By S. Thornton, J. Marion
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Extra info for Classical Dynamics of Particles and Systems
The generalized equation of Theorem 5 then reduces to y" = (GIG)y,2 + bZ(y) + [dIG 2]Z\y) If, for example, Z = yin, G = (kln)y' -n, then the connective condition of Theorem 5 is satisfied and the equation becomes y" = y,2/y + k,y'-n + k2y'-3n Duffing's, the elliptic equation, and therefore the equations of Theorem 5, are not solvable in terms of the normal functions we encounter. The solutions are obtained in terms of elliptic functions and elliptic integrals. We shall not delve too deeply into these matters here, but will refer the reader to Davis' text (ref.
2 Solving the Euler-Lagrange Equation Having defined the nonlinear differential equation (82), under the assumption that H is separable, we now seek its solution in tenns of the conditions of Theorem 2. Referring once again to (71), we note that the tenns equivalent to q(x)G(y) and r(x)Z(y) in (82) are missing. Hence, we conclude that q = r = 0, and since G(y) is not explicitly defined, we can define it as we so desire since it has a zero coefficient. Thus the connective condition (25) is satisfied and (82) can be solved by the method of Theorem 2.
2nd order Linear base equation ... 1storder Figure 2-3. Solving the Ricatti equation. See also Sugai' s paper (Related Literature, 4) for the generation of results which occur with the use of y = rulu' Furthennore, extension of the theory of linear first-order base equations to that of matrix equations is another natural step. 6 and Theorem 7 provide that extension, which is then applicable to the matrix Ricatti equation. For some further concepts applicable to the Ricatti equation, see Problem 9.