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We are interested in a Newtonian Hamiltnian system, H(x) = (1/2)y^2 + G(x), started at the origin. This equation can be written dx/dt = y, dy/dt = -g(x), where G is a primitive of g. The origin is surrounded by a continuous family of periodic orbits. Each orbit lies on an energy level, H(x,y) = c, and is uniquely determined by c. The period function T(c) is the minimal period of this orbit. ... Many author shave given conditiions on G and its derivatives for the period of the orbit to be monotone or to have critical points. Some of them give sufficient conditions for the period function to be monotonic, including Chicone [2], Chow and Wang [3], Rothe [4], and Shaaf [5]. Accordingly, we consider the following technical hypothesis: - There exist two nubers a < 0 < b such that 0
- G(0) = g(0) = 0.
- xg(x) > 0 if x <> 0, x \in (a, b).
- g'(0) > 0.
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