Dzhanibekov Effect

kinetic energy,

E

k

0.25

time, t

-100

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Ek<Ecr

Ek=Ecr

Ek>Ecr

This Demonstration illustrates the Dzhanibekov effect, discovered on a Russian space station in 1985. Later on it was rediscovered as an unexpected behavior of a tennis racket flipped into the air. When

E

k

≈

2

L

/2

I

2

for the kinetic energy of an asymmetric top, the top can exhibit periodic flipping. Here

L

is the angular momentum and

I

2

is the intermediate moment of inertia. In this Demonstration, the moments of inertia are chosen to be 1, 2, 3, and the magnitude of the angular momentum vector

L

is normalized to 1. Flipping occurs when the kinetic energy

E

k

is very close to 0.25. When

E

k

is exactly equal to 0.25, there is just one flip, here around

t=0

. The simulation is based on exact solutions of the Euler and Arnold equations for the angular momentum and the attitude matrix, using Mathematica's built-in elliptic functions JacobiSN, JacobiCN, JacobiDN, JacobiAmplitude and EllipticPi.