Lagrange's Equation

The Cartesian equations of motion of our system take the form

$$m_j\,\ddot{x}_j = f_j,$$ (670)

for $j=1,{\cal F}$, where $m_1, m_2, m_3$ are each equal to the mass of the first particle, $m_4, m_5, m_6$ are each equal to the mass of the second particle, etc. Furthermore, the kinetic energy of the system can be written

$$K = \frac{1}{2}\sum_{j=1,{\cal F}} m_j\,\dot{x}_j^{\,2}.$$ (671)

Now, since $x_j=x_j(q_1,q_2,\cdots, q_{\cal F},t)$, we can write

$$\dot{x}_j= \sum_{i=1,{\cal F}} \frac{\partial x_j}{\partial q_i}\,\dot{q}_i + \frac{\partial x_j}{\partial t},$$ (672)

for $j=1,{\cal F}$. Hence, it follows that $\dot{x}_j = \dot{x}_j(\dot{q}_1,\dot{q_2},\cdots, \dot{q}_{\cal F},q_1,q_2,\cdots,q_{\cal F},t)$. According to the above equation,

$$\frac{\partial \dot{x}_j}{\partial\dot{q}_i} = \frac{\partial x_j}{\partial q_i},$$ (673)

where we are treating the $\dot{q}_i$ and the $q_i$ as independent variables.

Multiplying Eq. (673) by $\dot{x}_j$, and then differentiating with respect to time, we obtain

$$\frac{d}{dt}\!\left(\dot{x}_j\,\frac{\partial \dot{x}_j}{\pa... ...\frac{d}{dt}\!\left( \frac{\partial x_j}{\partial q_i}\right).$$ (674)

Now,

$$\frac{d}{dt}\!\left(\frac{\partial x_j}{\partial q_i}\right)... ...,\dot{q}_k + \frac{\partial^2 x_j}{\partial q_i\,\partial t}.$$ (675)

Furthermore,

$$\frac{1}{2} \,\frac{\partial\dot{x}_j^{\,2}}{\partial \dot{q}_i} = \dot{x}_j\,\frac{\partial \dot{x}_j}{\partial \dot{q}_i},$$ (676)

and

$$\frac{1}{2}\,\frac{\partial \dot{x}_j^{\,2}}{\partial q_i} = \dot{x}_j\,\frac{\partial \dot{x}_j}{\partial q_i} = \dot{x}_j\,\frac{\partial}{\partial q_i}\!\left(\sum_{k=1,{\cal F... ...\partial x_j}{\partial q_k}\,\dot{q}_k + \frac{\partial x_j}{\partial t}\right)$$

$$= \dot{x}_j\left(\sum_{k=1,{\cal F}}\frac{\partial^2 x_j}{\partial ... ...artial q_k}\,\dot{q}_k + \frac{\partial^2 x_j}{\partial q_i\,\partial t}\right)$$

$$= \dot{x}_j\,\frac{d}{dt}\!\left(\frac{\partial x_j}{\partial q_i}\right),$$ (677)

where use has been made of Eq. (675). Thus, it follows from Eqs. (674), (676), and (677) that

$$\frac{d}{dt}\!\left(\frac{1}{2}\,\frac{\partial \dot{x}_j^{\... ... + \frac{1}{2}\,\frac{\partial \dot{x}_j^{\,2}}{\partial q_i}.$$ (678)

Let us take the above equation, multiply by $m_j$, and then sum over all $j$. We obtain

$$\frac{d}{dt}\!\left(\frac{\partial K}{\partial \dot{q}_i}\ri... ...\partial x_j}{\partial q_i} + \frac{\partial K}{\partial q_i},$$ (679)

where use has been made of Eqs. (670) and (671). Thus, it follows from Eq. (667) that

$$\frac{d}{dt}\!\left(\frac{\partial K}{\partial \dot{q}_i}\right) = Q_i + \frac{\partial K}{\partial q_i}.$$ (680)

Finally, making use of Eq. (669), we get

$$\frac{d}{dt}\!\left(\frac{\partial K}{\partial \dot{q}_i}\ri... ...rac{\partial U}{\partial q_i}+\frac{\partial K}{\partial q_i}.$$ (681)

It is helpful to introduce a function $L$, called the Lagrangian, which is defined as the difference between the kinetic and potential energies of the dynamical system under investigation:

$$L = K - U.$$ (682)

Since the potential energy $U$ is clearly independent of the $\dot{q}_i$, it follows from Eq. (681) that

$$\frac{d}{dt}\!\left(\frac{\partial L}{\partial \dot{q}_i}\right) -\frac{\partial L}{\partial q_i} =0,$$ (683)

for $i=1,{\cal F}$. This equation is known as Lagrange's equation.

According to the above analysis, if we can express the kinetic and potential energies of our dynamical system solely in terms of our generalized coordinates and their time derivatives, then we can immediately write down the equations of motion of the system, expressed in terms of the generalized coordinates, using Lagrange's equation, (683). Unfortunately, this scheme only works for conservative systems. Let us now consider some examples.