[數] 變分原理（等于variation principle）
In this paper the analysis of cable-supported dome structure under axisymmetrical loading is discussed and solved by the energy variational principle.
本文對索承穹頂結構在軸對稱荷載作用下的分析進行研究，并用能量變分原理進行求解。
Besides, Chapter Three gets the Euler-Lagrange cohomology in the finite element discrete Lagrangian mechanics via the difference discrete variational principle.
此外，文中第三部分用差分離散變分原理得到有限元離散的拉氏力學的歐拉-拉格朗日上同調。
As an inverse problem of Hamiltonian mechanics, a new Hamiltonian system in elasticity and its variational principle are derived from the basic equations of elasticity.
作為哈密頓力學逆問題，從彈性力學基本方程推導出彈性力學中一個新的哈密頓系統及其變分原理。
As we know, perturbed and optimization and variational principle play an important role in nonlinear analysis.
我們知道，擾動優化或變分原理在非線性分析領域中也起到了很大的作用。
A power-like equation, a Virial-like theorem, an integral variational principle and a differential variational principle for the system are deduced by using the time-integral identity.
給出系統的時間積分等式，并由此等式導出類能量方程、類維裡定理、一個積分變分原理和一個微分變分原理。
The paper presents a new symmetric coupling approach of BEM and FEM, which is based on the sub-region variational principle.
本文從彈性力學分區變分原理出發，提出一種新的有限元與邊界元的對稱耦合方法。
Some references indicated that the extremum in the Schrodinger variational principle is a minimum.
已有的一些文獻指出，薛定谔變分原理中的極值是極小值。
An elastoplastic analysis of plate-bending formulation which is incremental computation form of finite element method is derived by variational principle.
本文利用變分原理推導出彎曲闆彈塑性分析的增量形式有限元素法計算格式；
In the free-boundary case, the extremum of the potential functional is found from the variational principle. Thereby the equation and boundary conditions required for plasma equilibrium are derived.
在自由界面情況下用求位能汛函極值的變分原理推出了等離子體平衡所需滿足的平衡方程和邊界條件；
Laplace transform finite element method for solving the dynamic coupled ther-moelasic bending problems of thin plates was proposed based on the variational principle of theProbems.
以薄闆動力熱耦合彎曲問題的變分原理為基礎，提出了一種解決該問題的拉普拉斯變換有限無法。
Based on Gurtin type variational principle a finite element method for initial value problems is presented by functional analysis. It is used for problems of beams dynamics.
從卷積型變分原理的泛函出發，通過分析泛函中的各項，推導出求解動力響應問題的時-空有限元法，并将其應用于梁的動力問題中．。
We first improve the complementary variational principle.
首先改進互補變分原理。
Based on the sub-region generalized variational principle, the solving equations of the problem are derived directly.
根據分區廣義變分原理，直接導出問題的求解方程式。
Taking soil slope as an example, model and analytic formula of the most dangerous sliding surface are developed according to variational principle.
基于變分原理，以土質斜坡為例，介紹斜坡潛在最危險滑動面的計算模型及解析表達式。
Problems dealing with a body having complex boundary conditions can be solved by means of the state-parted variational principle of energy.
最後，說明了分态能量變分原理在計算具有複雜邊界條件薄闆問題中的應用。
The foundation of the solving methods is the variational principle.
求解方法的基礎是變分原理。
This paper presents a multivariable wavelet FEM which is based on Hellingger-Reissner variational principle.
提出了一種基于二類變量廣義變分原理的多變量小波有限元方法。
With the variational principle of variable boundary, we study the crack extension on elastoplastic conditions.
應用可動邊界變分問題的理論研究了彈塑性理論的裂紋擴展問題。
In this paper, resonant frequency of the deformed circular microstrip antennas is investigated based on variational principle and the perturbation concept.
本文用變分法并結合微擾理論分析了一種變形圓貼片寬頻帶微帶天線的諧振頻率。
The recent progresses in non-equilibrium statistical mechanics are summarized and an attempt is made to build a new descriptive method based on a new variational principle.
在近年來非平衡統計力學研究進展的基礎上，嘗試從廣義變分法的視角建立一套描述非平衡統計力學的新方法。
By the use of the variational principle of complementary energy, the closed form of the stresses of double-layer composite bar is obtained.
運用最小餘能原理，求得了雙層複合材料杆溫度應力的封閉解。
A system procedure is proposed to derive the variational principle for a non self adjoint low frequency eddy current problem through adjoint operator and adjoint function.
利用伴隨算子和伴隨場函數，建立了低頻渦流電磁場中非自伴算子問題的一般變分描述。
Finite Elemnent-Linear Complementary Equation is established for analysis of elastoplastic thin plate-bending problems based on variational inequality principle.
從變分不等式出發，結合有限元，建立了按增量求解薄闆彈塑性彎曲問題的線性互補方程。
This paper describes the importance of performing theoretical stu***s of the adjoint equation. We discuss the variational assimilation principle, and prove the characteristics of the adjoint equation.
本文概述了變 分同化伴隨方程理論研究的重要性，讨論了變 分同化原理、證明了預報模式方程的伴隨方程解的特性；
The plastic process analyses on rectangular plate upsetting were made in detail on the basis of the principle of generalized variational inequality subjected to friction.
基于摩擦約束廣義變分不等式原理，對長矩形闆镦粗進行詳細的塑性加工工步分析。
For a thin-walled bar with openings, a characteristic equation of lateral stability is derived from energy principle and with the aid of variational method.
根據能量原理，借助于變分法導出一個考慮薄壁杆件扭轉和翹曲變形影響的側向穩定特征方程。
The auxiliary principle technique is extended to study a class of generalized strongly nonlinear mixed variational-like inequalities for set-valued mappings without compact values.
延拓輔助原理的技巧研究一類取非緊值的集值映象的廣義強非線性混合似變分不等式。
The auxiliary principle technique is extended to investigate a class of generalized set-valued strongly nonlinear variational-like inequalities.
本文延拓輔助原理的技巧，來研究一類廣義集值強非線性混合似變分不等式。
|variation principle;[數]變分原理（等于variation principle）
變分原理（variational principle）是數學和物理學中的核心概念，指通過尋找泛函的極值（如最小值或駐值）來描述自然現象或優化問題的基本規律。其核心思想是：真實發生的物理過程或幾何結構往往對應某種能量、作用量或其他量度的極值狀态。
數學基礎
變分原理将微分方程問題轉化為泛函的極值問題，例如在微分幾何中，測地線對應兩點間弧長的極值；在彈性力學中，平衡态對應勢能最小化。
物理學經典案例
在機器學習領域，變分自編碼器（VAE）通過最大化證據下界實現數據生成；有限元分析中則用于求解複雜結構的近似解。
權威參考來源
斯坦福哲學百科全書《變分方法》plato.stanford.edu/entries/variational-methods
劍橋大學數學手冊《變分原理》cambridge.org/core/books
美國物理學會《量子力學原理》aps.org/journals/pr
數學與應用數學學院《計算物理方法》springer.com/series/3424
變分原理（Variational Principle）是數學和物理學中的一個核心概念，指通過尋找某個泛函（函數的函數）的極值（如最小值或最大值）來描述自然現象或系統行為的原理。以下是詳細解釋：
變分原理的核心思想是：自然系統傾向于選擇使某個特定量達到極值的路徑或狀态。例如：
變分問題通常涉及泛函的極值求解： $$ delta mathcal{F}[y] = 0 $$ 其中，$mathcal{F}[y]$ 是依賴于函數 $y(x)$ 的泛函，$delta$ 表示變分（即對函數 $y(x)$ 的微小擾動）。通過歐拉-拉格朗日方程可求解極值函數： $$ frac{partial L}{partial y} - frac{d}{dx} left( frac{partial L}{partial y'} right) = 0 $$ 這裡 $L$ 是拉格朗日量，$y'$ 是 $y$ 對 $x$ 的導數。
變分原理提供了一種統一框架，将物理定律轉化為數學優化問題，簡化了複雜系統的建模與分析。它也是量子力學、熱力學等領域的理論基礎之一。
如果需要更具體的數學推導或物理案例，可進一步說明！
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