General Quantum Variational Calculus

Svetlin G. Georgiev is a mathematician who has worked in various areas of mathematics. He currently focuses on harmonic analysis, functional analysis, partial differential equations, ordinary differential equations, Clifford and quaternion analysis, integral equations, and dynamic calculus on time scales.Khaled Zennir earned his PhD in mathematics in 2013 from Sidi Bel Abbès University, Algeria. In 2015, he received his highest diploma in Habilitation in mathematics from Constantine University, Algeria. He is currently an assistant professor at Qassim University in the Kingdom of Saudi Arabia. His research interests lie in the subjects of nonlinear hyperbolic partial differential equations: global existence, blowup, and longtime behavior.

• 1. Elements of the Multimensional General Quantum Calculus1.1 The Multidimensional General Quantum Calculus1.2 Line Integrals1.3 The Green Formula1.4 Advanced Practical Problems2. β-Differential Systems2.1 Structure of β-Differential Systems2.2 β-Matrix Exponential Function2.3 The β-Liouville Theorem2.4 Constant Coefficients2.5 Nonlinear Systems2.6 Advanced Practical Problems3. Functionals3.1 Definition for Functionals3.2 Self-Adjoint Second Order Matrix Equations3.3 The Jacobi Condition3.4 Sturmian Theory4. Linear Hamiltonian Dynamic Systems4.1 Linear Symplectic Dynamic Systems4.2 Hamiltonian Systems4.3 Conjoined Bases4.4 Riccati Equations4.5 The Picone Identity4.6 ”Big” Linear Hamiltonian Systems4.7 Positivity of Quadratic Functionals 5. The First Variation5.1 The Dubois-Reymond Lemma5.2 The Variational Problem5.3 The Euler-Lagrange Equation5.4 The Legendre Condition5.5 The Jacobi Condition5.6 Advanced Practical Problems6. Higher Order Calculus of Variations6.1 Statement of the Variational Problem6.2 The Euler Equation6.3 Advanced Practical Problems7. Double Integral Calculus of Variations7.1 Statement of the Variational Problem7.2 First and Second Variation7.3 The Euler Condition7.4 Advanced Practical Problems 8. The Noether Second Theorem8.1 Invariance under Transformations8.2 The Noether Second Theorem without Transformations of Time8.3 The Noether Second Theorem with Transformations of Time8.4 The Noether Second Theorem-Double Delta Integral CaseReferencesIndex