Extended Lagrange and Hamilton Formalism for Point Mechanics and Covariant Hamilton Field Theory

Contents
Preface
Foreword
List of Figures
Conventional Lagrange and Hamilton Formalism for Point Mechanics
	1. Conventional Lagrange Formalism
		1.1 Newton’s axioms
		1.2 Generalized coordinates
		1.3 Covariant and contravariant vectors and tensors
		1.4 Action principle, Lagrange function
		1.5 Euler-Lagrange equations of motion
		1.6 Energy function, autonomous Lagrangian systems
		1.7 Energy function for higher derivative Lagrangian systems
		1.8 Non-uniqueness of the Lagrangian
		1.9 Excursus: Non-relativistic quantum action principle
	2. Conventional Hamilton Formalism
		2.1 Legendre transformation in one dimension
		2.2 Legendre transformation for higher dimensions
		2.3 Transition from a Lagrangian to a Hamiltonian
		2.4 Excursus: Basics of symplectic geometry
		2.5 Action principle in the Hamiltonian formulation
		2.6 Hamilton’s equations of motion (canonical equations)
		2.7 Canonical equations for hybrid Hamiltons
		2.8 Autonomous (time-independent) Hamiltonian systems
	3. Canonical Transformation Theory
		3.1 Generating functions of canonical transformations
		3.2 Symmetry relations
		3.3 General form of Liouville’s theorem
		3.4 Canonical transformations as symplectic maps
		3.5 Canonical invariance of the Poisson brackets
		3.6 Poisson’s theorem
Extended Lagrange and Hamilton Formalisms for Point Mechanics
	4. Extended Lagrange Formalism
		4.1 The extended Lagrangian
		4.2 Extended set of Euler-Lagrange equations
		4.3 Correlation between L and Le
			4.3.1 Trivial extended Lagrangian
			4.3.2 Non-trivial extended Lagrangian
		4.4 Noether’s theorem in the extended Lagrangian formalism
		4.5 Excursus: Relativistic path integral with extended Lagrangians
	5. The Extended Hamiltonian in Point Mechanics
		5.1 Extended Hamiltonian from the generalized action functional
		5.2 Extended Hamiltonian as Legendre transform of an extended Lagrangian
		5.3 Extended set of canonical equations
		5.4 Canonical quantization in the extended Hamiltonian formalism
	6. Theory of Extended Canonical Transformations
		6.1 Extended canonical transformations
		6.2 Infinitesimal extended canonical transformations, generalized Noether theorem
		6.3 Conventional Noether theorem
		6.4 Energy-second-moment map
		6.5 Liouville’s theorem in the extended Hamilton description
		6.6 Galilei and Lorentz transformation as extended canonical transformations
			6.6.1 The Galilei transformation
			6.6.2 The Lorentz transformation
		6.7 Canonical invariance of extended Poisson brackets
		6.8 Extended Hamilton-Jacobi equation
Covariant Hamilton Field Theory under Fixed Spacetime
	7. Covariant Canonical Field Equations
		7.1 Euler-Lagrange equations of field theory
		7.2 Non-uniqueness of the Lagrangian
		7.3 Covariant canonical field equations
		7.4 Canonical energy-momentum tensor
		7.5 Non-uniqueness of the conjugate momentum fields πμI
		7.6 Hybrid Hamiltonians
	8. Canonical Transformations in Covariant Hamiltonian Field Theory
		8.1 Generating functions of type F1(ϕI, ΦI, x)
		8.2 Canonical transformation rule for the energy-momentum tensor
		8.3 Generating functions of type F2(ϕI, ΠI, x)
		8.4 Generating functions of type F3(ΦI, πI, x)
		8.5 Generating functions of type F4(πI, ΠI, x)
		8.6 Consistency checks of canonical transformation rules
		8.7 Poisson brackets
		8.8 Canonical invariance of Poisson brackets
		8.9 Liouville’s theorem of covariant Hamiltonian field theory
		8.10 Jacobi’s identity theorem in canonical field theory
		8.11 Poisson’s theorem in canonical field theory
		8.12 Hamilton-Jacobi equation
	9. Examples of Hamiltonians in Covariant Field Theory
		9.1 Ginzburg-Landau Hamiltonian
		9.2 Klein-Gordon Hamiltonian for a real scalar field
		9.3 Klein-Gordon Hamiltonian for complex fields
		9.4 Maxwell Hamiltonian
		9.5 Proca Hamiltonian
		9.6 Coupled Klein-Gordon-Maxwell system
		9.7 Covariant Dirac Hamiltonian
			9.7.1 Conventional Dirac Lagrangian
			9.7.2 Lorentz invariance of the Dirac equation
			9.7.3 Regularized Dirac Lagrangian, Dirac Hamiltonian
	10. Examples of Canonical Transformations in Covariant Hamiltonian Field Theory
		10.1 Point transformations
		10.2 Canonical shift of the conjugate momentum vector field πμI
		10.3 Global and local gauge transformation of the fields ϕI
		10.4 Interchange of canonical spinor variables
		10.5 Generalized Noether theorem in the realm of field theories
		10.6 Canonical transformation inducing an infinitesimal spacetime step
		10.7 Gauge invariance of the electromagnetic 4-potential
		10.8 Symmetry transformation of the coupled Klein-Gordon-Maxwell field and the pertaining conserved Noether current
	11. U(1) and SU(N) Gauge Theories in the Hamiltonian Formulation
		11.1 Gauge theories as canonical transformations
		11.2 U(1) gauge theory
		11.3 SU(N) gauge theory
			11.3.1 External gauge field
			11.3.2 Inclusion of the gauge field dynamics
			11.3.3 Hamiltonian of the free (uncoupled) gauge field
			11.3.4 Inserting the gauge-invariant Hamiltonian H3 into the action integral
			11.3.5 Locally gauge-invariant Lagrangian, Legendre transformation for a general system Hamiltonian
			11.3.6 Klein-Gordon type system Hamiltonian
			11.3.7 Hamiltonian of the Dirac system
			11.3.8 Comparison with Pauli’s amended Lagrangian
		11.4 Noether current of the SU(N) gauge transformation
Covariant Hamilton Field Theory with Spacetime as a Dynamical Variable
	12. General Spacetime Transformation of Systems of Scalar, Vector, and Tensor Fields
		12.1 Relative tensors and their transformation rules
			12.1.1 Connection coefficients and covariant derivative
		12.2 Covariant derivative of a tensor density
		12.3 Extended Lagrangians and Hamiltonians in covariant field theories
			12.3.1 Extended Klein-Gordon Lagrangian
			12.3.2 Extended Klein-Gordon Hamiltonian
			12.3.3 Extended Proca Lagrangian
			12.3.4 Extended Proca Hamiltonian
		12.4 Canonical transformations under a dynamic spacetime for real scalar and vector fields
	13. Gauge Theory of Gravity for Real Scalar and Vector Fields
		13.1 The generating function for diffeomorphisms
		13.2 Inclusion of the dynamics of the gauge fields
		13.3 Insertion of the enhanced Hamiltonian H2 into the action integral
		13.4 Addition of the “free gravity” Hamiltonian
		13.5 Canonical field equations
			13.5.1 Field equations for ϕ and πν
			13.5.2 Field equations for aμ and p μν
			13.5.3 Field equations for gαβ and kαβμ
			13.5.4 Field equations for γηαβ and qηαβν
			13.5.5 Summary of the coupled set of field equations
		13.6 Consistency relation
		13.7 Summary of the gauge procedure
	14. Applications of the Gauge Theory of Gravity
		14.1 Lagrangian description
		14.2 Sample HGr
		14.3 Zero vector field aμ, non-zero scalar field ϕ
		14.4 Classical vacuum solution for a Lagrangian LGr with linear Riemann tensor dependence
		14.5 Classical vacuum solution for a Lagrangian LGr with quadratic Riemann tensor dependence
		14.6 Covariant divergence of the quadratic Riemann tensor expression Q and of the Einstein tensor G
		14.7 Robertson-Walker metric, generalized Friedmann equation
	15. SU(N) × SO(3, 1) × Diff(M) Gauge Theory
		15.1 The SU(N) × SO(3, 1) symmetry group
		15.2 Generalization to the SU(N) × SO(3, 1) × Diff(M) symmetry group
Appendix A Solutions to the Exercises
Bibliography
Index