線型調和振動の拡張に関する注意
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(2) Vol. 5, No. 1 Journal of Hokkaido Qakugei University (B) Feb., 1954. Note on the Extension of the Linear Harmjnic Oscillation Junkichi SOMA Department of Physics, Hakodate Branch.. Masamichi O&I* Department of Physics, Faculty of Science, Hokkaido University. fflM^T1? • ^J^-TEM : ^^tl.WA<ffJj®^S^|g-i-%^.;@:. A few years ago, non-linear system which permits simple harmonic motion was. derived by J. E. Brock through the extension of the linear harmonic osci)lation.l) That is, the linear oscillation x+ci)2x=0 or x=A sin (io){+8), has the arbitrary constants A and 8 which are independent of w. Buh, by introducing- the condition. F(w\. As)=0. .. (1. ). Brock got the more general oscillation which satisfies the equation F c - C^A-), -z-2 - .z- • .^A-)] = o - (2) and permits a simple harmonic motion.. On the other hand, about thirteen years ago J. Chazy gave out the theory of the non-linear oscillation having the isochronism.2^ By his theory, if the oscillation is defined by the equ'ation. r. =. X(x). ^. (. 3. ). and the variable x is transformed by the equation x2=at-ut where a is a constant,. this oscillation has the isochronism in the case of / u)= const. + odd func., where f{u')=dxldu. And in the special case of /(")= const., this oscillation is the linear harmonic motion.. Above two extensions are discussed independently by the two authors from the separate standpoints, but the unfication of these two separate ones is possible by following consideration. That is, a linear harmonic motion is characterized by the two characters :. P It is expressed by slnusoidal function. ii) It has isochronism that is period (or frequency) is independent of the initial conditions.. Then, we can extend linear oscillation to the two different directions, each of which conserves either (i) or >i0 of the above characters respectively. 14 —.
(3) Junkichi Soiitn, Masajnichi 0^1. And it is clear that the works of Brock and Chazy just correspond to the two different extensions which we have mentioned above. Thus the situation of their extensions is clarified by our unification. Moreover, it is interesting that the second extension has not only the above mentioned meaning from a viewpoint of the unification, but also is applicable to the thermal expansion of solids.3'1. We thank cordially Prof. Y. Ikeda for his encouragement. ,* Now at Osaka City University. 1) J. E. Brock, J. Appl. Phys. 21, 238 (1950).. 2) J. Chazy, Comptes Eendus 211, 621 (1940). 3) J. Soma and M. Ogi, J. Phys. Soc. Japan 8, 6 (1953).. 15 —.
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