Author:
(1) V V Obukhov, Institute of research and development, Tomsk State Pedagogical University, Kievskaya str. 60, Tomsk 634061, Russia & International laboratory of theoretical cosmology, Tomsk State University of Control Systems and Radioelectronics, Lenina pr. 40, Tomsk 634050, Russia (E-mail: obukhov@tspu.edu.ru);
(2) K E Osetrin, Center for Mathematical and Computer Physics, Tomsk State Pedagogical University, Kievskaya str. 60, Tomsk 634061, Russia & International laboratory of theoretical cosmology, Tomsk State University of Control Systems and Radioelectronics, Lenina pr. 40, Tomsk 634050, Russia;
(3) A V Shapovalov, Department of Theoretical Physics, National Research Tomsk State University, Lenina pr. 36, Tomsk 634050, Russia & International laboratory of theoretical cosmology, Tomsk State University of Control Systems and Radioelectronics, Lenina pr. 40, Tomsk 634050, Russia. Table of Links Abstract and Introduction Conclusion, Statements and Declarations, and Bibliography Conclusion Summarising, we believe that the statement made in the papers [1, 2] about new scientific results in the problem of complete separation of variables in the HamiltonJacobi equation for geodesics does not correspond to reality. We did not find any new results in comparison with those contained in the theorem on necessary and sufficient conditions of Vladimir N. Shapovalov. The results of the article in question, unlike the original ones, have omissions, and the article contains incorrect statements. Thus, the methods and results proposed in the works [1, 2] do not contain scientific novelty and represent a special case of the theory of complete separation of variables in the free Hamilton-Jacobi equation, finally constructed in the 70s of the last century. Statements and Declarations Data availability statement All necessary data and references to external sources are contained in the text of the manuscript. All information sources used in the work are publicly available and refer to open publications in scientific journals and textbooks. Compliance with Ethical Standards The authors declare no conflict of interest. Bibliography [1] Katanaev M O, Complete separation of variables in the geodesic HamiltonJacobi equation in four dimensions, 2023 Physica Scripta 98 104001 URL https://dx.doi.org/10.1088/1402-4896/acf251 [2] Katanaev M O 2023 Complete separation of variables in the geodesic Hamilton-Jacobi equation (Preprint 2305.02222) URL https://arxiv.org/abs/2305.02222 [3] Obukhov V, Hamilton–Jacobi equation for a charged test particle in the St¨ackel space of type (2.0), 2020 Symmetry 12 URL https://www.mdpi.com/2073-8994/12/8/1289 [4] Obukhov V, Separation of variables in Hamilton–Jacobi equation for a charged test particle in the St¨ackel spaces of type (2.1), 2020 International Journal of Geometric Methods in Modern Physics 17 2050186 URL https://doi.org/10.1142/S0219887820501868 [5] Obukhov V, Separation of variables in Hamilton–Jacobi and Klein–Gordon–Fock equations for a charged test particle in the St¨ackel spaces of type (1.1), 2021 International Journal of Geometric Methods in Modern Physics 18 2150036 URL https://doi.org/10.1142/S0219887821500365 [6] Obukhov V V 2006 St¨ackel spaces in gravity theory (Tomsk: Tomsk State Pedagogical University) ISBN 978-5-89428-242-8 [7] Bagrov V G, Meshkov A G, Shapovalov V N and Shapovalov A V, Separation of variables in the Klein-Gordon equations. I, 1973 Soviet Physics Journal 16 1533–1538 URL https://doi.org/10.1007/BF00889957 [8] Shapovalov V N, Symmetry and separation of variables in Hamilton-Jacobi equations. I, 1978 Soviet Physics Journal 21 1124–1129 URL https://doi.org/10.1007/BF00894559 [9] Shapovalov V N, St¨ackel spaces, 1979 Siberian Mathematical Journal 20 790–800 URL https://doi.org/10.1007/BF00971844 [10] Bagrov V G, Obukhov V V and Shapovalov A V, Special St¨ackel electrovac spacetimes, 1986 Pramana - J. Phys. 26 93–108 URL https://doi.org/10.1007/BF02847629 [11] Bagrov V G and Obukhov V V, Classes of exact solutions of the Einstein–Maxwell equations, 1983 Annalen der Physik 495 181–188 URL https://doi.org/10.1002/andp.19834950402 [12] Bagrov V G, Obukhov V V and Osetrin K E, Classification of null St¨ackel electrovac metrics with cosmological constant, 1988 General Relativity and Gravitation 20 1141–1154 URL https://doi.org/10.1007/BF00758935 This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. Author: (1) V V Obukhov, Institute of research and development, Tomsk State Pedagogical University, Kievskaya str. 60, Tomsk 634061, Russia & International laboratory of theoretical cosmology, Tomsk State University of Control Systems and Radioelectronics, Lenina pr. 40, Tomsk 634050, Russia (E-mail: obukhov@tspu.edu.ru); (2) K E Osetrin, Center for Mathematical and Computer Physics, Tomsk State Pedagogical University, Kievskaya str. 60, Tomsk 634061, Russia & International laboratory of theoretical cosmology, Tomsk State University of Control Systems and Radioelectronics, Lenina pr. 40, Tomsk 634050, Russia; (3) A V Shapovalov, Department of Theoretical Physics, National Research Tomsk State University, Lenina pr. 36, Tomsk 634050, Russia & International laboratory of theoretical cosmology, Tomsk State University of Control Systems and Radioelectronics, Lenina pr. 40, Tomsk 634050, Russia. Author: Author: (1) V V Obukhov, Institute of research and development, Tomsk State Pedagogical University, Kievskaya str. 60, Tomsk 634061, Russia & International laboratory of theoretical cosmology, Tomsk State University of Control Systems and Radioelectronics, Lenina pr. 40, Tomsk 634050, Russia (E-mail: obukhov@tspu.edu.ru); (2) K E Osetrin, Center for Mathematical and Computer Physics, Tomsk State Pedagogical University, Kievskaya str. 60, Tomsk 634061, Russia & International laboratory of theoretical cosmology, Tomsk State University of Control Systems and Radioelectronics, Lenina pr. 40, Tomsk 634050, Russia; (3) A V Shapovalov, Department of Theoretical Physics, National Research Tomsk State University, Lenina pr. 36, Tomsk 634050, Russia & International laboratory of theoretical cosmology, Tomsk State University of Control Systems and Radioelectronics, Lenina pr. 40, Tomsk 634050, Russia. Table of Links Abstract and Introduction Abstract and Introduction Conclusion, Statements and Declarations, and Bibliography Conclusion, Statements and Declarations, and Bibliography Conclusion Summarising, we believe that the statement made in the papers [1, 2] about new scientific results in the problem of complete separation of variables in the HamiltonJacobi equation for geodesics does not correspond to reality. We did not find any new results in comparison with those contained in the theorem on necessary and sufficient conditions of Vladimir N. Shapovalov. The results of the article in question, unlike the original ones, have omissions, and the article contains incorrect statements. Thus, the methods and results proposed in the works [1, 2] do not contain scientific novelty and represent a special case of the theory of complete separation of variables in the free Hamilton-Jacobi equation, finally constructed in the 70s of the last century. Statements and Declarations Data availability statement Data availability statement All necessary data and references to external sources are contained in the text of the manuscript. All information sources used in the work are publicly available and refer to open publications in scientific journals and textbooks. Compliance with Ethical Standards Compliance with Ethical Standards The authors declare no conflict of interest. Bibliography [1] Katanaev M O, Complete separation of variables in the geodesic HamiltonJacobi equation in four dimensions, 2023 Physica Scripta 98 104001 URL https://dx.doi.org/10.1088/1402-4896/acf251 [2] Katanaev M O 2023 Complete separation of variables in the geodesic Hamilton-Jacobi equation (Preprint 2305.02222) URL https://arxiv.org/abs/2305.02222 [3] Obukhov V, Hamilton–Jacobi equation for a charged test particle in the St¨ackel space of type (2.0), 2020 Symmetry 12 URL https://www.mdpi.com/2073-8994/12/8/1289 [4] Obukhov V, Separation of variables in Hamilton–Jacobi equation for a charged test particle in the St¨ackel spaces of type (2.1), 2020 International Journal of Geometric Methods in Modern Physics 17 2050186 URL https://doi.org/10.1142/S0219887820501868 [5] Obukhov V, Separation of variables in Hamilton–Jacobi and Klein–Gordon–Fock equations for a charged test particle in the St¨ackel spaces of type (1.1), 2021 International Journal of Geometric Methods in Modern Physics 18 2150036 URL https://doi.org/10.1142/S0219887821500365 [6] Obukhov V V 2006 St¨ackel spaces in gravity theory (Tomsk: Tomsk State Pedagogical University) ISBN 978-5-89428-242-8 [7] Bagrov V G, Meshkov A G, Shapovalov V N and Shapovalov A V, Separation of variables in the Klein-Gordon equations. I, 1973 Soviet Physics Journal 16 1533–1538 URL https://doi.org/10.1007/BF00889957 [8] Shapovalov V N, Symmetry and separation of variables in Hamilton-Jacobi equations. I, 1978 Soviet Physics Journal 21 1124–1129 URL https://doi.org/10.1007/BF00894559 [9] Shapovalov V N, St¨ackel spaces, 1979 Siberian Mathematical Journal 20 790–800 URL https://doi.org/10.1007/BF00971844 [10] Bagrov V G, Obukhov V V and Shapovalov A V, Special St¨ackel electrovac spacetimes, 1986 Pramana - J. Phys. 26 93–108 URL https://doi.org/10.1007/BF02847629 [11] Bagrov V G and Obukhov V V, Classes of exact solutions of the Einstein–Maxwell equations, 1983 Annalen der Physik 495 181–188 URL https://doi.org/10.1002/andp.19834950402 [12] Bagrov V G, Obukhov V V and Osetrin K E, Classification of null St¨ackel electrovac metrics with cosmological constant, 1988 General Relativity and Gravitation 20 1141–1154 URL https://doi.org/10.1007/BF00758935 This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. This paper is available on arxiv under CC BY-NC-ND 4.0 DEED license. available on arxiv

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Revisiting Separation of Variables in Hamilton-Jacobi Equations: Conclusion, Statements and More

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