Note: Intro -- 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 -- 1.5.1 Vorticity Equation -- 1.5.2 Planar Flows with Constant Density -- 1.6 Elementary Symmetries of the Vorticity Equation -- 1.6.1 Continuous Symmetries -- 1.6.2 Discrete Symmetries -- 1.6.3 Breaking of Symmetries of the Vorticity Equation in the β Plane -- 1.7 Energy and Enstrophy Conservation -- 1.8 Conservation Laws -- 1.8.1 Kelvin's Circulation Theorem and Conservation of Circulation -- 1.8.2 Potential Vorticity and Ertel's Theorem -- 1.9 Conservation of Potential Vorticity and Models of Geophysical Flows -- 1.9.1 Shallow-Water Model with Primitive Equations -- 1.9.2 Quasi-geostrophic Shallow-Water Model -- 1.9.3 Energy and Enstrophy Conservation for the Quasi-geostrophic Shallow Water Model -- 1.9.4 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 Coordinates -- 2.5 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 Function -- 2.10.1 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 -- 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 -- 3 Variational Principles in Fluid Dynamics, Symmetries and Conservation Laws -- 3.1 Introduction: Lagrangian Coordinates and Labels -- 3.2 Lagrangian Density in Labelling Space -- 3.2.1 Hamilton's Equations -- 3.3 Hamilton's Principle for Fluids -- 3.4 Hamilton's Principle in the Eulerian Framework -- 3.4.1 Equivalence of the Lagrangian and Eulerian Forms of Hamilton's Principle -- 3.5 Symmetries and Conservation Laws -- 3.5.1 Preliminaries and Notation -- 3.5.2 Time Translations Symmetry -- 3.5.3 Particle Relabelling Symmetry -- 3.6 Bibliographical Note -- References -- 4 Variational Principles in Geophysical Fluid Dynamics and Approximated Equations -- 4.1 Introduction. , 4.2 Hamilton's Principle, Rotation and Incompressibility -- 4.2.1 Lagrangian Density in a Rotating Frame of Reference -- 4.2.2 Relabelling Symmetry in a Rotating Framework -- 4.2.3 Role of Incompressibility -- 4.3 A Finite Dimensional Example: Dynamics of Point Vortices -- 4.4 Approximated Equations -- 4.4.1 Rotating Shallow Water Equations -- 4.4.2 Two-Layer Shallow Water Equations -- 4.4.3 Rotating Green--Naghdi Equations -- 4.4.4 Shallow Water Semi-geostrophic Dynamics -- 4.4.5 Continuously Stratified Fluid -- 4.5 Selected Topics in Wave Dynamics -- 4.5.1 Potential Flows and Surface Water Waves -- 4.5.2 Luke's Variational Principle -- 4.5.3 Whitham's Averaged Variational Principle and Conservation of Wave Activity -- 4.5.4 Example 1: The Linear Klein--Gordon Equation -- 4.5.5 Example 2: The Nonlinear Klein--Gordon Equation -- 4.5.6 Example 3: The Korteweg--DeVries (KdV) Equation -- 4.6 Bibliographical Note and Suggestions for Further Reading -- References -- Appendix A Derivation of Equation 1.21.2) -- Appendix B Derivation of the Conservation of Potential Vorticity from Kelvin's Circulation Theorem -- Appendix C Some Simple Mathematical Properties of the Legendre Transformation -- Appendix D Derivation of Equation 2.142 -- Appendix E Invariance of the Equations of Motion 2.144 Under a Divergence Transformation -- Appendix F Functional Derivatives -- Appendix G Derivation of Equation 2.229 -- Appendix H Invariance of the Equations of Motion 2.217 Under a Divergence Transformation -- Appendix I Proofs of the Algebraic Properties of the Poisson Bracket -- Appendix J Some Identities Concerning the Jacobi Determinant -- Appendix K Derivation of 3.131 -- Appendix L Scaling the Rotating Shallow Water Lagrangian Density.