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CURVE WITH CONSTANT REACTION, L'HOSPITAL QUINTIC
The force applied by the marble on the support is constant
Problem of the "curva aequabilis pressionis" posed by
Jean Bernoulli in 1695, solved by L'Hospital in 1700.
Other names: curve with constant pressure, "loop the loop" curve 
Cartesian parametrization:
where .
The kinematic equation of motion is given by: . Curvilinear abscissa and radius of curvature: . 
For k = 1 (case of the L'Hospital quintic) : Cartesian parametrization: (). Cartesian equation: or . Polynomial quintic. Vertex (0, a/4); isolated point ; double point . Curvilinear abscissa: . Radius of curvature: . Parametrization of the evolute: . 
A curve with constant reaction is a curve such
that if a particle descends along it by the pull of gravity (the gravitational
field is supposed to be uniform), then the reaction of the curve on the
particle has a constant intensity; conversely, the force applied by the
particle on the curve has a constant intensity.

With Oy as the descending vertical, Newton's law,
projected on the normal, can be written: ,
where R is the intensity of the reaction force of the support, supposed
to be constant (see the notations).
The conservation of energy can be written: ,
with .
Writing the ratio of the reaction and the weight of the particle ,
we get the differential equation of the curve: .
Solving it (use )
gives the parametrization above.

In the case where k > 1, the curve is transcendental, ( can be integrated thanks to inverse trig functions); the curve is periodic. 


In the case where k = 1, the curve is algebraic. 

In the case where k < 1, the curve is transcendental again ( can be integrated thanks to inverse hyperbolic functions). 
The evolute of such a curve is the solution of the following
problem (also posed by Jean Bernoulli): determining a curve on which to
wind the wire of a pendulum so that the tension of the wire of this pendulum
remains constant.

For other curves of motion of a massive point in a gravitational field under certain conditions, see isochronous, brachistochrone and tautochronous.
Is this double looptheloop in Adventureland a curve with constant reaction?
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© Robert FERRÉOL 2017