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I've started reading The Variational Principles of Mechanics by Cornelius Lanczos; here is the concerned excerpt from p. 11:

The generalized coordinates $q_1,q_2,\ldots, q_n$ may or may not have a geometrical significance. It is necessary however that the functions $$x_1= f_1(q_1,q_2,\ldots, q_n),\\ .......................\\ .......................\\ z_N= f_{3N}(q_1,q_2,\ldots, q_n).$$ shall be finite, continuous and differentiable, and that the Jacobian of at least one combination of $n$ functions shall be different from zero. These conditions may be violated at certain singular points, which have to excluded from consideration. ...

While I could get that the functions must be 'finite, continuous and differentiable' but couldn't get the condition that the 'Jacobian of at least one combination of $n$ functions shall be different from zero'.

Can anyone tell me what is the necessity of this condition? What does this actually mean? Or why do the functions need to follow this?

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The conditions about

  • (i) differentiability of the functions and

  • (ii) the maximal rank of the corresponding rectangular Jacobian matrix

are regularization conditions imposed to simplify the mathematical analysis of the physical problem, in particular to legitimate the possible future use of the inverse function theorem. In the affirmative case, the functions are called independent. See also this related Phys.SE post.

Physical systems that do not meet these regularization conditions are more difficult to analyse.

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