Editing Generalized coordinates (section)

===Simple pendulum===
[[File:Pendulum constraint.svg|150px|thumb|Simple pendulum. Since the rod is rigid, the position of the bob is constrained according to the equation {{math|1=''f'' (''x'', ''y'') = 0}}, the constraint force {{math|'''C'''}} is the tension in the rod. Again the non-constraint force {{math|'''N'''}} in this case is gravity.]]

[[File:Simple pendulum generalized coordinates.svg|thumb|Dynamic model of a simple pendulum.]]
The relationship between the use of generalized coordinates and Cartesian coordinates to characterize the movement of a mechanical system can be illustrated by considering the constrained dynamics of a simple pendulum.<ref>{{cite book
 | last = Greenwood
 | first = Donald T.
 | year = 1987
 | title = Principles of Dynamics
 | edition = 2nd
 | publisher = Prentice Hall
 | isbn = 0-13-709981-9
}}</ref><ref>Richard Fitzpatrick, [http://farside.ph.utexas.edu/teaching/336k/Newton/node90.html Newtonian Dynamics].</ref>

A simple [[pendulum]] consists of a mass {{mvar|M}} hanging from a pivot point so that it is constrained to move on a circle of radius {{mvar|L}}.  The position of the mass is defined by the coordinate vector {{math|1='''r''' = (''x'', ''y'')}} measured in the plane of the circle such that {{mvar|y}} is in the vertical direction.  The coordinates {{mvar|x}} and {{mvar|y}} are related by the equation of the circle
:<math>f(x, y) = x^2+y^2 - L^2=0,</math>
that constrains the movement of {{mvar|M}}.  This equation also provides a constraint on the velocity components,
:<math> \dot{f}(x, y)=2x\dot{x} + 2y\dot{y} = 0.</math>

Now introduce the parameter {{mvar|θ}}, that defines the angular position of {{mvar|M}} from the vertical direction.  It can be used to define the coordinates {{mvar|x}} and {{mvar|y}}, such that
:<math> \mathbf{r}=(x, y) = (L\sin\theta, -L\cos\theta).</math>
The use of {{mvar|θ}} to define the configuration of this system avoids the constraint provided by the equation of the circle.

Notice that the force of gravity acting on the mass {{mvar|m}} is formulated in the usual Cartesian coordinates,
:<math> \mathbf{F}=(0,-mg),</math>
where {{mvar|g}} is the [[Gravitational acceleration|acceleration due to gravity]].

The [[virtual work]] of gravity on the mass {{mvar|m}} as it follows the trajectory {{math|'''r'''}} is given by
:<math> \delta W = \mathbf{F}\cdot\delta \mathbf{r}.</math>

The variation {{math|δ'''r'''}} can be computed in terms of the coordinates {{mvar|x}} and {{mvar|y}}, or in terms of the parameter {{mvar|θ}},
:<math> \delta \mathbf{r} =(\delta x, \delta y) = (L\cos\theta, L\sin\theta)\delta\theta.</math>
Thus, the virtual work is given by
:<math>\delta W = -mg\delta y = -mgL\sin(\theta)\delta\theta.</math>

Notice that the coefficient of {{math|δ''y''}} is the {{mvar|y}}-component of the applied force.  In the same way, the coefficient of {{math|δ''θ''}} is known as the [[generalized force]] along generalized coordinate {{mvar|θ}}, given by
:<math> F_{\theta} =  -mgL\sin\theta.</math>

To complete the analysis consider the kinetic energy {{mvar|T}} of the mass, using the velocity,
:<math> \mathbf{v}=(\dot{x}, \dot{y}) = (L\cos\theta, L\sin\theta)\dot{\theta},</math>
so,
:<math> T= \frac{1}{2} m\mathbf{v}\cdot\mathbf{v} = \frac{1}{2} m (\dot{x}^2+\dot{y}^2) = \frac{1}{2} m L^2\dot{\theta}^2.</math>

[[Virtual work#D'Alembert's form of the principle of virtual work|D'Alembert's form of the principle of virtual work]] for the pendulum in terms of the coordinates {{mvar|x}} and {{mvar|y}} are given by,
:<math> \frac{d}{dt}\frac{\partial T}{\partial \dot{x}} - \frac{\partial T}{\partial x} = F_{x} + \lambda \frac{\partial f}{\partial x},\quad \frac{d}{dt}\frac{\partial T}{\partial \dot{y}} - \frac{\partial T}{\partial y} = F_{y} + \lambda \frac{\partial f}{\partial y}.  </math>
This yields the three equations
:<math>m\ddot{x} = \lambda(2x),\quad m\ddot{y} = -mg + \lambda(2y),\quad x^2+y^2 - L^2=0,</math>
in the three unknowns, {{mvar|x}}, {{mvar|y}} and {{mvar|λ}}.

Using the parameter {{mvar|θ}}, those equations take the form
:<math>\frac{d}{dt}\frac{\partial T}{\partial \dot{\theta}} - \frac{\partial T}{\partial \theta} = F_{\theta},</math>
which becomes,
:<math> mL^2\ddot{\theta} = -mgL\sin\theta,</math>
or
:<math> \ddot{\theta} + \frac{g}{L}\sin\theta=0.</math>
This formulation yields one equation because there is a single parameter and no constraint equation.

This shows that the parameter {{mvar|θ}} is a generalized coordinate that can be used in the same way as the Cartesian coordinates {{mvar|x}} and {{mvar|y}} to analyze the pendulum.