Abstract:
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, Lagrange’s equations, transformation methods in mechanics.
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