Editing Generalized coordinates (section)

==Examples==

===Bead on a wire===

[[File:Bead on wire constraint.svg|thumb|200px|Bead constrained to move on a frictionless wire. The wire exerts a reaction force {{math|'''C'''}} on the bead to keep it on the wire. The non-constraint force {{math|'''N'''}} in this case is gravity. Notice the initial position of the wire can lead to different motions.]]

For a bead sliding on a frictionless wire subject only to gravity in 2d space, the constraint on the bead can be stated in the form {{math|1=''f'' ('''r''') = 0}}, where the position of the bead can be written {{math|1='''r''' = (''x''(''s''), ''y''(''s''))}}, in which {{mvar|s}} is a parameter, the [[arc length]] {{mvar|s}} along the curve from some point on the wire. This is a suitable choice of generalized coordinate for the system. Only ''one'' coordinate is needed instead of two, because the position of the bead can be parameterized by one number, {{mvar|s}}, and the constraint equation connects the two coordinates {{mvar|x}} and {{mvar|y}}; either one is determined from the other. The constraint force is the reaction force the wire exerts on the bead to keep it on the wire, and the non-constraint applied force is gravity acting on the bead.

Suppose the wire changes its shape with time, by flexing. Then the constraint equation and position of the particle are respectively

:<math>f(\mathbf{r}, t) = 0 \,,\quad \mathbf{r} = (x(s, t), y(s, t))</math>

which now both depend on time {{mvar|t}} due to the changing coordinates as the wire changes its shape. Notice time appears implicitly via the coordinates ''and'' explicitly in the constraint equations.

===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.

===Double pendulum===
[[File:Double-Pendulum.svg|thumb|right|A [[double pendulum]]]]
The benefits of generalized coordinates become apparent with the analysis of a [[double pendulum]].
For the two masses {{math|1=''m{{sub|i}}'' (''i'' = 1, 2)}}, let {{math|1='''r'''{{sub|''i''}} = (''x{{sub|i}}'', ''y{{sub|i}}''), ''i'' = 1, 2}} define their two trajectories.  These vectors satisfy the two constraint equations,
:<math>f_1 (x_1, y_1, x_2, y_2) = \mathbf{r}_1\cdot \mathbf{r}_1 - L_1^2 = 0</math>
and
:<math>f_2 (x_1, y_1, x_2, y_2) = (\mathbf{r}_2-\mathbf{r}_1) \cdot  (\mathbf{r}_2-\mathbf{r}_1) - L_2^2 = 0.</math>
The formulation of Lagrange's equations for this system yields six equations in the four Cartesian coordinates {{math|1=''x{{sub|i}}'', ''y{{sub|i}}'' (''i'' = 1, 2)}} and the two Lagrange multipliers {{math|1= ''λ{{sub|i}}'' (''i'' = 1, 2)}} that arise from the two constraint equations.

Now introduce the generalized coordinates {{math|1=''θ{{sub|i}}'' (''i'' = 1, 2)}} that define the angular position of each mass of the double pendulum from the vertical direction.  In this case, we have
:<math>\mathbf{r}_1 = (L_1\sin\theta_1, -L_1\cos\theta_1), \quad \mathbf{r}_2 = (L_1\sin\theta_1, -L_1\cos\theta_1) + (L_2\sin\theta_2, -L_2\cos\theta_2).</math>

The force of gravity acting on the masses is given by,
:<math>\mathbf{F}_1=(0,-m_1 g),\quad \mathbf{F}_2=(0,-m_2 g)</math>
where {{mvar|g}} is the acceleration due to gravity.  Therefore, the virtual work of gravity on the two masses as they follow the trajectories {{math|1='''r'''{{sub|''i''}} (''i'' = 1, 2)}} is given by
:<math> \delta W = \mathbf{F}_1\cdot\delta \mathbf{r}_1 + \mathbf{F}_2\cdot\delta \mathbf{r}_2.</math>

The variations {{math|1=δ'''r'''{{sub|''i''}} (''i'' = 1, 2)}} can be computed to be
:<math> \delta \mathbf{r}_1 = (L_1\cos\theta_1, L_1\sin\theta_1)\delta\theta_1, \quad \delta \mathbf{r}_2 = (L_1\cos\theta_1, L_1\sin\theta_1)\delta\theta_1 +(L_2\cos\theta_2, L_2\sin\theta_2)\delta\theta_2</math>

Thus, the virtual work is given by
:<math>\delta W =  -(m_1+m_2)gL_1\sin\theta_1\delta\theta_1 - m_2gL_2\sin\theta_2\delta\theta_2,</math>
and the generalized forces are
:<math>F_{\theta_1} =  -(m_1+m_2)gL_1\sin\theta_1,\quad F_{\theta_2} =  -m_2gL_2\sin\theta_2.</math>

Compute the kinetic energy of this system to be
:<math> T= \frac{1}{2}m_1 \mathbf{v}_1\cdot\mathbf{v}_1 + \frac{1}{2}m_2 \mathbf{v}_2\cdot\mathbf{v}_2 = \frac{1}{2}(m_1+m_2)L_1^2\dot{\theta}_1^2 + \frac{1}{2}m_2L_2^2\dot{\theta}_2^2 + m_2L_1L_2 \cos(\theta_2-\theta_1)\dot{\theta}_1\dot{\theta}_2.</math>

[[Euler–Lagrange equation]] yield two equations in the unknown generalized coordinates {{math|1=''θ{{sub|i}}'' (''i'' = 1, 2)}} given by<ref>Eric W. Weisstein, [http://scienceworld.wolfram.com/physics/DoublePendulum.html Double Pendulum], scienceworld.wolfram.com. 2007</ref>
:<math>(m_1+m_2)L_1^2\ddot{\theta}_1+m_2L_1L_2\ddot{\theta}_2\cos(\theta_2-\theta_1) + m_2L_1L_2\dot{\theta_2}^2\sin(\theta_1-\theta_2) = -(m_1+m_2)gL_1\sin\theta_1,</math>
and
:<math>m_2L_2^2\ddot{\theta}_2+m_2L_1L_2\ddot{\theta}_1\cos(\theta_2-\theta_1) + m_2L_1L_2\dot{\theta_1}^2\sin(\theta_2-\theta_1)=-m_2gL_2\sin\theta_2.</math>

The use of the generalized coordinates {{math|1=''θ{{sub|i}}'' (''i'' = 1, 2)}} provides an alternative to the Cartesian formulation of the dynamics of the double pendulum.

===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.