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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 Action-angle coordinates ： ウィキペディア英語版 Action-angle coordinates In classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems. The method of action-angles is useful for obtaining the frequencies of oscillatory or rotational motion without solving the equations of motion. Action-angle coordinates are chiefly used when the Hamilton–Jacobi equations are completely separable. (Hence, the Hamiltonian does not depend explicitly on time, i.e., the energy is conserved.) Action-angle variables define an invariant torus, so called because holding the action constant defines the surface of a torus, while the angle variables provide the coordinates on the torus. The Bohr–Sommerfeld quantization conditions, used to develop quantum mechanics before the advent of wave mechanics, state that the action must be an integral multiple of Planck's constant; similarly, Einstein's insight into EBK quantization and the difficulty of quantizing non-integrable systems was expressed in terms of the invariant tori of action-angle coordinates. Action-angle coordinates are also useful in perturbation theory of Hamiltonian mechanics, especially in determining adiabatic invariants. One of the earliest results from chaos theory, for the non-linear perturbations of dynamical systems with a small number of degrees of freedom is the KAM theorem, which states that the invariant tori are stable under small perturbations. The use of action-angle variables was central to the solution of the Toda lattice, and to the definition of Lax pairs, or more generally, the idea of the isospectral evolution of a system. ==Derivation== Action angles result from a type-2 canonical transformation where the generating function is Hamilton's characteristic function $W(\backslash mathbf)$ (''not'' Hamilton's principal function $S$). Since the original Hamiltonian does not depend on time explicitly, the new Hamiltonian $K(\backslash mathbf,\; \backslash mathbf)$ is merely the old Hamiltonian $H(\backslash mathbf,\; \backslash mathbf)$ expressed in terms of the new canonical coordinates, which we denote as $\backslash mathbf$ (the action angles, which are the generalized coordinates) and their new generalized momenta $\backslash mathbf$. We will not need to solve here for the generating function $W$ itself; instead, we will use it merely as a vehicle for relating the new and old canonical coordinates. Rather than defining the action angles $\backslash mathbf$ directly, we define instead their generalized momenta, which resemble the classical action for each original generalized coordinate :$$ J_ \equiv \oint p_k \, dq_k where the integration path is implicitly given by the constant energy function $E=E(q\_k,p\_k)$. Since the actual motion is not involved in this integration, these generalized momenta $J\_k$ are constants of the motion, implying that the transformed Hamiltonian $K$ does not depend on the conjugate generalized coordinates $w\_k$ :$$ \frac J_ = 0 = \frac where the $w\_k$ are given by the typical equation for a type-2 canonical transformation :$$ w_k \equiv \frac Hence, the new Hamiltonian $K=K(\backslash mathbf)$ depends only on the new generalized momenta $\backslash mathbf$. The dynamics of the action angles is given by Hamilton's equations :$$ \frac w_k = \frac \equiv \nu_k(\mathbf) The right-hand side is a constant of the motion (since all the $J$'s are). Hence, the solution is given by :$$ w_k = \nu_k(\mathbf) t + \beta_k where $\backslash beta\_k$ is a constant of integration. In particular, if the original generalized coordinate undergoes an oscillation or rotation of period $T$, the corresponding action angle $w\_k$ changes by $\backslash Delta\; w\_k\; =\; \backslash nu\_k\; (\backslash mathbf)\; T$. These $\backslash nu\_k(\backslash mathbf)$ are the frequencies of oscillation/rotation for the original generalized coordinates $q\_k$. To show this, we integrate the net change in the action angle $w\_k$ over exactly one complete variation (i.e., oscillation or rotation) of its generalized coordinates $q\_k$ :$$ \Delta w_k \equiv \oint \frac \, dq_k = \oint \frac \, dq_k = \frac \oint \frac \, dq_k = \frac \oint p_k \, dq_k = \frac = 1 Setting the two expressions for $\backslash Delta\; w\_$ equal, we obtain the desired equation :$$ \nu_k(\mathbf) = \frac The action angles $\backslash mathbf$ are an independent set of generalized coordinates. Thus, in the general case, each original generalized coordinate $q\_$ can be expressed as a Fourier series in ''all'' the action angles :$$ q_k = \sum_^\infty \sum_^\infty \cdots \sum_^\infty A^k_ e^ e^ \cdots e^ where $A^k\_$ is the Fourier series coefficient. In most practical cases, however, an original generalized coordinate $q\_k$ will be expressible as a Fourier series in only its own action angles $w\_k$ :$$ q_k = \sum_^\infty e^ 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「Action-angle coordinates」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.