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Russian Journal of Information Technology in Sports
Journals / Russian Journal of Information Technology in Sports / Towards a mathematical theory of athletic training. Part 3. The theory of adaptation of energy metabolism to physical loads by Nikolai Ivanovich Volkov
VAK Russia 5.8.5 JEL I12 UDC 796.012.1 CSCSTI 77.05 Russian Classification of Professions by Education 49.06.01 Russian Library and Bibliographic Classification 75 Russian Trade and Bibliographic Classification 5 BISAC SPO000000 VAK Russia 1.2.2 VAK Russia 1.1.10 VAK Russia 1.2.1 VAK Russia 1.2.3 VAK Russia 1.5.2 VAK Russia 1.5.5 VAK Russia 1.5.8 VAK Russia 3.3.9 VAK Russia 3.1.33 VAK Russia 5.2.3 VAK Russia 5.2.6 VAK Russia 5.3.2 VAK Russia 5.3.4 VAK Russia 5.8.4 VAK Russia 5.8.6 VAK Russia 5.8.7 VAK Russia 5.6.6 VAK Russia 5.12.1 VAK Russia 5.12.4 JEL C02 UDC 612.766.1 UDC 004 UDC 347.514.3 UDC 355.233.22 UDC 351.855.3 UDC 00 UDC 31 CSCSTI 77.03 CSCSTI 28.17 CSCSTI 34.39 CSCSTI 77.00 CSCSTI 14.00 CSCSTI 20.00 CSCSTI 83.00 Russian Classification of Professions by Education 49.04.01 Russian Classification of Professions by Education 01.03.02 Russian Classification of Professions by Education 12.03.04 Russian Classification of Professions by Education 02.00.00 Russian Classification of Professions by Education 06.00.00 Russian Classification of Professions by Education 09.00.00 Russian Classification of Professions by Education 32.00.00 Russian Classification of Professions by Education 39.00.00 Russian Classification of Professions by Education 44.00.00 Russian Library and Bibliographic Classification 3281 Russian Library and Bibliographic Classification 1 Russian Library and Bibliographic Classification 22 Russian Library and Bibliographic Classification 28 Russian Library and Bibliographic Classification 3 Russian Library and Bibliographic Classification 58 Russian Library and Bibliographic Classification 73 Russian Library and Bibliographic Classification 74 Russian Trade and Bibliographic Classification 7 BISAC SPO061000 BISAC COM004000 BISAC MAT003000 BISAC SEL BISAC COM014000 BISAC COM072000 BISAC COM017000 BISAC COM018000 BISAC COM062000 BISAC COM089000 BISAC COM021000 BISAC COM021030 BISAC COM023000 BISAC COM025000 BISAC COM074000 BISAC COM080000 BISAC COM079010 BISAC COM032000 BISAC COM031000 BISAC COM060000 BISAC COM039000 BISAC COM077000 BISAC COM051000 BISAC COM051300 BISAC COM012040 BISAC COM060170

TOWARDS A MATHEMATICAL THEORY OF ATHLETIC TRAINING. PART 3. THE THEORY OF ADAPTATION OF ENERGY METABOLISM TO PHYSICAL LOADS BY NIKOLAI IVANOVICH VOLKOV Towards a mathematical theory of athletic training. Part 3. The theory of adaptation of energy metabolism to physical loads by Nikolai Ivanovich Volkov

Published in Russian Journal of Information Technology in Sports · Pages 1–34 · Rubric: HISTORY OF INFORMATION TECHNOLOGY IN SPORTS
DOI: https://doi.org/10.62105/2949-6349-2026-3-4-e202616 · EDN: ASYPOU
Received: 20.08.2026 Accepted: 22.09.2026 Published: 30.09.2026
Authors
EG Egor Timme
Egor Timme
ORCID Scopus ResearcherID Scholar SPIN
ABSTRACT
Relevance. In the 1960s–1970s the Soviet Union saw the beginning of the active application of mathematical methods in sport. However, the contribution of Nikolai Ivanovich Volkov, Doctor of Biological Sciences, Professor, founder of the research field of “sport bioenergetics” and creator of the corresponding department at the GTSOLIFK, to the construction of a mathematical theory of adaptation to strenuous muscular activity has not yet been systematized. Meanwhile, it was he who was the first to describe the kinetics of bioenergetic processes by a system of differential equations and to introduce the criteria of power, capacity and efficiency of the energy supply mechanisms. Objectives of the work — systematization, interpretation and comparative analysis of the ideas, models and results of N.I. Volkov relating to the theory of adaptation and its mathematical expression; identification of the structure of the model he proposed and of its place among other approaches; establishment of a typology of relations between his constructions and modern models. Methods. The scientific works and biographical materials of N.I. Volkov of 1965–2013 were analysed, including the papers of 1965, 1966 and 1968, his doctoral dissertation (1990) and the monograph “Sport Bioenergetics” (2011). A comparison was made with the approaches of N.N. Engwer, E. Banister and V.N. Seluyanov and with modern quantitative models of the bioenergetics of muscular activity, and the relations between the models were classified. Results. The mathematical model of adaptation of N.I. Volkov has been systematized: 1) a compartmental kinetic model of energy metabolism in which the dynamics of the oxidizable substrate is described by a second-order linear differential equation; 2) biexponential equations of the kinetics of O₂ uptake during work and in recovery and an equation of lactate dynamics in the form of a difference of two exponentials; 3) a system of criteria of power, capacity and efficiency for the alactate, glycolytic and aerobic mechanisms; 4) an apparatus for optimizing the training process — target functions, the zone of optimal loads and the design of an extreme experiment. The approach of N.I. Volkov has been compared with the approaches of N.N. Engwer, E. Banister and V.N. Seluyanov and characterized as mechanistic, proceeding from the kinetics of metabolic processes. The solution of the equations of the model has been compared with current views: the fast and slow exponential components of the kinetics of O₂ uptake correspond to phase II and to the slow component, while the calculated time of the maximum lactate content corresponds to that observed at a load above the maximal lactate steady state. It is established that R. Morton relied on the 1966 paper of N.I. Volkov, and his three-component model is regarded in the literature as a development of the models of N.I. Volkov. A typology of relations is proposed: direct citation, structural coincidence, conceptual correspondence, analogy and commonality of the object. It is shown that the notion of critical power and of the limited work reserve above it corresponds to the criteria of power and capacity of N.I. Volkov, the maximal lactate steady state to the notion of a critical load, and the models of reconstitution of this reserve are analogous to the equations of recovery kinetics. Conclusion. N.I. Volkov created the first integral mathematical theory of the adaptation of energy metabolism to physical loads in Russian sport science, which completes the series of four mathematical models of athletic training considered in the present cycle of papers and supplies in it the mechanistic link — a description at the level of the kinetics of metabolic processes. His equations and criteria retain their significance as a basis for modern models of training process management, while the forecast he formulated — the development of mathematical models of adaptation and automated systems for controlling the physical condition of athletes — defines the research agenda at the present time as well.
KEYWORDS
mathematical modelling of the training process, theory of adaptation, kinetics of energy metabolism, power and capacity of the energy supply mechanisms, lactate, oxygen uptake, design of an extreme experiment, critical power, sport bioenergetics
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