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The main differential geometric structure of Hamiltonian mechanics is a symplectic 2-form, an antisymmetric function linear in each of two tangent vectors. A manifold/coordinate system given a symplectic form is known as a symplectic manifold, and the form is used to exhibit the duality between the position and momentum coordinates in phase space. Constraining one's geometric structure constrains the set of possible transformations to those preserving the structure, canonical or symplectic transformations, and thus simplifying computations giving stronger geometric results. Rather than summarizing the results, we indicate a few of these structures derived from the symplectic form: moment(um) mappings associated to finite dimensional Lie subgroups of the symplectic group, Lagrangian submanifolds of a symplectic manifold, periodic orbits of a given system. All of these help to determine the topology of a symplectic manifold, and can be honest representations of the manifold itself, as well has having some influence in the process of quantization. Quantization based on symplectic manifolds appears in [[Quantum_Mechanics_(Book)#Geometric_Quantization|geometric quantization]]. The momentum map and basic phase space quantization is in [[Quantum_Mechanics_(Book)#Quantum_Theory,_Groups_and_Representations|Woit's book]]. | The main differential geometric structure of Hamiltonian mechanics is a symplectic 2-form, an antisymmetric function linear in each of two tangent vectors. A manifold/coordinate system given a symplectic form is known as a symplectic manifold, and the form is used to exhibit the duality between the position and momentum coordinates in phase space. Constraining one's geometric structure constrains the set of possible transformations to those preserving the structure, canonical or symplectic transformations, and thus simplifying computations giving stronger geometric results. Rather than summarizing the results, we indicate a few of these structures derived from the symplectic form: moment(um) mappings associated to finite dimensional Lie subgroups of the symplectic group, Lagrangian submanifolds of a symplectic manifold, periodic orbits of a given system. All of these help to determine the topology of a symplectic manifold, and can be honest representations of the manifold itself, as well has having some influence in the process of quantization. Quantization based on symplectic manifolds appears in [[Quantum_Mechanics_(Book)#Geometric_Quantization|geometric quantization]]. The momentum map and basic phase space quantization is in [[Quantum_Mechanics_(Book)#Quantum_Theory,_Groups_and_Representations|Woit's book]]. | ||
The next three books introduce symplectic geometry and cover different aspects and applications. First, Vladimir Arnol'd initiated the field of symplectic topology - utilizing symplectic structure as an invariant of the space. In his book he elaborates on the foundations of the subject in modern geometric language. In contrast, Sternberg and Guillemin's book discusses the history of how the concept of symplectic geometry emerged and how it continues to evolve. Consequently, it doesn't get as far into geometry but reaches various parts of physics such as optics, a geometric definition of quantum mechanical particles, Yang-Mills Fields, and the moment map. We single the moment map out here, as its role in quantum mechanics is to show how orbits in a quantum mechanical phase space map to points in the convex space of underlying probability distributions. Sternberg and Atiyah independently showed this convexity with a general setting in a landmark result. Finally, Hermann's book develops applications in control theory, electronic circuits, and develops further geometry for his purposes. His books are unique and self-published, so rather than inserting them all here [[Robert_Hermann#Interdisciplinary_Mathematics_Series|we compiled a list]]. He engaged in engineering topics such as in the conferences at NASA Ames, and this resulted in a profound new geometric approach to engineering "applied pure mathematics." Noteably, among topics such as control theory, algebraic geometry, computing, soliton waves, he wrote about gauge theory and is the unsung source of the realization that gauge theory is the language of particle physics and the standard model that is now associated with physicists Weinberg, Salam, Wilson, T'Hooft and mathematicians Atiyah and Bott. | The next three books introduce symplectic geometry and cover different aspects and applications. First, Vladimir Arnol'd initiated the field of symplectic topology - utilizing symplectic structure as an invariant of the space. In his book he elaborates on the foundations of the subject in modern geometric language. In contrast, Sternberg and Guillemin's book discusses the history of how the concept of symplectic geometry emerged and how it continues to evolve. Consequently, it doesn't get as far into geometry but reaches various parts of physics such as optics, a geometric definition of quantum mechanical particles, Yang-Mills Fields, and the moment map. We single the moment map out here, as its role in quantum mechanics is to show how orbits in a quantum mechanical phase space map to points in the convex space of underlying probability distributions. Sternberg and Atiyah independently showed this convexity with a general setting in a landmark result. Finally, Hermann's book develops applications in control theory, electronic circuits, and develops further geometry for his purposes. His books are unique and self-published, so rather than inserting them all here [[Robert_Hermann#Interdisciplinary_Mathematics_Series|we compiled a list]]. He engaged in engineering topics such as in the conferences at NASA Ames, and this resulted in a profound new geometric approach to engineering "applied pure mathematics." Noteably, among topics such as control theory, algebraic geometry, computing, soliton waves, he wrote about gauge theory and is the unsung source of the realization that gauge theory is the language of particle physics and the standard model that is now associated with physicists Weinberg, Salam, Wilson, T'Hooft, and mathematicians Atiyah and Bott. | ||
Often when new techniques or structures are found in higher level formalisms such as quantum field theory, they are found to have simpler instantiations that were implicit in existing classical mechanics techniques. From the symplectic point of view, we use vector fields with underlying scalar functions called Hamiltonians to establish an ODE for time evolution. Dynamical systems theory takes this, and generalizes to 'a qualitative theory of ODEs' where the phase space need not be position and momentum but any dimension space where the evolution happens. Either with Hamiltonians or other ODEs of dynamical systems, we can take a continuous family of different systems depending on a parameter (introduced in Arnol'd's Dynamical Systems IV) and see that there are discontinuous changes or 'bifurcations' in the orbit structure of the system as the parameter varies. In QFT, symmetry breaking/the Higgs mechanism and goldstone bosons/superconductivity also depend on a parameter at which there is a discontinuous phase change and an appearance of new particles. Bifurcation theory can be seen as the distillation of this from infinite dimensional QFT into finite dimensions, and Kuznetsov's book covers the basic theoretical and numerical techniques. The dynamical systems series includes bifurcation theory in volume 5, but we felt that since its role is as an encyclopedia and Kuznetsov's book is more original to choose to include his. | |||
=== Applications === | === Applications === | ||