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14. Motion in a central field

On reducing the two-body problem to one of the motion of a single body, we arrive at the problem of determining the motion of a single particle in an external field such that its potential energy depends only on the distance r from some fixed point. This is called a central field. The force acting on the particle is \v{F} = \partial U(r)/\partial \v{r} = - (\dd{U}/\dd{r})\v{r}/r; its magnitude is likewise a function of r only, and its direction is everywhere that of the radius vector.

As has already been shown in 1/9, the angular momentum of any system relative to the centre of such a field is conserved. The angular momentum of a single particle is \v{M} = \v{r}\times\v{p}. Since \v{M} is perpendicular to \v{r}, the constancy of \v{M} shows that, throughout the motion, the radius vector of the particle lies in the plane perpendicular to \v{M}.

Thus the path of a particle in a central field lies in one plane. Using polar co-ordinates \v{r}, in that plane, we can write the Lagrangian as

1/14.1

see 1/4.5. This function does not involve the co-ordinate \phi explicitly. Any generalised co-ordinate q_i which does not appear explicitly in the Lagrangian is said to be cyclic. For such a co-ordinate we have, by Lagrange's equation, (\dd{}/\dd{t}) \partial L/\partial \dot{q}_i = \partial L/\partial q_i = 0, so that the corresponding generalised momentum p_i = \partial L/\partial \dot{q}_i is an integral of the motion. This leads to a considerable simplification of the problem of integrating the equations of motion when there are cyclic co-ordinates.

In the present case, the generalised momentum p_\phi=mr^2\dot{\phi} is the same as the angular momentum M_z=M (see 1/9.6), and we return to the known law of conservation of angular momentum:

1/14.2

This law has a simple geometrical interpretation in the plane motion of a single particle in a central field. The expression \mfrac{1}{2}r\cdot r\dd{\phi} is the area of the sector bounded by two neighbouring radius vectors and an element of the path 1/fig8