Advanced methods in applied mathematics, lecture course

Cover......Page 1 Contents......Page 5 Introduction......Page 7 1. Examples......Page 12 2. Free Vibrations......Page 14 3. Forced Vibrations......Page 18 4. General External Force f(t)......Page 20 5. Energy Balance......Page 22 6. Resonance Phenomena......Page 23 7. Recording Instruments......Page 26 3. Superposition of Different Vibrations with Random Phases......Page 29 1. Two Degrees of Freedom......Page 37 2. Systems with n Degrees of Freedom......Page 41 4. Influence of Friction......Page 47 1. The Frequencies as Successive Minima......Page 49 2. Constraints - Frequencies as Maximum -Minimum......Page 52 3. Loss of Degrees of Freedom by Continuous Processes. Behavior of the Spectrum under Changes of the System......Page 57 1. An Example of the Method......Page 59 2. The Method in General......Page 64 Introduction......Page 67 1 The String......Page 69 2. The Membrane......Page 73 3. The question of Small Deviations......Page 76 1. General Formulation......Page 77 2. The Boundary Value Problem for the Circular Membrane......Page 78 3. Integral Representation of the Solution for the Circle......Page 83 4. The RectangularMembrane......Page 85 1. The Laplace Difference Equation......Page 90 2. Boundary Value Problem in a Not......Page 93 3. Theoretical Remarks......Page 99 1. The Non-homogeneous and Homogeneous Equations......Page 102 2. The Boundary Value Problem for the Poisson Equation in a Square Domain......Page 105 3. Equilibrium under Concentrated External Forces......Page 108 4. Green's Function for the Elastic String......Page 111 5. Green's Function for the Elastic Membrane......Page 113 6. Further Examples of Green's Function......Page 116 7. Realization of Green's Function by Physical Means......Page 120 1. General Theory of Natural. Vibrations and Natural Modes......Page 122 2. The Vibrating String......Page 126 3. The Impulse Method......Page 131 4. The Non-homogeneous String......Page 135 5. Strings with Other Boundary Conditions......Page 137 6. The Square Membrane......Page 139 7. The Circular Membrane......Page 143 Introduction......Page 146 1. Orthogonality and Comnleteness of the Eigen-functions......Page 148 2. Equivalence of the Impulse and Expansion Methods......Page 154 3. Green's Function Expanded in Torms of Eigen-functions......Page 157 4. Asymptotic Behavior of the Eigen-value......Page 159 Part I: Introduction......Page 163 1. Homogeneous Quadratic "Functionals"......Page 168 2. Technicalities of the Calculus of Variations......Page 172 3. Natural Boundary Conditions......Page 178 4. Remarks Concerning Approximation of Functions......Page 186 1. Equilibrium Problems......Page 188 2. Further Examples and Remarks......Page 190 3. Applications to the Theory of Vibrations......Page 193 Part IV: Critical Remarks. Other Methods......Page 198