How many degrees of freedom does an asymmetric top have if it is rotating about a fixed point?What are the generalised coordinates used then?
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$\begingroup$ Is this top spinning with the influence of external forces like gravity? Is this purely rotational motion without any linear movement of the center of mass? $\endgroup$– Stevan V. SabanCommented May 22, 2022 at 16:24
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4 Answers
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I would say three degrees of freedom, and you can use, e.g., the Euler angles (http://en.wikipedia.org/wiki/Euler_angles ).
answered Aug 23, 2014 at 12:03
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In general, for a force free asymmetric top, it requires 3 principal moments of inertia, 3 angular velocities and conservation of the angular momentum and kinetic energy.
You need to specify 6 quantities to define the problem and you have 2 constraints.
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Here's my attempt. A rigid top can have a maximum of 6 DOFs. If the top is rotating about a fixed point, its center of mass cannot be moved vertically up or down (note vertical is defined by the direction of gravity), horizontally sideways or front or back, since in each case the point about which the top rotates will change, which is not allowed. This leaves us with rotation about each axis (vertical, horizontal and the last one perpendicular to both) being the only possible ones. The rotation about the vertical seems pretty obvious since it literally goes around in closed paths about the vertical. It will also rotate about the other two axes although the motion although not in closed paths (since it can't move through the floor). So, 3 DOFs.
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Let's work from the first principles. Assume that the top has $N$ particles. Choose the any one of them, say the one at the point at which the top is fixed. Let's call it $P$. That point can be described by three coordinates, which make up the three degrees of freedom. Choose the next particle, $Q$. Being a rigid body, the distance between $P$ and $Q$ is fixed. Therefore $Q$ can be described by two angles on a sphere centred at $P$. This takes our degrees of freedom to five. Now choose a third point, say $R$. Its distance from the previous two points is a constant. Therefore it can be described by one angle around a circle perpendicular to the line joining $P$ and $Q$. This adds the sixth degree of freedom. Once you know these, the rigid body constraint fixes the position of all the remaining particles.
Thus, in general, three Cartesian coordinates and three angles suffice to describe the motion of a rigid body.
In the case of a top fixed at one point, the three Cartesian coordinates are no longer needed. They do not appear as a quadratic term in the Lagrangian. Therefore we are left with just three generalised coordinates. They are all angles. We usually use the Euler angles as the generalised coordinates.
