Friday, December 6, 2024

Revisiting Differential Geometry (2)

 

Curves in 3-Space

As previously defined, every point in the space R 3 is uniquely determined by a position vector x = (x,  x2 , x3).  Then in order to introduce the concept of a curve we use a real vector function: x =  x(t)  and this will have components:

x=  x(t)      x=  x(t)      x=  x(t)

Which are single-valued functions of  real variable t defined in the interval:   I :   a  <  t  <   b.  To every value of t a point of R 3  is associated whose position vector is x(t).

Then the vector function determines a point set M in R 3   which we call a parametric representation of the set M, and the variable t is called the parameter of the representation. Given the above and that the trivial case of M = 1 point is discarded while the vector function must have multiple derivatives, we may assume:

1)  The functions x=  x(t) (i=1,2,3.....)  are  r (> 1) times continuously differentiable in I where the value of r will depend on the problem under consideration.

2) For every value of t in I, at least one of the three functions:

x(t) =   d x(t)/ dt

is different from zero.

DefinitionArc of a curve:




A point set in space  R 3  which can be represented by the allowable representations of an equivalence class is called an arc of a curve.  The functional correspondence of the points of an arc to the value of t- given by an allowable representation x = x(t) is continuous.   

 If an arc is simple the correspondence between the points and the values of the parameter t is one to one.  In this case not only is the functional relation of the points to the values of t continuous but also the inverse relation, i.e. the relation of the values of t to the points.

Definitioncurve:

A point set is called a curve if it can be represented by an equivalence class of the form x = x(t) whose interval I is not assumed to be closed or bounded, but is such that one always obtains the arc of a curve if the values of the parameter t are restricted to any closed and bounded subinterval of I.  

A curve is said to be closed if it possesses at least one representation which is periodic, i.e. of the form: 

x(t  +  w  ) =  x(t)

Thus the circle:  x(t) =  (r cos t, r sin t, 0)  with  r 2  =  4  is an  example of a simple closed curve.  

We have: x1 = r cos t ,  x 2 = r sin t,  x 3  = 0

->  x 1 2    +    x 2 2  =   r 2 (cos 2 t   +   sin 2 t) =  4

So, the circle is closed and in the x 1 x 2 -plane, e.g.

Special curves:  A few special curves and their representations are now examined.  The main proviso here is to point out that there are curves of the form  x =  x(t)  that cannot always be represented as a whole in the forms:   x 2  =  x 2  (x 1),   x 3  =  x 3  (x 1).  

This difficulty will occur when one value of the independent variable corresponds to several values of the dependent one, since the concept of a function requires a 1:1 onto correspondence of dependent to independent variable. Thus, in the case of the circle, a representation of the form just highlighted would be:

x 2  = + Ö( r 2   -   x 1 22)      x 3  = 0

If we choose the + sign, we have a semicircle in the upper half plane. If we choose the negative sign we have a semicircle in the other half plane.

Ellipse with center at origin of coordinate system in space.  In the parametric representation we can write (analogous to that for the circle):  x(t) =  (a cos t, b sin t, 0)  

Or:

x 1 2 / a 2 +    x 2  / b 2 = 1,   x 3  = 0

If the principal axes have lengths 2a and 2b (say with a = 4, b = Ö8), respectively and coincide with the x 1  and  x 2  axes, respectively,  we obtain the graph shown below:

Note that if a= b = r we just recover a circle:

 x(t) =  (r cos t, r sin t, 0)

The folium of Descartes

This can be expressed in parametric form as:  

 x(t) =  (3t/ 1 +  t,  3t2/ 1 +  t3  , 0)  

This curve will be found to lie in the 1st, 2nd and 4th quadrant  of the   x 1 -  x 2 plane as shown below:

By inspection the reader will also see it has a double point at (x 1 ,  x 2) = (0, 0) .  Note also that part of the curve in the 2nd quadrant corresponds to values of t from -1 to 0, while the loop in the first quadrant corresponds to values between 0 and  ¥.  In the 4th quadrant the t-values range from - ¥  to  -1.

The circular helix:  

The parametric form here is:  

x(t) =  (r cos t, r sin t, ct)   c ≠ 0. 

The orthogonal projection of the helix into the   x 1 x 2 - plane, e.g.



is the circle: 

x 1 2  + x 2     -  r 2  = 0,  x 3  = 0

which is the intersection of this plane with the cylinder of revolution on which the helix lies. Projecting the helix orthogonally into the x2 x 3 - plane we obtain the sine curve:

 x2 - r sin (x 3  /c)  = 0,  x = 0

A cosine curve will be obtained by projecting the helix orthogonally into the x 1 x 2  - plane.

Suggested Problems:

1) Sketch the graphs of;

x 2  = - Ö( 8 2   -   x 1 2)

And:

x 2  = - Ö( 16 2   -   x 1 2)

On the same Cartesian axes.

2)(a) Write the polar form of the equation of the line:

x 1   +   4 x 2   =  5

b)Determine the polar (r,  q) equation for :

x 1 2  + x 2   -  2ax 2  =  0,    a  ≠   0

And sketch the resulting curve.


"Trump a Slandered Victim Of The Left" - Playwright David Mamet's Inhabiting The Wrong Parallel Earth - In His WSJ Op-Ed

 

     The guy David Mamet believes is a slandered victim'

             Scene from Jan. 6, 2021 which too many have forgotten

It was damned near impossible for me to reconcile the author of 'Glengarry Glen Ross and The Verdict with the character (David Mamet) who wrote the unhinged pro-Trump screed op-ed ('Decline and Fall of American? Not Yet) in the Nov. 23-24 issue of The Wall Street Journal.

Mamet scribbles in two of the more odious and repellent passages:

"To the left, 'She isn't Trump' was sufficient reason to vote for Harris. Trump isn't the devil, but a rational consideration of him and his achievements was beyond the liberals' ability. Mr. Trump was the enemy because he represents a challenge they could not accept...

Why would sentient Americans vote away the freedoms of thought, conscience, assembly and expression guaranteed by our Constitution? It makes sense, as any suicide does, as an act of survival.  The suicide takes his own life to stop unbearable anguish.  It is the most desperate act of self-preservation."

Maybe Mamet has forgotten - or never processed-  that among Trump's "achievements" were:

- Being able to grab women's pussies ("They let ya do it!")

- Mocking a captive of the North Vietnamese (John McCain)

- Dismissing those U.S. soldiers who died in past wars as "losers and suckers".

- Declaring Nazis and white supremacists who marched in Charlottesville, VA as "some very fine people"

- Writing 'love letters' to one of the most bloodthirsty dictators (Kim Jung Un) on the planet

Asserting Gen. Mark Milley ought to be "hung for treason:"

Laying waste to an Ohio town and forcing closure of its schools, facilities, after bellowing in a debate that Haitian migrants were "eating the cats and the dogs".

- Inciting an insurrection (Jan. 6, 2021) and asserting it was about "love" when dozens of D.C. police had their heads bashed in.

So pardon me, but a rational consideration of Trump's 'achievements' - as listed above (and facts, not 'slander' as Mamet claims) leads one to conclude Trump is an egomaniacal - and possibly demented- psychotic, insurrectionist traitor, and rapist swine who has no business running a porty-potty far less the U.S. government.

As for 'sentient Americans' voting away the freedom of thought, well from where I stand it was the 74 million -odd yahoos and imbeciles who voted for Trump who are guilty as charged. As documented by one Canadian newspaper - which appears to have a vastly more objective perspective than Mamet- based on the ten examples below of voters' mental misfires.  Voting at cross-purposes to their own interests.:

COLUMN: Americans electing Trump like turkeys voting for Thanksgiving - Orillia News

  1. poor people voted for billionaires’ tax cuts
  2. immigrants voted for mass deportations
  3. seniors voted to gut social security and “Obamacare”
  4. men voted for their wives and daughters to possibly die from miscarriages
  5. value shoppers voted to raise duties on Chinese goods, which will increase Walmart and dollar-store prices
  6. police voted for a convicted felon
  7. “law and order” advocates voted for a convicted felon
  8. religious men and women voted for a man who often mocks religions
  9. “family values” people voted for a twice-divorced philanderer and adulterer
  10. taxpayers voted for a man apparently proud to pay little or no income tax

In an election like no other, many Americans behaved like turkeys voting to establish a new feast to be called “Thanksgiving.”

  By all measures of mental balance, rectitude and coherent judgment it is these Trumper voters who've committed electoral suicide - by voting to extinguish their own interests, welfare and security.  Don't believe me? Read through that list of mental misfires again and tell me with a straight face that Mamet has it right.

Ordinarily I'd just say Mamet was high on coke or magic shrooms when he wrote the piece, or maybe he's begun a descent into dementia. But other sections of the article show he was acutely aware of the dangerous tripe he'd spewed but just didn't give a fuck. In particular, when he dismissed all Trump's vows of retribution as mostly empty blather while we who challenged him were guilty of :

"horrors of the past 4 years, an appeasement of terror, assaults on free speech - seeming to me to be the descent into chaos which has been the end of every world power."

Methinks Mamet is living in another parallel planet - "Earth 2"- where the horrific attacks on D.C. police 4 years ago can be ignored just as the recent attacks on federal workers - thanks to Elon Musk's posts to his 32 million followers which incited their targeting . Like the targeting of Ashley Thomas - a director of climate diversification -who  "triggered an avalanche of memes and ridicule". (WSJ, 'Federal Workers Feel Ire Of Musk's Followers', p. A6, Nov. 23-24)

Meanwhile, blatherskite like Mamet’s has continued to feed into the skewed, ignorant beliefs of too many lazy Americans.  I.e. that Trump has been unfairly targeted in "unwarranted politically motivated cases". Beliefs based on propaganda ingested from excessive FOX watching, too much social media exposure and not enough independent research – including into the backgrounds of the cases.

So at the very least, it's incumbent on the remaining U.S. citizens - with active brain neurons-  to dismiss Mamet's tirade.  Dismiss it as misbegotten malarkey by a played out playwright whose best days are decades are behind him.  And who's now reaching for some belated fame by getting his name  'in lights' on the Rupert Murdoch- approved op-ed pages of  The Wall Street Journal.  All that after sucking the balls of Dotard Trump.

Sad, how another formerly dignified human and competent crafter of the arts has had his vestigial sentience flushed down Trump's toilet to screech out twaddle and codswallop on his unworthy behalf.  Or more aptly, from Shakespeare's Macbeth:

"A walking shadow, a poor player
That struts and frets his hour upon the stage
And then is heard no more: it is a tale
Told by an idiot, full of sound and fury,
Signifying nothing
."

See Also:

by Jeffrey C. Isaac | January 7, 2021 - 8:31am | permalink

And:

by Robert Weissman | January 7, 2021 - 7:25am | permalink

by Thom Hartmann | December 5, 2024 - 6:24am | permalink

— from The Hartmann Report

South Korean President Yoon Suk Yeol, whose declaration of a state of emergency yesterday shocked the world, has often been referred to (both within and outside of his country) as “South Korea’s Donald Trump.” A political outsider, he came to power with anti-establishment and often outrageously inflammatory rhetoric, trash talking women’s rights, “reforming” their healthcare system, and pushing hard for a neoliberal agenda that included raising the workweek from 52 to 69 hours.

In that, he reflects a growing trend among advanced democracies around the world, as decades of neoliberalism have weakened multiple nations’ abilities to sustain middle-class lifestyles while enriching an oligarch class that’s now reaching out—worldwid—to seize control of democratic governments to their own financial benefit.

Of all the events in world news over the past weeks—even more than the escalation of Russian President Vladimir Putin’s murderous crimes against Ukraine—President-elect Donald Trump and his authoritarian colleagues down at Mar-a-Lago are probably carefully watching what’s happening to Yoon and gaming out how a similar “emergency” action here in America might be recalibrated to have ultimate success.

» article continues...

Thursday, December 5, 2024

Parameters in the Orbital Plane : Expansion of the True Anomaly In Celestial Mechanics

 

Motion and angular variations from it in the orbital plane are a key aspect of celestial mechanics. The diagram below shows assorted "orbital elements" for a planet of mass m2 (the Sun is m1) and this can be used as a basis for orbital energy analysis and also to predict future positions. 



Energy constants in celestial mechanics are very useful for quickly coming to terms with specific properties of an orbit such as shown in the accompanying sketch- designating a generic orbit in x-y-z space. In the diagram, w   is the argument of the perihelion, W is the longitude of the ascending node, f is the true anomaly and i is the inclination of the orbit. The critical or key parameter here is h, the angular momentum vector for the orbiting system. It may be useful here to refer to the diagram (b) in the figure used for angular momentum vector (z) in an atomic system:


Getting specific, assuming r and r' are (radius vector) and d r/dt, respectively, the magnitude h, of the angular momentum vector is:

h = r x r’ =

(y z’ - z y’)

(z x’ - x z’) = (c1 c2 c3)

(x y’ - y z’)

so (r x r’) = (c1/ h, c2/ h, c3/h)

and inserting variables one finds:

c1/ h = sin W sin (i)

c2/ h = - cos W sin (i)

c3/h = cos(i)

Now since the inclination of Earth's orbit to the ecliptic  (i) is known (23.5 deg) and therefore cos(i) can be determined, then sin(i) can be as well. Also h can be determined, since:

 h = c3 / cos(i) = GMm a (1 – e 2) 1 /2 

where all the constants are known (a = semi-major axis of orbit, e = eccentricity of orbit)

The energy (vis viva) equation is: ½V 2 -  /r = c, and the c's - energy integration constants- are  found as shown below)  Also, h can be determined, since: h = c3 / cos(i) .  

Also h =  [c1 2 +  c2 2   +  c3 2]  ½   (We also know  W =  11.26 deg.)

To fix ideas the longitude of the ascending node, W  is one of the key orbital elements (difference between mean anomaly  M and argument of the perihelion,  )  where M can be obtained from a table based on observations, and w can be obtained using a Fourier expansion of the mean anomaly, M, e.g.:


  = M + (2e – e/ 4) sin M + 5 e2/4 sin 2M + ... etc.


Once W is known, and c1, c2 and c3 are computed, the astronomer is ready to compute the position of a planet, say Jupiter, forty or so years in the future.  The actual calculations are formidable but can be found in a good celestial mechanics text, or using the orbital element formulae can be worked out using a computer.



We end with the basis to compute the true anomaly f. We start with a series expansion of  f in terms of E:

tan f/2  =  Ö {(1 + e) / (1 - e)}   tan E/2

And then get an expansion in terms of complex logs. i.e.:  

log e Z = if,   log e W = iE  

Such that:

Ö {(1 + e) / (1 - e)}    =  (1  + b )/ (1 -   b)


Note:  exp (ix) = cos x + i sin x


cos (x) =  [exp  (ix) +  exp (-ix)]/ 2


sin (x) =  [exp  (ix) -  exp (-ix)] / 2


Then:   tan f/ 2  = 

1/i [exp  (if/2) -  exp (-if/2)/ exp  (if/2) +  exp (-if/2)]


Ö {(1 + e) / (1 - e)} 1/i [exp (iE/2) -  exp (-iE/2)/ exp  (iE/2) +
  exp (-iE/2)]

Where:   i f/ 2  =  ½   log Z

Then:  

exp(if/2) =  exp (log Z 1/2 )=   1/2 / exp  (iE/2) =  W 1/2


In complex format:

Ö {(1 + e) / (1 - e)} 1/i [exp (iE/2) -  exp (-iE/2)/ exp  (iE/2) +
  exp (-iE/2)]

Þ Ö Z  -  1/ Ö Z /  Ö Z  +  1/ Ö Z  =   (Z - 1)/ ( Z + 1)

=  (1  + b )/ (1 -   b  (W - 1)/ ( W + 1)

Cross multiplying:

(Z - 1) (1 -   b)( W + 1)  =  (Z + 1)  (1  + b ) (W - 1)

So that:    Z =  (W -  b)/  ( 1 -  Wb)


The final form for the expansion is then:

Z =  W ( 1 -  W -1 b) ( 1 -  Wb) -1


Taking the logs of both sides:

log Z = log W + log (1 - W -1 ) + log (1 - W) -1

Whence:  if  = iE


Suggested Problems:

1) The Earth's aphelion distance is 1.01671 AU and its perihelion distance is 0.98329 AU. Given its eccentricity e = 0.016, then use this information and any other (from previous posts) to find:

a) the velocities at aphelion and perihelion

b) the energy constants C(A) and C(P) at each of these points, and h, the 'specific relative angular momentum'.

c) Hence or otherwise, use the vis viva equation to confirm the results you obtained in (a)


2)Explain the significance of the expression:

Z =  (W -  b)/  ( 1 -  Wb)


3)What  can you deduce regarding the relationship between: f,  E  and  w ?  What common factor unites them?