Next: Motion in a Central
Up: Lagrangian Dynamics
Previous: Generalized Forces
The Cartesian equations of motion of our system take
the form
![\begin{displaymath}
m_j\,\ddot{x}_j = f_j,
\end{displaymath}](img1581.png) |
(600) |
for
, where
are each equal to the mass of the
first particle,
are each equal to the mass of the
second particle, etc. Furthermore, the kinetic energy of the
system can be written
![\begin{displaymath}
K = \frac{1}{2}\sum_{j=1,{\cal F}} m_j\,\dot{x}_j^{\,2}.
\end{displaymath}](img1584.png) |
(601) |
Now, since
, we can write
![\begin{displaymath}
\dot{x}_j= \sum_{i=1,{\cal F}} \frac{\partial x_j}{\partial q_i}\,\dot{q}_i
+ \frac{\partial x_j}{\partial t},
\end{displaymath}](img1586.png) |
(602) |
for
.
Hence, it follows that
. According to the
above equation,
![\begin{displaymath}
\frac{\partial \dot{x}_j}{\partial\dot{q}_i} = \frac{\partial x_j}{\partial q_i},
\end{displaymath}](img1588.png) |
(603) |
where we are treating the
and the
as independent
variables.
Multiplying Equation (603) by
, and then differentiating
with respect to time, we obtain
![\begin{displaymath}
\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).
\end{displaymath}](img1591.png) |
(604) |
Now,
![\begin{displaymath}
\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}.
\end{displaymath}](img1592.png) |
(605) |
Furthermore,
![\begin{displaymath}
\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},
\end{displaymath}](img1593.png) |
(606) |
and
where use has been made of Equation (605). Thus, it follows
from Equations (604), (606), and (607) that
![\begin{displaymath}
\frac{d}{dt}\!\left(\frac{1}{2}\,\frac{\partial \dot{x}_j^{\...
... + \frac{1}{2}\,\frac{\partial \dot{x}_j^{\,2}}{\partial q_i}.
\end{displaymath}](img1598.png) |
(608) |
Let us take the above equation, multiply by
, and then sum over all
.
We obtain
![\begin{displaymath}
\frac{d}{dt}\!\left(\frac{\partial K}{\partial \dot{q}_i}\ri...
...\partial x_j}{\partial q_i} + \frac{\partial K}{\partial q_i},
\end{displaymath}](img1600.png) |
(609) |
where use has been made of Equations (600) and (601). Thus, it follows from Equation (597) that
![\begin{displaymath}
\frac{d}{dt}\!\left(\frac{\partial K}{\partial \dot{q}_i}\right) = Q_i + \frac{\partial K}{\partial q_i}.
\end{displaymath}](img1601.png) |
(610) |
Finally, making use of Equation (599), we get
![\begin{displaymath}
\frac{d}{dt}\!\left(\frac{\partial K}{\partial \dot{q}_i}\ri...
...rac{\partial U}{\partial q_i}+\frac{\partial K}{\partial q_i}.
\end{displaymath}](img1602.png) |
(611) |
It is helpful to introduce a function
, called the Lagrangian, which
is defined as the difference between the kinetic and potential energies of the dynamical system under investigation:
![\begin{displaymath}
L = K - U.
\end{displaymath}](img1604.png) |
(612) |
Since the potential energy
is clearly independent of the
, it follows from Equation (611) that
![\begin{displaymath}
\frac{d}{dt}\!\left(\frac{\partial L}{\partial \dot{q}_i}\right) -\frac{\partial L}{\partial q_i} =0,
\end{displaymath}](img1605.png) |
(613) |
for
. 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, (613).
Unfortunately, this scheme only works for conservative systems.
Let us now consider some examples.
Next: Motion in a Central
Up: Lagrangian Dynamics
Previous: Generalized Forces
Richard Fitzpatrick
2011-03-31