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Mechanics by Landau and Lifshitz is the start of a profound transformation of the reader from basic calculus to all of mathematics and physics. The seeds of nearly all modern subjects of interest can be found here and in the following volumes starting with the principle of least action in a coordinate invariant manner. Coordinate invariance means there is no artificial distinction between linear and angular momenta, because they are simply the momenta measured along different (cartesian vs spherical) coordinates. Further, this sets the way for physical reasoning by focusing on the geometrical qualities of the trajectories of particles which is expanded upon in mathematics as variational calculus on manifolds and symplectic geometry respectively, to the beginning and end of the book. Fundamental formulae such as Newton's laws, Kepler's formulae of celestial mechanics, and the formulae for energy are derived simply from the principle of least action. Prerequisites are minimal, but strength of courage and to interpret physics simply are required. | Mechanics by Landau and Lifshitz is the start of a profound transformation of the reader from basic calculus to all of mathematics and physics. The seeds of nearly all modern subjects of interest can be found here and in the following volumes starting with the principle of least action in a coordinate invariant manner. Coordinate invariance means there is no artificial distinction between linear and angular momenta, because they are simply the momenta measured along different (cartesian vs spherical) coordinates. Further, this sets the way for physical reasoning by focusing on the geometrical qualities of the trajectories of particles which is expanded upon in mathematics as variational calculus on manifolds and symplectic geometry respectively, to the beginning and end of the book. Fundamental formulae such as Newton's laws, Kepler's formulae of celestial mechanics, and the formulae for energy are derived simply from the principle of least action. Prerequisites are minimal, but strength of courage and to interpret physics simply are required. | ||
Since Landau, the most significant change to the foundation of the subject other than the rewriting of the same laws in differential geometric language is the replacement of informal infinitesimal arguments with Lie Group and Lie Algebra constructions. Simply, a Lie Group represents a parametrized family of finite-change transformations such as rotations by an angle or translations by a vector. The Lie Algebra in an instantiation of a Lie Group transformation is then a vector field representing the transformation infinitesimally. The method is powerful because one can check if a quantity is conserved by checking infinitesimally, if the derivative in the direction of the vector field is 0. This can be done for higher dimensional quantities, not just scalars, through commutators or the Lie derivative - commutators are not unique to quantum mechanics. What it does is replaces some of the proofs in Landau's text with one line of Lie-algebraic manipulations. Since then, Lie Algebras have become a foundation of geometry as exhibited in Arnol'd's geometric perspective on ordinary differential equations and on the mathematical structure of classical mechanics. Olver's books applies these methods more generally to PDE theory and variational problems, giving radical simplifications in the form of otherwise difficult to solve equations via their Lie symmetries. As a side effect of the approach aiming to give geometric constructions of dynamical and variational problems, powerful computational techniques useful for applications appear. | |||
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. | |||
=== Applications === | === Applications === | ||