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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Darwin Lagrangian ： ウィキペディア英語版 Darwin Lagrangian The Darwin Lagrangian (named after Charles Galton Darwin, grandson of the naturalist) describes the interaction to order $\backslash frac$ between two charged particles in a vacuum and is given by〔 pp. 596-598〕 :$L\; =\; L\_\backslash text\; +\; L\_\backslash text,$ where the free particle Lagrangian is :$L\_\backslash text\; =\; \backslash frac\; m\_1\; v\_1^2\; +\; \backslash frac\; m\_1\; v\_1^4\; +\; \backslash frac\; m\_2\; v\_2^2\; +\; \backslash frac\; m\_2\; v\_2^4,$ and the interaction Lagrangian is :$L\_\backslash text\; =\; L\_\backslash text\; +\; L\_\backslash text,$ where the Coulomb interaction is :$L\_\backslash text\; =\; -\backslash frac,$ and the Darwin interaction is :$L\_\backslash text\; =\; \backslash frac\; \backslash frac\; \backslash mathbf\; v\_1\; \backslash cdot\; \backslash left(1\; +\; \backslash mathbf\; \backslash mathbf\backslash right)\; \backslash cdot\; \backslash mathbf\; v\_2.$ Here ''q''1 and ''q''2 are the charges on particles 1 and 2 respectively, ''m''1 and ''m''2 are the masses of the particles, v1 and v2 are the velocities of the particles, ''c'' is the speed of light, r is the vector between the two particles, and $\backslash hat$ is the unit vector in the direction of r. The free Lagrangian is the Taylor expansion of free Lagrangian of two relativistic particles to second order in ''v''. The Darwin interaction term is due to one particle reacting to the magnetic field generated by the other particle. If higher-order terms in ''v''/''c'' are retained, then the field degrees of freedom must be taken into account, and the interaction can no longer be taken to be instantaneous between the particles. In that case retardation effects must be accounted for. ==Derivation of the Darwin interaction in a vacuum== The relativistic interaction Lagrangian for a particle with charge q interacting with an electromagnetic field is〔 Jackson, pp. 580-581. 〕 :$L\_\backslash text\; =\; -q\backslash Phi\; +\; \backslash mathbf\; u\; \backslash cdot\; \backslash mathbf\; A,$ where u is the relativistic velocity of the particle. The first term on the right generates the Coulomb interaction. The second term generates the Darwin interaction. The vector potential in the Coulomb gauge is described by〔 Jackson, p. 242. 〕 (Gaussian units) :$\backslash nabla^2\; \backslash mathbf\; A\; -\; =\; -\; \backslash mathbf\; J\_t$ where the transverse current J''t'' is the solenoidal current (see Helmholtz decomposition) generated by a second particle. The divergence of the transverse current is zero. The current generated by the second particle is :$\backslash mathbf\; J\; =\; q\_2\; \backslash mathbf\; v\_2\; \backslash delta\; \backslash left(\; \backslash mathbf\; r\; -\; \backslash mathbf\; r\_2\; \backslash right),$ which has a Fourier transform :$\backslash mathbf\; J\backslash left(\; \backslash mathbf\; k\; \backslash right)\; \backslash equiv\; \backslash int\; d^3r\; \backslash exp\backslash left(\; -i\backslash mathbf\; k\; \backslash cdot\; \backslash mathbf\; r\; \backslash right)\; \backslash mathbf\; J\backslash left(\; \backslash mathbf\; r\; \backslash right)\; =\; q\_2\; \backslash mathbf\; v\_2\; \backslash exp\backslash left(\; -i\backslash mathbf\; k\; \backslash cdot\; \backslash mathbf\; r\_2\; \backslash right).$ The transverse component of the current is :$\backslash mathbf\; J\_t\backslash left(\; \backslash mathbf\; k\; \backslash right)\; =\; q\_2\; \backslash left(\backslash mathbf\; 1\; -\; \backslash mathbf\; \backslash mathbf\; \backslash right)\; \backslash cdot\; \backslash mathbf\; v\_2\; \backslash exp\backslash left(\; -i\backslash mathbf\; k\; \backslash cdot\; \backslash mathbf\; r\_2\; \backslash right).$ It is easily verified that :$\backslash mathbf\; k\; \backslash cdot\; \backslash mathbf\; J\_t\backslash left(\; \backslash mathbf\; k\; \backslash right)\; =\; 0,$ which must be true if the divergence of the transverse current is zero. We see that :$\backslash mathbf\; J\_t\backslash left(\; \backslash mathbf\; k\; \backslash right)$ is the component of the Fourier transformed current perpendicular to k. From the equation for the vector potential, the Fourier transform of the vector potential is :$$ \mathbf A \left( \mathbf k \right) = \left(\mathbf 1 - \mathbf \mathbf \right ) \cdot \mathbf v_2 \exp\left( -i\mathbf k \cdot \mathbf r_2 \right) where we have kept only the lowest order term in v/c. The inverse Fourier transform of the vector potential is :$\backslash mathbf\; A\; \backslash left(\; \backslash mathbf\; r\; \backslash right)$ =\int \; \mathbf A \left( \mathbf k \right) \; = \left(1 + \mathbf \mathbf\right ) \cdot \mathbf v_2 where :$\backslash mathbf\; r\; =\; \backslash mathbf\; r\_1\; -\; \backslash mathbf\; r\_2$ (see Common integrals in quantum field theory ). The Darwin interaction term in the Lagrangian is then :: \mathbf v_1 \cdot \left(1 + \mathbf \mathbf\right ) \cdot \mathbf v_2 |} where again we kept only the lowest order term in v/c. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Darwin Lagrangian」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.