Abstract: The problem of Birkhoff symmetry for generalized Birkhoffian systems is studied, and the corresponding conserved quantities are given. A theorem known for nonsingular equivalent Lagrangians is generalized to the generalized Birkhoffian systems. We prove that under certain conditions the matrix Λ, which is related with the generalized Birkhoffian equations obtained from two groups of dynamical functions B,Rμ,Λμ and B,Rμ,Λμ, has the property that the traces of all its integer powers are the conserved quantities of the system. An example is given to illustrate the application of the results.

Keywords: 
• generalized Birkhoffian system /
• Birkhoff symmetry /
• conserved quantity /
• trace of matrix