Torque-free motion of a symmetric top
The simulation displays the evolution of a torque-free ellipsoid around
its center of mass. The Lagrangian is as follows:
L = ½ Ix
(ωx2 + ωy2)
+ ½ Iz ωz2 ,
where
the components of the angular velocity in the body frame are
ωx
= φ' sin θ sin ψ + θ' cos ψ
ωy
= φ' sin θ cos ψ - θ' sin ψ
ωz
= φ' cos θ + ψ'
(we use ' for the
time derivative).
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To see the definition of the Euler angles ( θ,φ,ψ),
click on Variables in the simulation below.
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There you can also select the initial values (since φ
and ψ are cyclic, there is no
restriction in taking φ0
= ψ0
= 0), as well as the quotient of principal moments of
inertia Iz/Ix.
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Units are arbitrary.
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Check Velocities to see the evolution of the angular velocity
components in the body frame (ωx,ωy,ωz)
and in the space frame (ω'x,ω'y,ω'z),
where.
ω'x = θ' cos φ
+ ψ' sin φ sin θ
ω'y
= θ' sin φ - ψ' cos φ
sin θ
ω'z = φ' + ψ'
cos θ
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The point of view of the three-dimensional projections can be changed
with the cursors or the mouse.
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Each whole image can be moved with the mouse while pressing Ctrl.
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To change the zoom in the projections, press Shift
when moving up or down the mouse pointer.
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Put the mouse pointer over an element to get information about it.
Reference: H. Goldstein, Ch. Poole and J. Safko, Classical
Mechanics, 3rd edition, Addison-Wesley, San Francisco
Activities
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Press Start and get familiarized with the display options below
the images.
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Discuss the evolution of the angular momentum and velocity in the
space frame and in the body frame. (it may be helpful to enable a
combination of Velocities, Axes, Vectors and Trajectories.
This is an English translation of the Basque original for a course
on mechanics, oscillations and waves.
It requires Java
1.5 or newer and was created by Juan
M. Aguirregabiria with Easy
Java Simulations (Ejs) by Francisco
Esquembre. I thank Wolfgang Christian and Francisco Esquembre for
their help.