Consider a
particle moving in a circle as shown below:
(Readers may want to examine this earlier post before going on:
http://brane-space.blogspot.com/2010/08/introduction-to-polar-coordinates-and.html )
We have the constraint on the motion:
r = (x2 + y2)½
By looking at such
constraints we can reduce the number of coordinates required to specify a
mechanical system.
We may write
generalized coordinates as:
q 1 , q 2 …….. q
n
This would apply to some system composed
of n particles with position vectors:
r 1 , r
2 …….. r n
Such that we may
write: å n i q i
In
all such cases we require that the Jacobian determinant* J ≠ 0 else
no legitimate set of generalized coordinates can be defined in a specific form. Consider the case
of the above for x,y variables with stationary axis: x = r cos q, y =
r sin q
And when the axis
is moving: x= r cos (wt + f) and: y = r sin (wt + f)
Show that (x, y)
or (r, q) can be used as generalized
coordinates q 1 , q 2 …….. q
n
Solution:
Let x =
x(q 1 , q
2 , t)
Then write: x’ =
dx/ dt =
(¶ x/¶ q
1)
¶
q 1/¶ t +(¶ x/¶ q
2)
¶
q 2/¶t + ….¶ x/¶ t
dx/ dt =
(¶ x/¶ q
1)
q 1’ + = (¶ x/¶ q 2) q 2’ + ….….¶ x/¶ t
And:
x’ = r’ cos q - r
sin q q’
y’ = r’ sin
q + r’
cos q q’
(Rem: r’ = dr/ dt ; q’ = dq/ dt )
So
the coordinates work in either system, for circular motion
Generalized Momenta can be angular or linear:
For generalized
forces: consider a particle which has moved an incremental amount d r where:
d r = d x i
+
d y j + d jk
This is an actual
displacement but so small that the forces don’t change, say for a vertical
displacement. For N particles we have:
d W = å N i (F i x d x +
F i y d y
+ F i z d z )
Given:
¶ x/¶ q’ = p q
this can be referred to generalized coordinates: q 1 , q 2 …….. q n
d x = (¶ x/¶ q
1) d
q 1 + (¶ x/¶ q
2) d
q 2 +
(¶ x/¶ q 3)
d q 3
Or in polar
coordinates:
d x = cos q
d r -
r sin d q
d y = sin q
d r -
r cos d q
In general, we may
write: d
W = Q k d q k
= Q r
y r + Q q d q
Consider now the
transformation of:
F = F x i
+ F y
j To:
= F r
r + F q q
The generalized
force is:
Q r = - ¶ V /¶ r =
- ¶ V /¶ x
(- ¶ x /¶ r )
- ¶ V /¶ y (- ¶ x /¶ y )
= F x (¶ x /¶ r )
+ F y
(¶ y /¶ r )
= F x cos q + F y
sin q
= F r
We may now introduce the
diagram below:
To derive generalized angular force Q q :
Q q
= F x (¶ x /¶ q ) + F y
(¶ y /¶ q )
= F x r sin q + F y
r cos q
= r F q
Note from the
diagram this is a component in the direction of q ^
It is of interest
here to obtain the Lagrangian in polar coordinates. We know:
x = r cos q and y
= r
sin q
So that:
x’ = r’ cos q - r
sin q q’
y’ = r’ sin
q + r’
cos q q’
The kinetic energy
T is:
T = ½ m
( r’ 2 + r
2 q ‘2 )
V = mg r sin q
Therefore:
L = T -
V =
½ m ( r’ 2 + r
2 q ‘2 ) - mg r sin q
It is also useful
to consider the unit vectors associated with polar coordinates in central force
problems: n pointing in direction of
increasing r, and l, in the
direction of increasing q .
The velocity
components can then be written:
v r = dr/
dt, v q
= r dq /dt
Using the polar
coordinate unit vectors (n, l) this
can be rewritten as:
v
= dr/ dt
n +
r dq /dt l
For the change in
the radial coordinate alone:
v
= dr/ dt
= d(r n)/ dt = ( dr/ dt n + r
dn /dt)
Where the last
derivative can be evaluated from:
dn /dt
= dn / dq (dq
/dt)
To evaluate dn / dq one makes use of the fact that by radian
measure of angles:
D n = D q
and in the limit as D q ® 0,
‖D
n
‖ / ‖D q ‖ = 1
So: dn / dq = l
i.e.
As D q ® 0, D n
becomes perpendicular to n and
assumes the direction of l .
In an analogous
way we may obtain the relation:
dl / dq = -n
Finally, we have: dn
/dt =
dq /dt
l
And: dl
/dt =
- dq /dt n
Problems:
1. Using one or more of the preceding unit vector relations,
obtain an expression for the acceleration in two dimensions of polar coordinates.
2.A particle of mass m moves in a plane under the influence of a force F = - kr, directed toward the origin. Sketch a polar coordinate system (r, q ) to describe the motion of the particle and thereby obtain the Lagrangian (L = T - V, i.e. difference in kinetic and potential energy).
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Here is another article to view => Coordinates In Space – Formula Collection | Mathematics Class 12
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