathrm dt)_{\rm space}=(\mathrm d\mathbf u/\mathrm dt)_{\rm frame}+\mathbf\Omega\times\mathbf u

to write its velocity in the non-rotating frame in terms of

\dot x,\dot y,\dot z

and

\Omega

, and hence write its Lagrangian per unit mass

L

in the rotating frame. ((b)) Show that

L

has no explicit time dependence, so the Jacobi integral

h=\sum_k\dot q_k\,\partial L/\partial\dot q_k-L

is conserved, and evaluate

h

to show it equals

C_{\rm J}=\tfrac12|\dot{\mathbf r}'|^2+\Phi_{\rm eff}

with

\Phi_{\rm eff}=-G_{\rm N}m_1/r_1-G_{\rm N}m_2/r_2-\tfrac12\Omega^2(x^2+y^2)

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