Mathematical Physics Vol 1

5.10 Examples

297

Thus

2 ) − xE

2 )= E

E 2 α ( x

2 α , α + 1 ( x

α ( − x ) .

which was to be proved.

Elliptic functions

Problem 214 Determine the final equations of motion of a heavy point M , ofmass m , along a smooth circular vertical fixed line - the mathematical pendulum.

Figure 5.11: Mathematical pendulum.

Solution Motion of a heavy point along a vertical fixed circular line, without friction, is called the mathematical pendulum (Fig. 5.11). The pendulum moves in a vertical plane, say x − z , with downward direction of the z -axis (direction of action of gravitational force). In this case it is convenient to use the polar coordinate system, where the coordinate origin is set at the centre of the circle. Note first that the position of the point is determined by the angle ϕ , because the point is always at the same distance R from the coordinate origin. In order to solve this problem, we will use the law of change of kinetic energy d A = d E k , where d A = ∑ i F i d r i is the work element, F is the force, and E k = 1 2 mv 2 the kinetic energy. In our case, two forces act on the point: gravity G = mg k and the reaction resulting from R = R n n (smooth line, so there is no reaction in the direction of the tangent), and thus d A = G · r + R · r = mg k · ( d x i − d z k )+ R n n · d r t = − mg d z ⇒ A = Z z R mg d z = mg ( z − R ) .