Editing Analytical mechanics (section)

===Difference between [[Curvilinear coordinates|curvillinear]] and [[generalized coordinates]]===
Generalized coordinates incorporate constraints on the system. There is one generalized coordinate ''q<sub>i</sub>'' for each [[Degrees of freedom (physics and chemistry)|degree of freedom]] (for convenience labelled by an index ''i'' = 1, 2...''N''), i.e. each way the system can change its [[Configuration space (physics)|configuration]]; as curvilinear lengths or angles of rotation. Generalized coordinates are not the same as curvilinear coordinates. The number of ''curvilinear'' coordinates equals the [[dimension]] of the position space in question (usually 3 for 3d space), while the number of ''generalized'' coordinates is not necessarily equal to this dimension; constraints can reduce the number of degrees of freedom (hence the number of generalized coordinates required to define the configuration of the system), following the general rule:<ref name="autogenerated1">''Analytical Mechanics'', L.N. Hand, J.D. Finch, Cambridge University Press, 2008, {{ISBN|978-0-521-57572-0}}</ref>{{dubious|date=January 2024}}

{{block indent | em = 1.5 | text = ''['''dimension of position space''' (usually 3)] × [number of '''constituents''' of system ("particles")] − (number of '''constraints''')''}}
{{block indent | em = 1.5 | text = ''= (number of '''degrees of freedom''') = (number of '''generalized coordinates''')''}}

For a system with ''N'' degrees of freedom, the generalized coordinates can be collected into an ''N''-[[tuple]]:
<math display="block">\mathbf{q} = (q_1, q_2, \dots, q_N) </math>
and the [[time derivative]] (here denoted by an overdot) of this tuple give the ''generalized velocities'':
<math display="block">\frac{d\mathbf{q}}{dt} = \left(\frac{dq_1}{dt}, \frac{dq_2}{dt}, \dots, \frac{dq_N}{dt}\right) \equiv \mathbf{\dot{q}} = (\dot{q}_1, \dot{q}_2, \dots, \dot{q}_N) .</math>