變分學可看待為微積分某方向的一個延續,也是應用數學的基本技巧之一;其他領域如幾何、微分方程、物理力學等和變分法的互動也有一段很長的歷史;最近也發現,變分法也可應用到經濟學和電機學上。雖然這門學問源於應用問題,但它在數學上的想法、觀念和技巧都具有它優雅的一面。在這個課程中,我們會探討變分法的基礎理論和觀念,訓練學生在這方面的計算技巧和欣賞一些古典問題如何在變分法中得到解答。課程內容大概如下:
-
Introductory
problems: The Catenary, Brachystochrone, Dido (Isoperimetric), Geodesics,
Optimal Harvest Strategy problems.
-
The First
Variation: The Euler-Lagrange Equation.
-
Isoperimetric
problems: Lagrange Multipliers, single and multiple constraints, abnormal
problems.
-
Holonomic and
Nonholonomic constrains, variable endpoints.
-
The Hamiltonian
Formulation: Legrendre transforms, Hamiltonian equations.
-
The Second
Variation: Legendre and Jacobi conditions, a sufficient condition for minimizers,
conjugate points.
以下為常用之參考文獻:(前兩本為常用教科書)
·
Van Brunt: The
Calculus of Variations.
·
Troutman:
Variational Calculus and Optimal Control: Optimization with Elementary
Convexity.
·
Giaquinta &
Hildebrandt: Calculus of Variations I, II.
·
Bliss: Calculus
of Variations.
·
Gelfand & Fomin:
Calculus of Variations.
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