Editing Generalized coordinates (section)

===Holonomic constraints===

{{multiple image
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|image1 = Generalized coordinates open curved path 3d 2df.svg
|caption1 = Open curved surface {{math|1=''F''(''x'', ''y'', ''z'') = 0}}
|image2 = Generalized coordinates closed curved path 3d 2df.svg
|caption2 = Closed curved surface {{math|1=''S''(''x'', ''y'', ''z'') = 0}}
|footer=Two generalized coordinates, two degrees of freedom, on curved surfaces in 3D. Only two numbers {{math|(''u'', ''v'')}} are needed to specify the points on the curve, one possibility is shown for each case. The full three [[Cartesian coordinates]] {{math|(''x'', ''y'', ''z'')}} are not necessary because any two determines the third according to the equations of the curves.
}}

For a system of {{mvar|N}} particles in 3D [[real coordinate space]], the [[position vector]] of each particle can be written as a 3-[[tuple]] in [[Cartesian coordinates]]:

:<math>\begin{align}
& \mathbf{r}_1 = (x_1,y_1,z_1), \\
& \mathbf{r}_2 = (x_2,y_2,z_2), \\
& \qquad \qquad \vdots \\
& \mathbf{r}_N = (x_N,y_N,z_N)
\end{align}</math>

Any of the position vectors can be denoted {{math|'''r'''{{sub|''k''}}}} where {{math|1=''k'' = 1, 2, …, ''N''}} labels the particles. A ''[[holonomic constraint]]'' is a ''constraint equation'' of the form for particle {{mvar|k}}<ref>{{harvnb|Goldstein|Poole|Safko|2002|page=12}}</ref>{{efn|Some authors set the constraint equations to a constant for convenience with some constraint equations (e.g. pendulums), others set it to zero. It makes no difference because the constant can be subtracted to give zero on one side of the equation. Also, in Lagrange's equations of the first kind, only the derivatives are needed.}}

:<math>f(\mathbf{r}_k, t) = 0</math>

which connects all the 3 spatial coordinates of that particle together, so they are not independent. The constraint may change with time, so time {{mvar|t}} will appear explicitly in the constraint equations. At any instant of time, any one coordinate will be determined from the other coordinates, e.g. if {{mvar|x{{sub|k}}}} and {{mvar|z{{sub|k}}}} are given, then so is {{mvar|y{{sub|k}}}}. One constraint equation counts as ''one'' constraint. If there are {{mvar|C}} constraints, each has an equation, so there will be {{mvar|C}} constraint equations. There is not necessarily one constraint equation for each particle, and if there are no constraints on the system then there are no constraint equations.

So far, the configuration of the system is defined by {{math|3''N''}} quantities, but {{mvar|C}} coordinates can be eliminated, one coordinate from each constraint equation. The number of independent coordinates is {{math|1=''n'' = 3''N'' − ''C''}}. (In {{mvar|D}} dimensions, the original configuration would need {{mvar|ND}} coordinates, and the reduction by constraints means {{math|1=''n'' = ''ND'' − ''C''}}). It is ideal to use the minimum number of coordinates needed to define the configuration of the entire system, while taking advantage of the constraints on the system. These quantities are known as '''generalized coordinates''' in this context, denoted {{math|''q{{sub|j}}''(''t'')}}. It is convenient to collect them into an {{mvar|n}}-[[tuple]]

:<math>\mathbf{q}(t) = (q_1(t),\ q_2(t),\ \ldots,\ q_n(t)) </math>

which is a point in the ''[[Configuration space (physics)|configuration space]]'' of the system. They are all independent of one other, and each is a function of time. Geometrically they can be lengths along straight lines, or [[arc length]]s along curves, or angles; not necessarily Cartesian coordinates or other standard [[orthogonal coordinates]]. There is one for each [[degrees of freedom (physics and chemistry)|degree of freedom]], so the number of generalized coordinates equals the number of degrees of freedom, {{mvar|n}}. A degree of freedom corresponds to one quantity that changes the configuration of the system, for example the angle of a pendulum, or the arc length traversed by a bead along a wire.

If it is possible to find from the constraints as many independent variables as there are degrees of freedom, these can be used as generalized coordinates.<ref name="Kibble 2004 page=232">{{harvnb|Kibble |Berkshire|edition=5th|2004|page=232}}</ref> The position vector {{math|'''r'''{{sub|''k''}}}} of particle {{mvar|k}} is a function of all the {{mvar|n}} generalized coordinates (and, through them, of time),<ref>{{harvnb|Torby|1984|page=260}}</ref><ref>{{harvnb|Goldstein|Poole|Safko|2002|page=13}}</ref><ref name="Hand 1998 page=15">{{harvnb|Hand|Finch|1998|page=15}}</ref><ref name="Kibble 2004 page=232"/><ref group = nb>Some authors e.g. Hand & Finch take the form of the position vector for particle {{mvar|k}}, as shown here, as the condition for the constraint on that particle to be holonomic.</ref>

:<math>\mathbf{r}_k = \mathbf{r}_k(\mathbf{q}(t)) \,, </math>

and the generalized coordinates can be thought of as parameters associated with the constraint.

The corresponding time derivatives of {{math|'''q'''}} are the generalized velocities,

:<math>\dot{\mathbf{q}} = \frac{d\mathbf{q}}{dt} = (\dot{q}_1(t),\ \dot{q}_2(t),\ \ldots,\ \dot{q}_n(t)) </math>

(each dot over a quantity indicates one [[time derivative]]). The velocity vector {{math|'''v'''{{sub|''k''}}}} is the [[total derivative]] of {{math|'''r'''{{sub|''k''}}}} with respect to time

:<math>\mathbf{v}_k = \dot{\mathbf{r}}_k = \frac{d\mathbf{r}_k}{dt} = \sum_{j=1}^n \frac{\partial \mathbf{r}_k}{\partial q_j}\dot{q}_j \,.</math>

and so generally depends on the generalized velocities and coordinates. Since we are free to specify the initial values of the generalized coordinates and velocities separately, the generalized coordinates {{mvar|q{{sub|j}}}} and velocities {{math|''dq{{sub|j}}''/''dt''}} can be treated as ''independent variables''.