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PARACENTRIC CURVE

Problem posed by Leibniz in 1689 and solved by Jacques Bernoulli in 1694. |

With Oy as the descending vertical axis, the conservation of energy gives ,
and the condition associated to the problem can be written , hence the differential equation of the curve: .
In the special case where , this equation simplifies to , which gives the following polar equation: for , with . Kinematic equations of motion: . Polar parametrization: , which gives, using the elliptic functions of Jacobi, . |

A paracentric curve is a curve such that if a particle descends along it by the pull of gravity, the distance between the particle and a fixed point O is proportional to time, when the gravitational field is supposed to be uniform; in other words, the particle moves away from (or comes closer to) O at constant speed. |
When the particle descends along the red curve, the length of the black segment line increases at constant speed. |

Only a special case was solved above; in the other cases, the curve is a spiral around *O*.

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© Robert FERRÉOL
2017