When 3rd year physics students are first introduced to theoretical mechanics they are generally taken aback when confronted by the concepts of generalized coordinates, momenta and forces. In this post I examine aspects of this and hope to show why such generalizations manifest practical use and power in solving problems in theoretical mechanics.
We may write generalized coordinates as:
q 1 , q 2 …….. q n
This would apply to some system composed of n particles with associated 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 over time, t with phase angle f):
x= r cos (wt + f) and: y = r sin (wt + f)
Example Problem:
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 (p q , p f ) or linear ( p x, p y , p z ):
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 the 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
dl / dq = -n
Suggested 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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