Relationships and significance of lactate minimum, critical velocity, heart rate deflection and 3 000 m track-tests for running

The running velocities associated to lactate minimum (V-lm), heart rate deflection (V-HRd), critical velocity (CV), 3000 M (V-3000) and 10000 m performance (V-10km) were compared. Additionally the ability of V-lm and VHRd on identifying sustainable velocities was investigated.Methods. Twenty runners (28.5 +/- 5.9 y) performed 1) 3000 m running test for V3000; 2) an all-out 500 in sprint followed by 6x800 m incremental bouts with blood lactate ([lac]) measurements for V-lm; 3) a continuous velocity-incremented test with heart rate measurements at each 200 m for V-HRd; 4) participants attempted to 30 min of endurance test both at V-lm(ETVlm) and V-HRd(ETVHRd). Additionally, the distance-time and velocity-1/time relationships produced CV by 2 (500 m and 3000 m) or 3 predictive trials (500 m, 3000 m and distance reached before exhaustion during ETVHRd), and a 10 km race was recorded for V-10km.Results. The CV identified by different methods did not differ to each other. The results (m(.)min(-1)) revealed that V-.(lm) (281 +/- 14.8)< CV (292.1 +/- 17.5)=V-10km (291.7 +/- 19.3)< V-HRd (300.8 +/- 18.7)=V-3000 (304 +/- 17.5) with high correlation among parameters (P < 0.001). During ETVlm participants completed 30 min of running while on the ETVHRd they lasted only 12.5 +/- 8.2 min with increasing [lac].Conclusion. We evidenced that CV and Vim track-protocols are valid for running evaluation and performance prediction and the parameters studied have different significance. The V-lm reflects the moderate-high intensity domain (below CV), can be sustained without [lac] accumulation and may be used for long-term exercise while the V-HRd overestimates a running intensity that can be sustained for long-time. Additionally, V-3000 and V-HRd reflect the severe intensity domain (above CV).

Peculiar trajectories around the Lagrangian equilateral points

In the present work we analyse the behaviour of a particle under the gravitational influence of two massive bodies and a particular dissipative force. The circular restricted three body problem, which describes the motion of this particle, has five equilibrium points in the frame which rotates with the same angular velocity as the massive bodies: two equilateral stable points (L-4, L-5) and three colinear unstable points (L-1, L-2, L-3). A particular solution for this problem is a stable orbital libration, called a tadpole orbit, around the equilateral points. The inclusion of a particular dissipative force can alter this configuration. We investigated the orbital behaviour of a particle initially located near L4 or L5 under the perturbation of a satellite and the Poynting-Robertson drag. This is an example of breakdown of quasi-periodic motion about an elliptic point of an area-preserving map under the action of dissipation. Our results show that the effect of this dissipative force is more pronounced when the mass of the satellite and/or the size of the particle decrease, leading to chaotic, although confined, orbits. From the maximum Lyapunov Characteristic Exponent a final value of gamma was computed after a time span of 10(6) orbital periods of the satellite. This result enables us to obtain a critical value of log y beyond which the orbit of the particle will be unstable, leaving the tadpole behaviour. For particles initially located near L4, the critical value of log gamma is -4.07 and for those particles located near L-5 the critical value of log gamma is -3.96. (c) 2006 Elsevier B.V. All rights reserved.

A peculiar stable region around Pluto

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)