A ball of mass m and negligible ra- dius can slide freely and without fric- tion around a torus of Mass M radius a, at an angular velocity 6'. At the same time the torus spinning around a vertical axis passing through it's cen- ter at an angular velocity o'. Given the moment of inertia of the torus around the vertical axis is equal to Ma? and the zero potential energy reference passes through the center of the torus as shown in the figure.
A ball of mass m and negligible ra- dius can slide freely and without fric- tion around a torus of Mass M radius a, at an angular velocity 6'. At the same time the torus spinning around a vertical axis passing through it's cen- ter at an angular velocity o'. Given the moment of inertia of the torus around the vertical axis is equal to Ma? and the zero potential energy reference passes through the center of the torus as shown in the figure.
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Transcribed Image Text:Hamiltonian
A ball of mass m and negligible ra-
dius can slide freely and without fric- Torus (mass M)
tion around a torus of Mass M radius
ball mass (m)
asin(0)
a, at an angular velocity 0'. At the
same time the torus spinning around a
vertical axis passing through it's cen-
ter at an angular velocity o'. Given
the moment of inertia of the torus
a
E, = 0
around the vertical axis is equal to
Ma? and the zero potential energy
reference passes through the center of
the torus as shown in the figure.
(a) Knowing that the lagrangian of the system is:
1
L =
= =ma (0^ + sin° (0)6) +(Ma²)d? – mgacos(0)
Find the generalized momenta
(b) Determine the Hamiltonian function
(c) Derive the set of canonical equations of motion
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