**theoretical mechanics**. Its integration of advanced mathematics into practical problems has directly led to many other disciplines - from biology to economics - to try to copy it with mixed results. In this post I want to look at how its concepts and approaches can be applied to familiar areas of astronomy I've covered in earlier, simpler post - mainly to do with circular motion, as well as planetary motion.

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 = (x

^{2 }+ y^{2})^{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 j**k**
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**q ‘**^{2}^{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 d

**n**/dt)

Where the last
derivative can be evaluated from:

d

**n**/dt = d**n**/ 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: d

**n**/ 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

**:**
d

**l**/ dq =**-n**
Finally, we have: d

**n**/dt = dq /dt**l**

And: d

**l**/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).

*****

## 2 comments:

Here is another article to view => Coordinates In Space – Formula Collection | Mathematics Class 12

Thanks for the article!

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