Lecture 17: Lagrangian Mechanics
We use the approach of Lagrangian mechanics to provide an easy way of
deriving equations of motions for complicated interconnected systems.
All we need to know to use this approach is the kinetic energy
T
of
the system and its potential energy
V
. Once we know these, we form
L
=
T

V
and take some derivatives in a way defined by the socalled
EulerLagrange equations to obtain equations of motion for the system.
Next we describe how those equations are derived.
The coordinates used for the description of motion of the system,
that are not the regular Cartesian coordinates
x, y, z
in the three
dimensional space are referred to as
generalized coordinates
. One exam
ple is the polar coordinates that you have used in description of many
systems that rotate around an axis.
The systems for which internal
Figure 1:
coordinates (think angle
θ
describing rotation of a disk) completely de
termine the position of the system in space. A disk restricted to rotate
around a pin is a holonomic system. A disk rotating on a floor is not.
For holonomic systems: the number of generalized coordinates is
equal to the number of Cartesian coordinates minus the number of
constraint equations.
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 Winter '08
 Mezic,I
 Polar coordinate system, Ri, Lagrangian mechanics, ri ∂ ri

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