Geodesic
On a Change of Variables in Lagrange’s Equations
Russian journal of nonlinear dynamics, Tome 18 (2022) no. 4, pp. 473-480.

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This paper studies a material system with a finite number of degrees of freedom the motion of which is described by differential Lagrange’s equations of the second kind. A twice continuously differentiable change of generalized coordinates and time is considered. It is well known that the equations of motion are covariant under such transformations. The conventional proof of this covariance property is usually based on the integral variational principle due to Hamilton and Ostrogradskii. This paper gives a proof of covariance that differs from the generally accepted one. In addition, some methodical examples interesting in theory and applications are considered. In some of them (the equilibrium of a polytropic gas sphere between whose particles the forces of gravitational attraction act and the problem of the planar motion of a charged particle in the dipole force field) Lagrange’s equations are not only covariant, but also possess the invariance property.
Keywords: analytical mechanics, transformation methods in mechanics.
Mots-clés : Lagrange’s equations 
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A. P. Markeev. On a Change of Variables in Lagrange’s Equations. Russian journal of nonlinear dynamics, Tome 18 (2022) no. 4, pp. 473-480. http://geodesic.mathdoc.fr/item/ND_2022_18_4_a1/

[1] Fichtenholz, G. M., Differential- und Integralrechnung: In 3 Vols., Hochschulbücher für Mathematik, 61–63, Deutsch, Frankfurt am Main, 2006, 317 pp.

[2] Markeev, A. P., Theoretical Mechanics, R Dynamics, Institute of Computer Science, Izhevsk, 2007, 592 pp. (Russian)

[3] Aizerman, M. A., Classical Mechanics, Nauka, Moscow, 1980, 368 pp. (Russian)

[4] Yakovenko, G. N., A Short Course in Analytical Dynamics, Binom, Moscow, 2004, 240 pp. (Russian)

[5] Kotkin, G. L., Serbo, V. G., and Chernykh, A. I., Lectures on Analytical Mechanics, R Dynamics, Institute of Computer Science, Izhevsk, 2010, 236 pp. (Russian)

[6] Belenky, I. M., Introduction to Analytical Mechanics, Vysshaya Shkola, Moscow, 1964, 324 pp. (Russian)

[7] Pars, L. A., A Treatise on Analytical Mechanics, Heinemann, London, 1965, 641 pp.

[8] Sansone, G., Equazioni differenziali nel campo reale: Parte seconda, 2nd ed., Zanichelli, Bologna, 1949, xvi+475 pp.

[9] Tamm, I. E., Fundamentals of the Theory of Electricity, Mir, Moscow, 1979, 684 pp.