Editing Generalized coordinates (section)

{{Short description|System configuration relative to another}}
{{Classical mechanics|cTopic=Formulations}}

In [[analytical mechanics]], '''generalized coordinates''' are a set of parameters used to represent the state of a system in a [[Configuration space (physics)|configuration space]]. These parameters must uniquely define the configuration of the system relative to a reference state.<ref name=Ginsberg>{{harvnb |Ginsberg |2008 }},
[https://books.google.com/books?id=je0W8N5oXd4C&pg=PA397 p. 397, &nbsp;§7.2.1 Selection of generalized coordinates]</ref> The '''generalized velocities''' are the time [[derivative]]s of the generalized coordinates of the system. The adjective "generalized" distinguishes these parameters from the traditional use of the term "coordinate" to refer to [[Cartesian coordinates]].

An example of a generalized coordinate would be to describe the position of a pendulum using the angle of the pendulum relative to vertical, rather than by the x and y position of the pendulum.

Although there may be many possible choices for generalized coordinates for a physical system, they are generally selected to simplify calculations, such as the solution of the [[equations of motion]] for the system. If the coordinates are independent of one another, the number of independent generalized coordinates is defined by the number of [[Degrees of freedom (mechanics)|degrees of freedom]] of the system.<ref name=Amirouche>{{cite book |title=Fundamentals of multibody dynamics: theory and applications |author=Farid M. L. Amirouche |chapter-url=https://books.google.com/books?id=_nlEcQYldeIC&pg=PA46 |page=46 |chapter=§2.4: Generalized coordinates |publisher=Springer |isbn=0-8176-4236-6 |year=2006}}</ref><ref name= Scheck>{{cite book |title=Mechanics: From Newton's Laws to Deterministic Chaos |author=Florian Scheck |chapter-url=https://books.google.com/books?id=yUDo7VptDgIC&pg=PA286 |page=286 |chapter=§5.1 Manifolds of generalized coordinates |isbn=978-3-642-05369-6 |publisher=Springer |edition =5th |year=2010}}</ref>

Generalized coordinates are paired with generalized momenta to provide [[canonical coordinates]] on [[phase space]].