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Lie Groups and Lie Algebras are ubiquitous in all of physics, typically introduced later in analogous situations in classical field theory or quantum mechanics, but are just as simply introduced in mechanics and make the analogies between the Hamiltonian formalisms in each area manifest. | Lie Groups and Lie Algebras are ubiquitous in all of physics, typically introduced later in analogous situations in classical field theory or quantum mechanics, but are just as simply introduced in mechanics and make the analogies between the Hamiltonian formalisms in each area manifest. | ||
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 variable 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. | ||