4 L. D. Faddeev and O. A. Yakubovski˘ı We find a differential equation that the function ft(q, p) satisfies. To this end, we differentiate the identity fs+t(µ) = ft(Gsµ) with respect to the variable s and set s = 0: ∂fs+t(µ) ∂s s=0 = ∂ft(µ) ∂t , ∂ft(Gsµ) ∂s s=0 = ∇ft(µ) · v(µ) = ∂ft ∂q ∂H ∂p ∂ft ∂p ∂H ∂q . Thus, the function ft(q, p) satisfies the differential equation (13) ∂ft ∂t = ∂H ∂p ∂ft ∂q ∂H ∂q ∂ft ∂p and the initial condition (14) ft(q, p)|t=0 = f(q, p). The equation (13) with the initial condition (14) has a unique solu- tion, which can be obtained by the formula (12) that is, to construct the solutions of (13) it suffices to know the solutions of the Hamil- tonian equations. We can rewrite (13) in the form (15) dft dt = {H, ft}, where {H, ft} is the Poisson bracket of the functions H and ft. For arbitrary observables f and g the Poisson bracket is defined by {f, g} = ∂f ∂p ∂g ∂q ∂f ∂q ∂g ∂p , and in the case of a system with n degrees of freedom {f, g} = n i=1 ∂f ∂pi ∂g ∂qi ∂f ∂qi ∂g ∂pi . We list the basic properties of Poisson brackets: 1) {f, g + λh} = {f, g} + λ{f, h} (linearity) 2) {f, g} = −{g, f} (skew symmetry) 3) {f,{g, h}} + {g,{h, f}} + {h,{f, g}} = 0 (Jacobi identity) 4) {f, gh} = g{f, h} + {f, g}h.
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