~e generalize a theorem known for one-dimensional nonsingular equivalent Lagrangians (L and L) to the multidimensional case. In particular, we prove that the matrix A, which relates the lefthand sides of the Euler-Lagrange equations obtained from Land L, is such that the trace of all its integer powers are constants of the motion. We construct several multidimensional examples in which the elements of A are functions of position, velocity, and time, and prove that in some cases equivalence prevails even if detA = O. P ACS numbers: 03.20. + i
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