On geometric formulation of mechanics

Geometry of symplectic and Poisson manifolds is well known to provide the adequate mathematical formulation of autonomous Hamiltonian mechanics. The literature on this subject is extensive. Our recent book

 Geometric Formulation of Classical and Quantum Mechanics (WS, 2010)

presents the advanced geometric formulation of classical and quantum non-relativistic mechanics in a general setting of time-dependent coordinate and reference frame transformations. This formulation of mechanics as like as that of classical field theory (Advanced Classical Field Theory) lies in the framework of general theory of dynamic systems, Lagrangian and Hamiltonian formalism on fibre bundles.

Classical non-relativistic mechanics is formulated as a particular field theory on smooth fibre bundles over the time axis R. Quantum non-relativistic mechanics is phrased in the geometric terms of Banach and Hilbert bundles and connections on these bundles. A quantization scheme speaking this language is geometric quantization. Relativistic mechanics is adequately formulated as particular classical string theory of one-dimensional submanifolds.

Non-autonomous dynamic systems, Newtonian systems, Lagrangian and Hamiltonian non-relativistic mechanics, relativistic mechanics, quantum non-autonomous mechanics are considered.

The concept of a connection is the central ingredient in this geometric formulation. Connections on a configuration space of non-relativistic mechanics describe reference frames. Holonomic connections on a velocity space define non-relativistic dynamic equations. Hamiltonian connections in Hamiltonian non-relativistic mechanics define the Hamilton equations. Evolution of quantum systems is described in terms of algebraic connections. A connection on a prequantization bundle is the main ingredient in geometric quantization.

Our book provides a detailed exposition of theory of partially integrable and superintegrable systems and their quantization, classical and quantum non-autonomous constraint systems, Lagrangian and Hamiltonian theory of Jacobi fields, classical and quantum mechanics with time-dependent parameters, the technique of non-adiabatic holonomy operators, formalism of instantwise quantization and quantization with respect to different reference frames.

Reference:

G.Giachetta, L.Mangiarotti, G.Sardanashvily, Advanced mechanics. Mathematical introduction, arXiv: 0911.0411