**Overview of Optimization Calculus of Vibrations** |

Lecture 01 - Classification of Optimization Problems and the Place of Calculus of Variations in it |

Lecture 02 - Classification of Optimization Problems and the Place of Calculus of Variations in it (cont.) |

Lecture 03 - Genesis of Calculus of Variations |

Lecture 04 - Genesis of Calculus of Variations (cont.) |

Lecture 05 - Formulation of Calculus of Variations Problems in Geometry and Mechanics |

Lecture 06 - Formulation of Calculus of Variations Problems in Geometry and Mechanics (cont.) |

**Summary of Finite Variable Optimization** |

Lecture 07 - Unconstrained Minimization in One and Many Variables |

Lecture 08 - Unconstrained Minimization in One and Many Variables (cont.) |

Lecture 09 - Constrained Minimization KKT Conditions |

Lecture 10 - Constrained Minimization KKT Conditions (cont.) |

Lecture 11 - Sufficient Conditions for Constrained Minimization |

Lecture 12 - Sufficient Conditions for Constrained Minimization (cont.) |

**Mathematical Preliminaries for Calculus of Variations** |

Lecture 13 - Function and Functional, Metrics and Metric Space, Norm and Vector Spaces |

Lecture 14 - Function and Functional, Metrics and Metric Space, Norm and Vector Spaces (cont.) |

Lecture 15 - Function Spaces and Gateaux Variation |

Lecture 16 - First Variation of a Functional Frechet Differential and Variational Derivative |

Lecture 17 - Fundamental Lemma of Calculus of Variations and Euler-Lagrange Equation |

Lecture 18 - Fundamental Lemma of Calculus of Variations and Euler-Lagrange Equation (cont.) |

**Euler-Lagrange Equation with and without Constraints** |

Lecture 19 - Extension of Euler-Lagrange Equation to Multiple Derivatives |

Lecture 20 - Extension of Euler-Lagrange Equation to Multiple Functions in a Functional |

Lecture 21 - Global Constraints in Calculus of Variations |

Lecture 22 - Global Constraints in Calculus of Variations (cont.) |

Lecture 23 - Local (Finite Subsidiary) Constraints in Calculus of Variations |

Lecture 24 - Local (Finite Subsidiary) Constraints in Calculus of Variations (cont.) |

**Size Optimization of a Bar for Maximum Stiffness for Given Volume** |

Lecture 25 - Size Optimization of a Bar for Maximum Stiffness for Given Volume I |

Lecture 26 - Size Optimization of a Bar for Maximum Stiffness for Given Volume II |

Lecture 27 - Size Optimization of a Bar for Maximum Stiffness for Given Volume III |

Lecture 28 - Calculus of Variations in Functionals involving Two and Three Independent Variables |

Lecture 29 - Calculus of Variations in Functionals involving Two and Three Independent Variables (cont.) |

**Advanced Concepts and General Framework for Optimal Structural Design** |

Lecture 30 - General Variation of a Functional, Transversality Conditions; Broken Examples, Weierstrass-Erdmann Corner Conditions |

Lecture 31 - General Variation of a Functional, Transversality Conditions; Broken Examples, Weierstrass-Erdmann Corner Conditions (cont.) |

Lecture 32 - Variational (Energy) Methods in Statics; Principles of Minimum Potential Energy and Virtual Work |

Lecture 33 - General Framework of Optimal Structural Designs |

Lecture 34 - General Framework of Optimal Structural Designs (cont.) |

Lecture 35 - Optimal Structural Design of Bars and Beams using the Optimality Criteria Method |

**First Integrals, Invariants, and Noether's Theorem and Minimum Characterization of Eigenvalue Problems** |

Lecture 36 - Invariants of Euler-Lagrange Equation and Canonical Forms |

Lecture 37 - Noether's Theorem |

Lecture 38 - Minimum Characterization of Sturm-Liouville Problems |

Lecture 39 - Rayleigh Quotient for Natural Frequencies and Mode Shapes of Elastic Systems |

Lecture 40 - Stability Analysis and Buckling using Calculus of Variations |

**Optimal Structural Design and Inverse of Euler-Lagrange Equation** |

Lecture 41 - Strongest (Most Stable) Column |

Lecture 42 - Dynamic Compliance Optimization |

Lecture 43 - Electro-thermal-elastic Structural Optimization |

Lecture 44 - Formulating the Extremization Problem |