Analytical dynamics, being a synopsis of leading topics in the analytical theory of dynamics with numerous examples and selections from Newton's Principia and other sources, by Arthur S. Hathaway.

change in the direction of motion, except by collision or sudden
impulse.
13. An orbit in a central field is concave toward an attractive
center, convex towards a repulsive one.
Definition. An apse of an orbit in a central field is a point
*of the orbit which is at a maximum or a minimum distance from
the center. The line joining an apse to the center is an apsidal
line; the distance from the center to an apse is an apsidal distance.
14. An apsidal line is perpendicular to the orbit at its apse;
if it is an axis of symmetry of the field, it is an axis of symmetry
of the orbit (eg., when the acceleration varies as a function of the.distance only). An apse cannot be a point of inflection.
15. If a central field varies as a function of the distance
*only, then the arcs of an orbit between successive apises are equal
and similar, and subtend equal angles at the center; alternate
apsidal distances are equal so that there are not more than two
apsidal distances. The orbit is closed only if the angle between
successive apsidals is commensurable with four right angles.
16. Find the equation for determining apsidal distances.
Ans. If +(r) be the potential at distance r, (HI/r)2+2((r)
-c where c is determined by initial conditions.
17. The periodic time of a planet which is double the earth's
distance from the sun is about 1033 cays.
18. Prove from the hodograph of a planet that its velocity
can be resolved into two constant velocities, c and cc, perpendicular
respectively to the radius vector and the major axis.
19. Find the locus of the empty focus and the center of the
orbit of a planet which is projected from a given point with given'
speed.
20. Find the locus of the apses of the orbit above, in any
central field.
21. Show that there are two directions of projection of' a
planet from a given point, so that it will pass through another
given point.
II. DYNAMICS
Mass is quantity of matter. For homogeneous matter, mass is
proportional to volume; a certain volume is unit mass, and the
mass of a unit volume is density so that mass density times
volume.
A particle is a quantity of matter concentrated in a geometrical
point; it is a fiction which is convenient and almost necessary in
the development of dynamics. Its actual function in the dynamics
of finite bodies will be considered later.

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