International Scholarly Research Notices

Research Article

Rigid Body Trajectories in Different 6D Spaces

Table 1

Velocity functions for the various groups.
ℝ 3 ⋉ 𝑆 𝑂 ( 3 ) 𝑆 𝑂 ( 4 ) 𝑆 𝐸 ( 3 ) 𝑆 𝑂 ( 1 , 3 )
𝑣 1 ( 𝑡 ) 𝑣 𝑥 𝑣 𝑥 𝑣 𝑥 𝑣 𝑥
𝑣 2 ( 𝑡 ) 𝑣 𝑦 c o s ( 𝑓 ) 𝑣 𝑦 c o s ( 𝑤 4 𝑡 + 𝑓 ) 𝑣 𝑦 c o s ( 𝑤 5 𝑡 + 𝑓 ) 𝑣 𝑦 c o s ( 𝑤 6 𝑡 + 𝑓 )
𝑣 3 ( 𝑡 ) 𝑣 𝑦 s i n ( 𝑓 ) 𝑣 𝑦 s i n ( 𝑤 4 𝑡 + 𝑓 ) 𝑣 𝑦 s i n ( 𝑤 5 𝑡 + 𝑓 ) 𝑣 𝑦 s i n ( 𝑤 6 𝑡 + 𝑓 )
𝑤 1 ( 𝑡 ) 𝑤 𝑤 𝑤 𝑤
𝑤 2 ( 𝑡 ) 0 0 0 0
𝑤 3 ( 𝑡 ) 0 0 0 0
𝑤 4 = ( 1 − 𝑜 1 / 𝑚 ) 𝑤 𝑤 5 = 𝑤 𝑤 6 = ( 1 + 𝑜 1 / 𝑚 ) 𝑤