## 11 July 2015

### Weightlifting and the 2/3 power law

At least since Haldane's 1926 paper [1], we know that the various characteristics of an organism scale according to different power laws. For instance, its mass increases as the volume, i.e. as the cube of the length: $$M \sim L^3$$. The strength $$S$$, however, should be proportional to the muscle section, and thus increase as $$S \sim L^2$$. We therefore get $$S \sim M^{2/3}$$.

This law has been verified in the 50s by Lietzke [2] for the world-record weightlifting total and found to hold with very good precision (exponent 0.6748, instead of 2/3). How have things evolved since then?

 Figure 1: World records in Olympic weightlifting (totals), in 1972 (diamonds; snatch+ clean & jerk + press) and current values (open dots; only snatch+ clean & jerk). The lines are power law fits to the data, with exponents 0.7 (for 1972) and 0.54 (for current values).
In Figure 1 I plotted the records for the total, in 1972 (which include the press event) and the current ones (only snatch and clean & jerk); this explains the difference in magnitude. There is however a significant difference in the exponent: 0.7 for the 1972 data (within the error bar from the 2/3 value) but only 0.54 (±0.03) for the current values!

I then did the same analysis for the individual events, see Figure 2 for the 1972 values and Figure 3 for the current ones. The discrepancy holds, although the 1972 snatch and clean & jerk records are not well described by a power law.

 Figure 2: World records for the individual events in 1972 and power-law fits, with exponents 0.64 ± 0.07 for the snatch and clean & jerk and 0.76 ± 0.05 for the press.
 Figure 3: Current world records for the individual events and power-law fits, with exponents 0.54 ± 0.05 and 0.55 ± 0.03 for men and 0.66 ± 0.05 and 0.75 ± 0.03 for women (snatch and C & J, respectively).
In Figure 3 I also plotted the values for the women's records in the two events, with exponents 0.66 ± 0.05 and 0.75 ± 0.03 for the snatch and C & J, respectively! For all the fits, I assigned the record value to the maximum weight in the class and I did not include the heaviest class, which is not well defined.

What does this all mean?
• First of all, the power-law model does not describe the data all that well.
• Second, the 2/3 exponent is not very robust and has not persisted in time, as Lietzke had predicted in his paper.
• Finally, I have no idea how to explain the flatter dependence for current men's records with respect to the 1972 ones and also to the women's records.

[1] J. B. S. Haldane, "On being the right size" in Possible Words and Other Papers (Harper & Brothers, New York), pp. 20-28, (1928).
[2] M. H. Lietzke, "Relation between Weight-Lifting Totals and Body Weight", Science 124, pp. 486-487, (1956).