Let formula_24 be the position, formula_25 be the kinetic energy, formula_13 the potential energy, both independent of time formula_27. Lagrange's equation of motion readsformula_28Here, the dot notation for the time derivative is used, formula_29. Poisson set formula_30. He argued that if formula_13 is independent of formula_32, he could writeformula_33giving formula_34He introduced an explicit formula for momenta,formula_35Thus, from the equation of motion, he gotformula_36Poisson's text influenced the work of William Rowan Hamilton and Carl Gustav Jacob Jacobi. A translation of Poisson's Treatise on Mechanics was published in London in 1842. Let formula_37 and formula_38 be functions of the canonical variables of motion formula_39 and formula_40. Then their Poisson bracket is given byformula_41Evidently, the operation anti-commutes. More precisely, formula_42. By Hamilton's equations of motion, the total time derivative of formula_43 isformula_44where formula_45 is the Hamiltonian. In terms of Poisson brackets, then, Hamilton's equations can be written as formula_46 and formula_47.