Quaternion Solution for the Rock'n'roller: Box Orbits, Loop Orbits and Recession

We consider two types of trajectories found in a wide range of mechanical systems, viz. box orbits and loop orbits. We elucidate the dynamics of these orbits in the simple context of a perturbed harmonic oscillator in two dimensions. We then examine the small-amplitude motion of a rigid body, the rock'n'roller, a sphere with eccentric distribution of mass. The equations of motion are expressed in quaternionic form and a complete analytical solution is obtained. Both types of orbit, boxes and loops, are found, the particular form depending on the initial conditions. We interpret the motion in terms of epi-elliptic orbits. The phenomenon of recession, or reversal of precession, is associated with box orbits. The small-amplitude solutions for the symmetric case, or Routh sphere, are expressed explicitly in terms of epicycles; there is no recession in this case

Preschool hyperactivity specifically elevates long-term mental health risks more strongly in males than females: a prospective longitudinal study through to young adulthood

Evidence of continuities between preschool hyperactivity and adult mental health problems highlight the potential value of targeting early identification and intervention strategies. However, specific risk factors are currently unclear. This large-scale prospective longitudinal study aimed to identify which hyperactive preschoolers are at greatest long-term risk of poor mental health. One hundred and seventy children (89 females) rated as hyperactive by their parents and 88 non-hyperactive controls (48 females) were identified from a community sample of 4,215 3 year-olds. Baseline data relating to behavioral/emotional problems and background characteristics were collected. Follow-up mental health and functional impairment outcomes were collected between 14 and 25 years of age. At age 3 years, males and females in the hyperactive group had similarly raised levels of hyperactivity and other behavior problems. In adolescence/young adulthood, these individuals showed elevated symptoms of ADHD, conduct disorder, mood disorder, anxiety and autism, as well as functional impairment. Preschool hyperactivity was strongly predictive of poor adolescent/adult outcomes for males across domains with effects being specifically driven by hyperactivity. For females, the effects of preschool hyperactivity were smaller and dropped to non-significant levels when other preschool problems were taken into account. Environmental risk factors also differed between the sexes, although these may also have been mediated by genetic risk. In conclusion, these results demonstrate marked sex differences in preschool predictors of later adolescent/adult mental health problems. Future research should include a measure of preschool inattention as well hyperactivity. The findings highlight the potential value of tailored approaches to early identification strategies