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.