next curve  previous curve  2D curves  3D curves  surfaces  fractals  polyhedra 
ISOCHRONOUS CURVE OF LEIBNIZ
Curve studied by Leibniz in 1687 and by Jacques Bernoulli in 1690.
Wilhelm Gottfried Leibniz (1646  1716): German philosopher and scholar. 

With Ox as the descending vertical axis and a starting point at O with zero horizontal speed, the conservation of energy gives , i.e.
since .
Hence the kinematics equations of motion: that can be integrated to: , hence the equation of the trajectory: . 
An isochronous curve of Leibniz is a curve such that if a particle comes down along it by the pull of gravity, the vertical component of the speed is constant, when the gravitational field is supposed to be uniform.
The solution is a semicubical parabola, as shown in the above derivation. Note that the initial speed at the vertex of the parabola must be equal to for the semicubical parabola to be isochronous; the vertical component is then a constant equal to (and the horizontal component is proportional to the square root of time).
In this derivation, the acceleration vector of the gravity force was supposed to be constant; if it is only supposed to have constant norm, but is oriented in the direction of a point at finite distance, then we get the curves called isochronous curves of Varignon.
For other curves of motions of a massive point in a gravitational field subject to certain conditions, see the
isochronous curves of Huygens, the paracentric isochronous curve, the tautochronous curves, the synchronous curves and the curve of constant reaction.
next curve  previous curve  2D curves  3D curves  surfaces  fractals  polyhedra 
© Robert FERRÉOL 2017