Editing Generalized coordinates (section)

===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.