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Foreword; Preface; Acknowledgements; Contents; 1 Fundamental Equations of Fluid and Geophysical Fluid Dynamics; 1.1 Introduction; 1.2 The Continuum Hypothesis; 1.3 Derivation of the Equations of Motion; 1.3.1 Conservation of Mass; 1.3.2 Incompressibility and Density Conservation; 1.3.3 Momentum Equation in an Inertial Frame of Reference; 1.4 Elementary Symmetries of the Euler's Equation; 1.4.1 Continuous Symmetries; 1.4.2 Discrete Symmetries; 1.4.3 Role of Gravity in Breaking the Symmetries of the Euler's Equation; 1.5 Momentum Equation in a Uniformly Rotating Frame of Reference
1.9.3 Energy and Enstrophy Conservation for the Quasi-geostrophic Shallow Water Model1.9.4 Quasi-geostrophic Model of a Density Conserving Ocean; 1.9.5 Quasi-geostrophic Model of a Potential Temperature-Conserving Atmosphere; 1.9.6 Conservation of Pseudo-Enstrophy in a Baroclinic Quasi-geostrophic Model; 1.9.7 Surface Quasi-geostrophic Dynamics; 1.10 Bibliographical Note; References; 2 Mechanics, Symmetries and Noether's Theorem; 2.1 Introduction; 2.2 Hamilton's Principle of Least Action; 2.3 Lagrangian Function, Euler
Lagrange Equations and D'Alembert's Principle
2.4 Covariance of the Lagrangian with Respect to Generalized Coordinates2.5 Role of Constraints; 2.6 Canonical Variables and Hamiltonian Function; 2.7 Hamilton's Equations; 2.8 Canonical Transformations and Generating Functions; 2.8.1 Phase Space Volume as Canonical Invariant: Liouville's Theorem and Poisson Brackets; 2.8.2 Casimir Invariants and Invertible Systems; 2.9 Noether's Theorem for Point Particles; 2.9.1 Mathematical Preliminary; 2.9.2 Symmetry Transformations and Proof of the Theorem; 2.9.3 Some Examples
2.10 Lagrangian Formulation for Fields: Lagrangian Depending on a Scalar Function2.10.1 Hamiltonian for Scalar Fields; 2.11 Noether's Theorem for Fields with the Lagrangian Depending on a Scalar Function; 2.11.1 Mathematical Preliminary; 2.11.2 Linking Back to the Physics; 2.12 Lagrangian Formulation for Fields: Lagrangian Density #x83;; 2.12.1 Hamilton's Equations for Vector Fields; 2.12.2 Canonical Transformations and Generating Functionals for Vector Fields; 2.13 Noether's Theorem for Fields: Lagrangian Density Dependent on Vector Functions; 2.14 Bibliographical Note; References
1.9.3 Energy and Enstrophy Conservation for the Quasi-geostrophic Shallow Water Model1.9.4 Quasi-geostrophic Model of a Density Conserving Ocean; 1.9.5 Quasi-geostrophic Model of a Potential Temperature-Conserving Atmosphere; 1.9.6 Conservation of Pseudo-Enstrophy in a Baroclinic Quasi-geostrophic Model; 1.9.7 Surface Quasi-geostrophic Dynamics; 1.10 Bibliographical Note; References; 2 Mechanics, Symmetries and Noether's Theorem; 2.1 Introduction; 2.2 Hamilton's Principle of Least Action; 2.3 Lagrangian Function, Euler
Lagrange Equations and D'Alembert's Principle
2.4 Covariance of the Lagrangian with Respect to Generalized Coordinates2.5 Role of Constraints; 2.6 Canonical Variables and Hamiltonian Function; 2.7 Hamilton's Equations; 2.8 Canonical Transformations and Generating Functions; 2.8.1 Phase Space Volume as Canonical Invariant: Liouville's Theorem and Poisson Brackets; 2.8.2 Casimir Invariants and Invertible Systems; 2.9 Noether's Theorem for Point Particles; 2.9.1 Mathematical Preliminary; 2.9.2 Symmetry Transformations and Proof of the Theorem; 2.9.3 Some Examples
2.10 Lagrangian Formulation for Fields: Lagrangian Depending on a Scalar Function2.10.1 Hamiltonian for Scalar Fields; 2.11 Noether's Theorem for Fields with the Lagrangian Depending on a Scalar Function; 2.11.1 Mathematical Preliminary; 2.11.2 Linking Back to the Physics; 2.12 Lagrangian Formulation for Fields: Lagrangian Density #x83;; 2.12.1 Hamilton's Equations for Vector Fields; 2.12.2 Canonical Transformations and Generating Functionals for Vector Fields; 2.13 Noether's Theorem for Fields: Lagrangian Density Dependent on Vector Functions; 2.14 Bibliographical Note; References