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Today I read a book in mechanics and encountered a funny proposition about rigid body with fixed point. Perhaps somebody will be interested to propose it to students as a task. This proposition is almost correct:)
Consider a rigid body with a fixed point . Let be a coordinate frame connected with this rigid body. Consider a unit sphere with center at as well. Now let us move the body from the initial position such that the axis $Oz$ describes a closed curve (without self-crossings) on the sphere and the projection of body's angular velocity on is equal to zero identically. It turns out that when the axis comes to the initial position other two axes will be rotated relative their initial position. The angle of rotation equals (up to the sign) the area of a figure drawn by the axis on the sphere.
Consider a rigid body with a fixed point
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