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Degrees of freedomIn two dimensions ω is always directed orthogonalto the plane and therefore can be considered to be ascalar. One angle is therefore enough to describe allrotations. One consequently has three degrees offreedom: 2 translational and 1 rotational.In three dimensions ω is a three dimensionalvector and one needs three correspondinggeneralized coordinates (angles), so one has sixdegrees of freedom: 3 translational and 3 rotational.75Two dimensionsf t2I r ()r dA• rI TAmoment of inertiacenter of massTAf () r rdAttorqueequation of motion for rotation :76
Two dimensionsTime evolution of the rotation angle φusing the Verlet algorithm:2 ()t t 2 t t t t TtIwithjATt () f y () td x () t f x () td y () tj j j jώ = angularaccelerationx-component of thevector connecting the center of mass to the mass element j77Two dimensionsTime evolution of the rigid bodyusing the Verlet algorithm: xtt xt xttt M f t2 12j( )jA tt t tt t I T t2 12 ( )78
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