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CURVE OF THE SPHERICAL PENDULUM
Curve studied by Clairaut in 1735, Lagrange (mécanique
analytique), and Puiseux
in 1842.
Other name: curve of the conical pendulum. See: Paul Appell : cours de mecanique rationnelle, page 530. Wikipedia article |
Differential equation of the motion (derived from
Newton's second law):
where q = longitude, j = colatitude and First integral: Spherical differential equation of the curve: i.e., with where note the tiny difference with the spherical catenary, for which |
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The curve of the spherical pendulum is the curve described
by the end of a simple massive pendulum attached to a fixed point, that
can move in three dimensions, and placed in a uniform gravitational field
(here ).
This curve is traced on a sphere, and is none other than a flow line of this sphere: it can be physically obtained by making a ball roll inside a sphere.
Like the case of the spherical catenaries, we get curves composed of a sequence of undulations joining alternatively two parallels (obtained for the values at which the polynomial P above cancels), and images of one another by rotations around Oz. The curve is either closed, or dense in the zone between the two parallels.
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These acrobat
bikers of the Shanghai circus describe such curves.
These curves can be generalized by considering the Coriolis
force, which give the following differential equation of the motion: .
When the pendulum is dropped without an initial speed,
we get the curve of the Foucault pendulum, which, for small oscillations,
can be approached by a hypocycloid.
Without the Coriolis force, the curve would amount to
an arc of a circle.
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Here is a view of the curve described by a pendulum in forced uniform rotation around Oz.
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See the spinning top
festoons, which are another generalization of the curves of the spherical
pendulum.
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© Robert FERRÉOL 2018