Abstract
It is shown that any nonsingular Lagrangian describing the motion of a particle on a semisimple Lie group possesses a Fundamental Poisson bracket Relation (FPR) and consequently charges in involution. This property is independent of the dynamics of the model and can be derived in a quite simple and general way from the geometric and algebraic structures of the group manifold. The conditions a Hamiltonian has to satisfy in order those charges are to be conserved are discussed. These conditions lead to an algebra which plays an important role in the construction of conserved charges. In the second paper of the series, this work is extended to the coset spaces which are symmetric spaces.
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