On a Change of Variables in Lagrange’s Equations
Received 31 May 2022; accepted 12 July 2022; published 26 July 2022
2022, Vol. 18, no. 4, pp. 473-480
Author(s): Markeev A. P.
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.
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