Editing Generalized coordinates (section)

===Spherical pendulum===

[[File:Spherical pendulum Lagrangian mechanics.svg|thumb|200px|Spherical pendulum: angles and velocities.]]

For a 3D example, a [[spherical pendulum]] with constant length {{mvar|l}} free to swing in any angular direction subject to gravity, the constraint on the pendulum bob can be stated in the form

:<math>f(\mathbf{r}) = x^2 + y^2 + z^2 - l^2 = 0 \,, </math>

where the position of the pendulum bob can be written

:<math>\mathbf{r} = (x(\theta,\phi),y(\theta,\phi),z(\theta,\phi)) \,, </math>

in which {{math|(''θ'', ''φ'')}} are the [[spherical coordinates|spherical polar angles]] because the bob moves in the surface of a sphere. The position {{math|'''r'''}} is measured along the suspension point to the bob, here treated as a [[point particle]]. A logical choice of generalized coordinates to describe the motion are the angles {{math|(''θ'', ''φ'')}}. Only two coordinates are needed instead of three, because the position of the bob can be parameterized by two numbers, and the constraint equation connects the three coordinates {{math|(''x'', ''y'', ''z'')}} so any one of them is determined from the other two.