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<<Mentor's note: this is spin-off this thread>>

Zampieri, Gaetano. "Nonholonomic versus vakonomic dynamics." Journal of Differential Equations 163.2 (2000): 335-347.

https://www.researchgate.net/profile/Gaetano_Zampieri/publication/224039915_Nonholonomic_versus_Vakonomic_Dynamics/links/544fb9a50cf201441e934bcd.pdf [Broken]

One error I'm aware of in LL vol. I is the claim on integrability. But what's wrong with LL's treatment of anholonomous constraints (in sectin 38 in my German edition)? It just leads to the usual equations with Lagrange parameters you also get from d'Alembert's principle ("virtual displacements"). There are wrong statements about this in the literature, e.g., in Goldstein, where he uses (38.2) instead of (38.4), leading to the socalle "vaconomic mechanics", which is considered wrong. On this issue, see"Mechanics" by LL contains errors. For example, it is well known that the equations of nonholonomic mechanics do not follow from the variational principle of Hamilton. This fact has serious physical and geometric reasons. For details see for example

[Nonholonomic Mechanics and Control (Interdisciplinary Applied Mathematics), Anthony Bloch, et al]. Nevertheless LL claim that they obtain equations of nonholonomic mechanics from the variational principle.

Zampieri, Gaetano. "Nonholonomic versus vakonomic dynamics." Journal of Differential Equations 163.2 (2000): 335-347.

https://www.researchgate.net/profile/Gaetano_Zampieri/publication/224039915_Nonholonomic_versus_Vakonomic_Dynamics/links/544fb9a50cf201441e934bcd.pdf [Broken]

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