Just some remarks.
1) ##b_{ij}x_j## is not a tensor at least this expression does not keep its shape under changes of variables
2) the operation ##\partial/\partial x_i## takes tensors to not-tensors
3) if only linear changes are considered ##x_i=c_{ij}x'_j## then everything is ok
yes, that is the point
that's a key mistake. They are not equivalent. In the case of nonholonomic constraints the Hamilton principle implies vaconomic equations. I can only again refer you to the books cited above.
There is a funny story as well. I think only Russians know it. Very long ago before the famous many valued Landau and Lifshitz textbook appeared, Landau and Pitaevsky wrote a textbook on classical mechanics. This book contained many errors and it was completely smashed by Fok in his article...
the solution ##x_*(t)## to the variational problem described in #11 satisfies the equation
$$\frac{d}{dt}\frac{\partial \mathcal L}{\partial \dot x^k}-\frac{\partial \mathcal L}{\partial x^k}=0,\quad \mathcal L=L+\lambda_k(t)a^k_s\dot x^s$$
this equation contains ##\dot\lambda## and it is not...
There is an assertion that follows from very general theorem directly and I do not understand if this assertion trivial or it may be of some interest. The assertion is enclosed below please comment
Landau and Lifshitz put in the basis of mechanics the Hamilton principle. They particularly say: let ##x_*(t)## be a critical point of the functional
$$x(\cdot)\mapsto \int_{t_1}^{t_2}L(x(t),\dot x(t))dt$$ in a class of functions ##x(t)## that satisfy nonholonomic constraints:
$$a_i^j(x)\dot...
nope
for details see A. M. Bloch, J. Baillieul, P. Crouch, J. Marsden: Nonholonomic Mechanicsand Control (Interdisciplinary Applied Mathematics). Springer, 2000
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by the way, I am sorry for self citing but here I tried to write a clear text with explanations about nonholonomic variational principle https://arxiv.org/abs/2104.03913
nothing new just an essence from textbooks