THE TRACTRIX AND THE PENDULUM Exercises ... - The Live Toad

THE TRACTRIX AND THE PENDULUM 3We can integrate this once by first multiplying both sides of the differentialequation by θ ′ . When we do we obtain12 (θ′ ) 2 = C + g L cos(θ).(d) Interpret the constant of integration C above.(e) If we use Leibniz notation θ ′ = dθ/dt and separate variables in thedifferential equation above then we find that∫dθt = √ ,a + b cos(θ)where a and b are certain constants. (What are they?)(f) The integral above cannot be evaluated using only standard functions(polynomials, radicals, exponentials, logarithms, trigonometric functions,etc.). It defines a new kind of of function, called an ellipticfunction, in the same way that the integral of 1/u defines log(u). Weexhibit the connection with elliptic curves as follows. If we substitutex = − cos(θ) then we can rewrite the integral above in the form∫ dxywhere y 2 = f(x) and f(x) is some cubic in x. Work out the details.University of Toledo