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