1st Edition

# General Quantum Variational Calculus

Quantum calculus is the modern name for the investigation of calculus without limits. The quantum calculus or q-calculus began with FH Jackson in the early twentieth century, but this kind of calculus had already been worked out by Euler and Jacobi.

Recently, it arose interest due to high demand of mathematics that model quantum computing. The q-calculus appeared as a connection between mathematics and physics. It has a lot of applications in different mathematical areas such as number theory, combinatorics, orthogonal polynomials, basic hypergeometric functions and other sciences quantum theory, mechanics, and the theory of relativity. Recently, the concept of general quantum difference operators that generalize the quantum calculus are defined.

This book is specially designed for those who wish to understand general quantum variational calculus. This book encompasses recent developments of general quantum variational calculus. The text material of this book is presented in highly readable, mathematically solid format. Many practical problems are illustrated displaying a wide variety of solution techniques.

This book is addressed to a wide audience of specialists such as mathematicians, physicists, engineers and biologists. It can be used as a textbook at the graduate level and as a reference book for several disciplines

1. Elements of the Multimensional General Quantum Calculus

1.1 The Multidimensional General Quantum Calculus

1.2 Line Integrals

1.3 The Green Formula

1.4 Advanced Practical Problems

2. β-Differential Systems

2.1 Structure of β-Differential Systems

2.2 β-Matrix Exponential Function

2.3 The β-Liouville Theorem

2.4 Constant Coefficients

2.5 Nonlinear Systems

2.6 Advanced Practical Problems

3. Functionals

3.1 Definition for Functionals

3.2 Self-Adjoint Second Order Matrix Equations

3.3 The Jacobi Condition

3.4 Sturmian Theory

4. Linear Hamiltonian Dynamic Systems

4.1 Linear Symplectic Dynamic Systems

4.2 Hamiltonian Systems

4.3 Conjoined Bases

4.4 Riccati Equations

4.5 The Picone Identity

4.6 ”Big” Linear Hamiltonian Systems

4.7 Positivity of Quadratic Functionals

5. The First Variation

5.1 The Dubois-Reymond Lemma

5.2 The Variational Problem

5.3 The Euler-Lagrange Equation

5.4 The Legendre Condition

5.5 The Jacobi Condition

5.6 Advanced Practical Problems

6. Higher Order Calculus of Variations

6.1 Statement of the Variational Problem

6.2 The Euler Equation

6.3 Advanced Practical Problems

7. Double Integral Calculus of Variations

7.1 Statement of the Variational Problem

7.2 First and Second Variation

7.3 The Euler Condition

7.4 Advanced Practical Problems

8. The Noether Second Theorem

8.1 Invariance under Transformations

8.2 The Noether Second Theorem without Transformations of Time

8.3 The Noether Second Theorem with Transformations of Time

8.4 The Noether Second Theorem-Double Delta Integral Case

References

Index

### Biography

**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 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.