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Generalized Momenta
Consider the motion of a single particle moving in one dimension. The
kinetic energy is
![\begin{displaymath}
K = \frac{1}{2}\,m\,\dot{x}^{\,2},
\end{displaymath}](img1656.png) |
(649) |
where
is the mass of the particle, and
its displacement.
Now, the particle's linear momentum is
. However,
this can also be written
![\begin{displaymath}
p = \frac{\partial K}{\partial \dot{x}}= \frac{\partial L}{\partial\dot{x}},
\end{displaymath}](img1658.png) |
(650) |
since
, and the potential energy
is independent of
.
Consider a dynamical system described by
generalized coordinates
, for
. By analogy with the above expression, we can
define generalized momenta of the form
![\begin{displaymath}
p_i = \frac{\partial L}{\partial\dot{q}_i},
\end{displaymath}](img1661.png) |
(651) |
for
. Here,
is sometimes called the momentum conjugate to the coordinate
. Hence, Lagrange's equation (613) can be written
![\begin{displaymath}
\frac{d p_i}{dt} = \frac{\partial L}{\partial q_i},
\end{displaymath}](img1663.png) |
(652) |
for
. Note that a generalized momentum does not necessarily have
the dimensions of linear momentum.
Suppose that the Lagrangian
does not depend explicitly on some coordinate
. It follows from Equation (652) that
![\begin{displaymath}
\frac{d p_k}{dt} = \frac{\partial L}{\partial q_k}=0.
\end{displaymath}](img1665.png) |
(653) |
Hence,
![\begin{displaymath}
p_k = {\rm const.}
\end{displaymath}](img1666.png) |
(654) |
The coordinate
is said to be ignorable in this case.
Thus, we conclude that the generalized momentum associated with
an ignorable coordinate is a constant of the motion.
For example, in Section 9.5, the Lagrangian (615) for a
particle moving in a central potential is independent of the angular
coordinate
. Thus,
is an ignorable coordinate,
and
![\begin{displaymath}
p_\theta = \frac{\partial L}{\partial\dot{\theta}} = m\,r^2\,\dot{\theta}
\end{displaymath}](img1667.png) |
(655) |
is a constant of the motion. Of course,
is the angular momentum
about the origin. This is conserved because a central force exerts no torque
about the origin.
Again, in Section 9.7, the Lagrangian (642) for a mass
sliding down a sliding slope is independent
of the Cartesian coordinate
. It follows that
is an ignorable coordinate,
and
![\begin{displaymath}
p_x = \frac{\partial L}{\partial \dot{x}} = M\,\dot{x} + m\,(\dot{x}+\dot{x}'\,\cos\theta)
\end{displaymath}](img1669.png) |
(656) |
is a constant of the motion. Of course,
is the total linear momentum in the
-direction. This is conserved because there is no external force acting on
the system in the
-direction.
Next: Spherical Pendulum
Up: Lagrangian Dynamics
Previous: Sliding down a Sliding
Richard Fitzpatrick
2011-03-31