@inbook{bedcd07ef1f54725971961c70083b17a,

title = "A Primer on Noncommutative Classical Dynamics on Velocity Phase Space and Souriau Formalism",

abstract = "We present a comprehensive survey on dynamics of the motion of particles with noncommutative Poisson structure. We use Souriau{\textquoteright}s method of orbit to study this exotic mechanics on the tangent bundle of the configuration space or velocity phase space. We consider Feynman-Dyson{\textquoteright}s proof of Maxwell{\textquoteright}s equations using Jacobi identity on the velocity phase space. In this review we generalize the Feynman-Dyson{\textquoteright}s scheme by incorporating the non-commutativity between various spatial coordinates along with the velocity coordinates. This allows us to study a generalized class of Hamiltonian systems. We explore various dynamical flows associated to the Souriau form associated to this generalized Feynman-Dyson{\textquoteright}s scheme. Moreover, using the Souriau form we show that these new classes of generalized systems are volume preserving mechanical systems.",

keywords = "Feynman-Dyson{\textquoteright}s method, Generalized Hamiltonian dynamics, Kostant-Kirillov two form, Noncommutativity, Poisson manifolds, Schouten-Nijenhuis bracket, Souriau form",

author = "Cari{\~n}ena, {Jos{\'e} F.} and H{\'e}ctor Figueroa and Partha Guha",

note = "Publisher Copyright: {\textcopyright} 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.",

year = "2023",

doi = "10.1007/978-3-031-39334-1_12",

language = "British English",

series = "STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics and Health",

publisher = "Springer Nature",

pages = "533--568",

booktitle = "STEAM-H",

address = "United Kingdom",

}