We consider Lagrangians in Hamilton's principle defined on the tangent space $TG$ of a Lie group $G$. Invariance of such a Lagrangian under the action of $G$ leads to the symmetry-reduced Euler-Lagrange equations called the Euler-Poincar\'e equations. In this case, the invariant Lagrangian is defined on the Lie algebra of the group and its Euler-Poincar\'e equations are defined on the dual Lie algebra, where dual is defined by the operation of taking variational derivative. On the Hamiltonian side, the Euler-Poincar\'e equations are Lie-Poisson and they possess accompanying momentum maps, which encode both their conservation laws and the geometry of their solution space. The standard Euler-Poincar\'e examples are treated, including particle dynamics, the rigid body, the heavy top and geodesic motion on Lie groups. Additional topics deal with Fermat's principle, the $\mathbb{R}^3$ Poisson bracket, polarized optical traveling waves, deformable bodies (Riemann ellipsoids) and shallow water waves, including the integrable shallow water wave systems associated with geodesic motion on the diffeomorphisms. The lectures end with the semidirect-product Euler-Poincar\'e reduction theorem for ideal fluid dynamics. This theorem introduces the Euler--Poincar\'e variational principle for incompressible and compressible motions of ideal fluids, with applications to geophysical fluids. It also leads to their Lie-Poisson Hamiltonian formulation.
Deep Dive into Applications of Poisson Geometry to Physical Problems.
We consider Lagrangians in Hamilton’s principle defined on the tangent space $TG$ of a Lie group $G$. Invariance of such a Lagrangian under the action of $G$ leads to the symmetry-reduced Euler-Lagrange equations called the Euler-Poincar'e equations. In this case, the invariant Lagrangian is defined on the Lie algebra of the group and its Euler-Poincar'e equations are defined on the dual Lie algebra, where dual is defined by the operation of taking variational derivative. On the Hamiltonian side, the Euler-Poincar'e equations are Lie-Poisson and they possess accompanying momentum maps, which encode both their conservation laws and the geometry of their solution space. The standard Euler-Poincar'e examples are treated, including particle dynamics, the rigid body, the heavy top and geodesic motion on Lie groups. Additional topics deal with Fermat’s principle, the $\mathbb{R}^3$ Poisson bracket, polarized optical traveling waves, deformable bodies (Riemann ellipsoids) and shallow wa
These being lecture notes for a summer school, one should not seek original material in them. Rather, the most one could hope to find would be the insight arising from incorporating a unified approach (based on reduction by symmetry of Hamilton's principle) with some novel applications. I hope the reader will find insight in the lecture notes, which are meant to be informal, more like stepping stones than a proper path.
Many excellent encyclopedic texts have already been published on the foundations of this subject and its links to symplectic and Poisson geometry. See, for example, [AbMa1978], [Ar1979], [GuSt1984], [JoSa98], [LiMa1987], [MaRa1994], [McSa1995] and many more. In fact, the scope encompassed by the modern literature on this subject is a bit overwhelming. In following the symmetry-reduction theme in geometric mechanics from the Euler-Poincaré viewpoint, I have tried to select only the material the student will find absolutely necessary for solving the problems and exercises, at the level of a beginning postgraduate student. The primary references are [Ma1992], [MaRa1994], [Le2003], [Bl2004], [RaTuSbSoTe2005]. Other very useful references are [ArKh1998] and [Ol2000]. The reader may see the strong influences of all these references in these lecture notes, but expressed at a considerably lower level of mathematical sophistication than the originals.
The scope of these lectures is quite limited: a list of the topics in geometric mechanics not included in these lectures would fill volumes! The necessary elements of calculus on smooth manifolds and the basics of Lie group theory are only briefly described here, because these topics were discussed in more depth by other lecturers at the summer school. Occasional handouts are included that add a bit more depth in certain key topics. The main subject of these lecture notes is the use of Lie symmetries in Hamilton’s principle to derive symmetry-reduced equations of motion and to analyze their solutions. The Legendre transformation provides the Hamiltonian formulation of these equations in terms of Lie-Poisson brackets.
For example, we consider Lagrangians in Hamilton’s principle defined on the tangent space T G of a Lie group G. Invariance of such a Lagrangian under the action of G leads to the symmetry-reduced Euler-Lagrange equations called the Euler-Poincaré equations. In this case, the invariant Lagrangian is defined on the Lie algebra of the group and its Euler-Poincaré equations are defined on the dual Lie algebra, where dual is defined by the operation of taking variational derivative. On the Hamiltonian side, the Euler-Poincaré equations are Lie-Poisson and they possess accompanying momentum maps, which encode both their conservation laws and the geometry of their solution space.
The standard Euler-Poincaré examples are treated, including particle dynamics, the rigid body, the heavy top and geodesic motion on Lie groups. Additional topics deal with Fermat’s principle, the R 3 Poisson bracket, polarized optical traveling waves, deformable bodies (Riemann ellipsoids) and shallow water waves, including the integrable shallow water wave system known as the Camassa-Holm equation. The lectures end with the semidirect-product Euler-Poincaré reduction theorem for ideal fluid dynamics. This theorem introduces the Euler-Poincaré variational principle for incompressible and compressible motions of ideal fluids, with applications to geophysical fluids. It also leads to their Lie-Poisson Hamiltonian formulation.
Some of these lectures were first given at the MASIE (Mechanics and Symmetry in Europe) summer school in 2000 [Ho2005]. I am grateful to the MASIE participants for their helpful remarks and suggestions which led to many improvements in those lectures. For their feedback and comments, I am also grateful to my colleagues at Imperial College London, especially Colin Cotter, Matthew Dixon, J. D. Gibbon, J. Gibbons, G. Gottwald, J.T. Stuart, J.-L. Thiffeault, Cesare Tronci and the students who attended these lectures in my classes at Imperial College. After each class, the students were requested to turn in a response sheet on which they answered two questions. These questions were, “What was this class about?” and “What question would you like to see pursued in the class?” The answers to these questions helped keep the lectures on track with the interests and understanding of the students and it enfranchised the students because they themselves selected the material in several of the lectures.
I am enormously grateful to many friends and colleagues whose encouragement, advice and support have helped sustained my interest in this field over the years. I am particularly grateful to J. E. Marsden, T. S. Ratiu and A. Weinstein for their faithful comraderie in many research endeavors. Outlook: The variational principles and the Poisson brackets for the rigid body and the heavy top provide models of a general construction associated to Euler-Poincaré reduc
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