By R.F. Nagaev

This e-book provides a rational scheme of study for the periodic and quasi-periodic resolution of a wide type of difficulties inside technical and celestial mechanics. It develops steps for the decision of sufficiently common averaged equations of movement, that have a transparent actual interpretation and are legitimate for a extensive type of weak-interaction difficulties in mechanics. the standards of balance relating to desk bound options of those equations are derived explicitly and correspond to the extremum of a different "potential" functionality. a lot attention is given to functions in vibrational know-how, electric engineering and quantum mechanics, and a few effects are provided which are instantly valuable in engineering perform. The booklet is meant for mechanical engineers, physicists, in addition to utilized mathematicians focusing on the sector of standard differential equations.

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**Example text**

30), takes the form L H= = m 2~ [(mj;+~Axf + (mY+~AYf + (mi+~Azf] +eU(x,y,z). 36) becomes H 1 2 2 = ('yx, 0, 0). Then the Hamiltonian e 2 ,2 ey 2 2 = -2 (Px +Py +pz) - -xpx + - 22 x - eEx. 37) have the following general integral pz pz const, Z = - t + zo, Px = --Rsmcp, m e mEe x Rcoscp + xo, p y = -, - + -Rcoscp, e e, , Ee Rcos cp + - t + Yo, y where the fast phase cp = wt Vxa tan cp = --;E=;'"e-"---, - vYa vXa e, . 39) with = j; (0) = Y(0). 38) describe a helix whose axis is coincident with axis x.

57) as a partial differential equation for M. In the general case, this equation does not admit single-valued analytical solutions of arguments q and p. 54) and eventually integrate the original system. For this reason, M is referred to as Jacobi's last multiplier. 56) for Q and P into the right hand side of eq. 53). The resulting expressions form Pfaff's system of equations 1 of . 58) Due to eqs. 58) velocity v (q,p) of the representative point in the phase plane (q, p) satisfies the scalar equation div (Mv) = O.

55) 0 which, generally speaking, can not be expressed in terms of the elementary functions. For rotations to exist, it is necessary that inequality 2h > al +a2 holds. 53), which is necessary for constructing dependence q (t), presents considerable difficulties. In the particular case of rotation at ultra-high energy this problem is solvable using power series in small parameters. 56) for h into eq. 51) and expanding the right hand side as a series in powers of 8, we obtain r::::::L [ ( y mho 1 hI - II + 1;;2 + (h2 ho - 11)2) (hI 2h6 1 84 l) + .