Wednesday, September 29, 2021

Would Repukes Really Crash The Economy To Try To Win In Midterms? Of Course! That's What Traitors DO!

 "Mitch McConnell just had Republicans in the Senate declare a filibuster against a simple piece of legislation that would raise the debt ceiling to keep the federal government running, telling Democrats if they want to avoid a massive and destructive government shutdown they’re going to have to raise the debt ceiling using reconciliation.

It appears that this is suicidal behavior. Why would they be willing to blow up the entire country along with themselves and their reputations?

On its surface, strapping an explosive vest to yourself and blowing yourself up seems pointless, an exercise in futility or even nihilism. But only a very small percentage of suicide bombers are simply suicidal.

Mitch McConnell’s Republican Senate suicide bombers are, like those who blew themselves up to try to drive Americans out of Iraq or Israel out of Gaza, actually on a mission with clear purposes and objectives.

Not that you’d know it from the media. Just like the grievances of Middle Eastern suicide bombers are usually ignored or glossed over, the actual reasons why McConnell’s Republicans would strap on the debt-ceiling suicide vest is almost never mentioned in mainstream American reporting." - Thom Hartmann, Republican Suicide Bombers Threaten America,  smirkingchimp.com

As I noted in my Aug. 6th post, the GOP has ceased to be a viable, functional party and has now morphed into a fascist, pro-authoritarian cult.   At every turn these misfits and criminals have militated against the common good and national welfare, from spreading their Covid lies and support of traitor militias to now hurling the nation under the bus financially. This is in respect to blocking the Democrats from raising the debt ceiling, which means paying off our existing debts - NOT starting new ones.

 We can also cite the GOOp Senators' role in the insurrection and sedition of January 6th, and the (recently revealed)  traitor Jeffrey Clark's effort to overturn the general election.   These rats have some nerve claiming Gen. Mark Miley was "committing treason" at a Senate hearting on the Afghan withdrawal yesterday, when they are committing treason in betraying the nation's critical  financial interests - including the cessation of pay to our soldiers.  This is manifest in their refusal to allow raising of the debt  ceiling. 

 Contrary to widely circulated myths a debt-ceiling increase does not authorize new spending; it just raises the arbitrary cap on how much the government can borrow to pay off bills to which it has already committed. Given that Congress generally spends more than it collects in revenue, it must allow the Treasury to borrow money to make up the difference. If Treasury can’t borrow - say to pay VA benefits and Social Security - a lot of bad stuff happens which is where the GOP terror enters.

If. the government can’t borrow to pay bills that come due, it would have to suspend certain pension payments, withhold or cut the pay of soldiers and federal workers, or delay interest payments, which would constitute default. Unless Congress raises the debt ceiling, the Treasury could be forced to cut payments by more than 40%, including to millions of U.S. households now depending on the child tax credit.

 Uncle Sam would also have trouble paying civilian salaries, Social Security benefits, government contractors, domestic and international creditors, and almost every other bill already racked up. If we default on these IOUs, even briefly, that not only hurts those denied their promised payments. it also makes it more expensive for the government to borrow going forward. 

Since GOP fiscal hawks are always yapping about "saddling our great grand kids with paying huge interest on the debt" it's kind of choice these roaches would stoop to making it even worse.  But they would because they're fiscal terrorists.  It's bad enough, as Rachel Maddow put it Monday night- they have  zero interest in responsible governance - but to also block the Democrats from paying our just debts is beyond the pale, bordering on fiscal terrorism.  In this case the will to crash the still fragile economy and banking system - like Osama bin Laden hoped to do in the 9/11 attacks.

But the terror doesn't stop there. A U.S. default could trigger a worldwide financial crisis, possibly a depression. That’s because financial markets currently treat U.S. debt as the safest of assets, with all other assets around the world benchmarked against us. Our default would send waves of financial panic cascading through lots of other markets, too.   If. the government can’t borrow to pay bills that come due, it would have to suspend certain pension payments, withhold or cut the pay of soldiers and federal workers, or delay interest payments, which would constitute default. Unless Congress raises the debt ceiling, the Treasury could be forced to cut payments by more than 40%, including to some U.S. households, according to one estimate from Goldman Sachs.      

To refresh readers' memories, in 2011, Standard & Poor’s stripped the U.S. of its triple-A credit rating for the first time after the Treasury came within days of being unable to pay certain benefits. In other words, a credit downgrade merely for coming close to default.  Now imagine what an actual default at the hands of Republican terrorists would do.

It is little wonder that business groups (e.g. Business Roundtable, U.S. Chamber of Commerce, etc.), current and former Treasury officials and Wall Street firms have raised alarms in recent weeks over the prospect of a government default, which they say would be disastrous for financial markets and the U.S. 

As proven vermin all these odious grifters and Trump parasites care about is raw power and paying homage to the worst traitor in U.S. history - Dotard J Trump.  A maggot my Revolutionary war ancestors would have had hung long ago. 

Already Treasury Secretary Janet Yellen has warned lawmakers that if debt ceiling suspension isn't approved by Oct. 18th calamitous effects will follow.  These include missing $20 b in Social Security payouts on Oct. 20th, as well as inability to pay $6 b in IRS tax refunds.  More specifically, up to 50 million Americans will miss one or more Social Security payments and 30 million families will lose out on collecting any child tax credit monies.  Federal workers will also face extended layoffs as "Moscow Mitch" McConnell's  Reep traitors take the country hostage.

  But that is only the beginning of the calamity as we risk also a major recession as U.S. default leads to credit ratings downgraded, and capital reserves to pay existing obligations vanish.  Look for interest rates to spike, including on mortgages as well as credit cards.   And then watch out for international credit markets and stock markets to be roiled, even crash. Already the nasty effects are rebounding through short term money market funds as investors went Treasury assurances they won't be caught a day late and dollar short if a default happens.  Basically, they demand the same protections as with holding longer term money markets. 

Do the Repukes care about any of the financial catastrophe they are on the cusp of creating?  Of course not, they are laughing all the way to Mar-a-Lago, to prove to traitor Trump what they have wrought in his name.  These despicable degenerates actually get off on destroying the nation's financial stability and seeing millions suffer when critical benefits - whether for VA support or Social Security - are not received.   And then to have the audacity to try to pin it on the Democrats!

When you wake up Saturday morning and find the U.S. in deadbeat status and all our interest rates jacked up - from cars to credit cards- blame the Repukes.  A cult of degenerates who actually believe if they crash the U.S. economy by letting the U.S default for the first time in history they can prevail in next year's midterms.  Oh, and get this:  On the basis of being more "fiscally prudent" than the Dems.  No, you can't make this shit up.  That's what they believe.  Investigate it for yourself!

 See Also;

by Thom Hartmann | September 30, 2021 - 7:14am | permalink

 

Monday, September 27, 2021

Delaunay Variable Problem Solution

 First obtain the perturbation term R in terms of Legendre functions:

R=  k 2  3    [ 1/  r 3 +  ½  2/ r 3 3 - 3/2  2/ r 3 3  cos 2 S 

If we take  m =  mass of Jupiter,  m = mass  of Earth, and a =  semi-major axis of Jupiter we can calculate the first order perturbations in L, G,  ℓ and g using the reference Hamiltonian:

  =  

 -  m 2 / 2 L 2  -  2  3    [1/  r 3 +  ½  2/ r 3 3 - 3/2  2/ r 3 3  cos 2 S 

We thereby obtain a functional Hamiltonian:

 (L, G,  ℓ,  g, m 2 , m 3  ,  a 3  ,  t)  

And can write out the differential equations to solve the problem.  One such equation would be:

dL/ dt =      /  ℓ         

  Integration yielding:

L  -  L o =   ò t  o      F (ℓ)   dt    

Where F (ℓ)  =  F(L, G,  ℓ ,  g, constants, t) 

We  then substitute for each of the variables: L, G etc. Earth and Jupiter values, and also:    

 ℓ   -  ℓ  o  ,   g   -   g  o  ,   etc.  leaving everything else constant and taking the specific integral in each case.  Do this for L, G,  ℓ and g

Using the  mass values for Jupiter and Earth  expressed in terms  of solar (m )
 
m =  1/ (1047.355 m ☉ )

m =  1/( 32930 m ☉ )   

   =  5.2 AU  

 

On computation using the preceding, we get an error in the reference Hamiltonian:  

  =  2  3    10   -2

For an error magnitude e  »   0.012

Is There Really A "Missing White Woman" Syndrome In The U.S. Media ?

 For some time now a number of writers have pointed to a "missing white woman" dynamic in much of the U.S. media, most recently highlighted in a NY Times piece by columnist Charles Blow, e.g.
 

In 2004, at the Unity journalists of color convention in Washington, Gwen Ifill coined the phrase “missing white woman syndrome,” joking that “if there is a missing white woman you’re going to cover that every day.”. It is not that these white women should matter less, but rather that all missing people should matter equally. Race should not determine how newsroom leaders assign coverage, especially because those decisions often lead to disproportionate allocation of government resources, as investigators try to solve the highest-profile cases.

The obsessive fascination with missing white women also leads to a slanting in sympathies. All missing-persons stories are human tragedies, and because we are all human we empathize with the people we see. But this also erases the trauma of other missing people, as if nonwhite people never go missing, when they absolutely do. 

 It all becomes cyclical: Media raises the profile; law enforcement engages because of that high profile; the public becomes invested; then the media continues its coverage because of the massive law enforcement response and widespread public interest.

Just like that, we have all been manipulated into playing a part in the white damsel ideology, that young white women, often attractive, are the very epitome of innocence and virtue.

  It should be noted that the basis for Blow's piece is generally correct, except that hundreds of white women and teen girls also go missing each year also get no press coverage.  The latter are often teen runaways who somehow end up in sex trafficking rings or otherwise fall out of sight.  The women are usually from lower or working class backgrounds and lack the family connections and know how to use the media ecosystem to push for attention.  They have the resources that the families of other white missing females lack. 

This is not, however, to dismiss or minimize the disappearance of hundreds of women of color each year - as Mr. Blow notes  - numbering 700 or more.  It is only to point out that the peculiar emphasis of the media on certain higher profile white women must not be construed to mean that it is an exclusively white female bias.  In fact, it is more a general class bias - applicable as much to poorer whites as to women of color and indigenous women, girls.

  In 2020, according to a CNN report, an estimated 500,000 Americans went missing of which 300,000 were whites.  Most of those whites you never heard word one about in the media. Nor will you.  True, 200,000 non-whites were included in those missing - but neither will you hear much about any of them. Why?

It is understandable that the recent near wall-to-wall coverage of Gabrielle ("Gabby") Petito's disappearance has again focused attention on missing white females.  But this wasn't merely because she was white but because of the macabre circumstances surrounding the case. Thus, the fact she had been traveling with a fiancé (Brian Laundrie) who left her alone and drove back home himself attracted attention, and so did his own subsequent disappearance.    Another factor that helped enormously in Petito's case was her profile on Instagram in which she shared living her dream of traveling  the country.  This enabled other users to connect, including another traveling couple that spotted the couple's white van and conveyed the info to police.  In an age of social media contacts this is likely critical and also means if a person like me vanished - who never carries or owns a cell - it is doubtful I would be found either.  Just something to consider.

This harkens back to perhaps the most infamous missing white woman case in recent years, that of Shanann Watts.  Readers may recall all the drama that erupted when Colorado husband Chris Watts reported his wife Shanann - and two young daughters, Bella and Celeste-   went missing in November, 2018.  He swore he had no clue how or why this disappearance happened.  (Though Shanann's cell was never taken with her.) This went on for some time before it was discovered he had strangled his wife, and then killed his two daughters before tossing their bodies into an oil tank. (At a location near where he worked.)

  Again, it was the macabre circumstances of the case that drove media attention.  The intensity of that coverage as well as that of Gabby  Petito -also mirrored the fascination of social media users - who also pored over photos, videos.    One other NY Times writer, Katie Robertson, also observed that the "disappearances of women of color tend not to generate the same volume of media interest despite occurring at a higher rate."

But then again we must ask why?  I submit there are two overriding reasons: 1) The resources at the disposal of the missing woman's family (in the preceding case that of Shanann Watts in NC) and 2) the macabre events surrounding the disappearance.  The other factor that works against vanishing women (or men) of any color is their sheer numbers.  The corporate media finds a similar dynamic operating when a story focusing on one notable or tragic death - say of a crippled child - grabs clicks or readers dwarfing the account of mass deaths of those killed in an earthquake.   In effect, we have become accustomed to a focus on the few exposed to perdition, as opposed to the many.  But is this the fault of the media, or of human nature - and specifically the human brain itself?  

Well, I suspect it's partly of both but mainly the biological structure and tendencies of the brain which is unable to assimilate the scope of mass death (or mass disappearance), and feel the same degree of empathy as for single victims.  My post doc psychology niece Shayl confirms this, noting: 

"It's just too big an ask for the typical human to  psychologically process a mass death event, say like 200,000 -plus dying in the tsunami that struck in the Indian Ocean in 2004.  Our limited brains aren't equipped to extend sympathy like we can for a murdered child, like Bella who screamed "No, daddy no!" When Chris Watts smothered her with  a pillow."

I would suggest the same thing would apply if every missing female - white, brown or black- had feature stories written about her. In the end the clicks and reads would only accumulate for the few, those missing in macabre or unusual circumstances and whose families have the resources to tap the interest of the media. 

Stewart Coles -  another post doc (in communications)- argues that the public interest in the Petito case helped drive the media coverage - which makes sense.  Go to any news web site, whether WSJ, WAPO, FT or NYT, and it is obvious reads, clicks are tracked.  Media websites seeing more clicks, reads for a specific story will continue to run with it or find new angles.  Coles argues, however, this isn't the full story, e.g.  

"We have to consider how sometimes choices about hat stories are read and what we know are based on what gatekeepers within that media think people want to know about. So if those gatekeepers think that readers are more interested in a missing white woman, they are going to give more information, news about that."

Which makes sense, but as MSNBC's Joy Reid pointed out in 'Reid Out' on Friday, gatekeepers could still be persuaded to cover more disappearance cases of prominent women (and men, like geologist Daniel Robinson) of color.  This would at least be a start in balanced coverage and commensurate with the excess coverage of certain white missing people - namely women.  But it would require a conscious effort, not merely an obsession with click stats. 

Even with such an initiative there's no guarantee that the disappearances of people of color will ever generate the same volume of media interest - but at least it's a start and an improvement over what we behold now.  Yes, there is indeed a  missing persons attention problem in this country but it extends beyond white women.  It extends to all those who may not have the same access to social media or a profile that engages others of their race. Also, to those whose class may not command the resources or PR  savvy to enlist the full breadth of the legal infrastructure to pursue leads to locate a missing loved one.

  

Friday, September 24, 2021

Lessons Learned Re: The Black Death In Our Last Visit To Germany

 


Plague mask - top- and ironwork of plague reaper in Oberammagau.
                                                                           

Front wall and entrance to chapel in Garmisch showing plague art, ca. 1640.


In this epoch of rampant plague (Covid-Delta variant) it is useful to ponder a much earlier plague which laid waste to one third the population of western Europe in the middle ages.  When we visited Bavaria, Germany (Garmisch, Partenkirchen, Oberammergau,) in May, 2013,  we observed multifold echoes resonant from this horrific bubonic plague, most often engraved for posterity - or paintings-  as on church or chapel walls. For example, at St. Sebastian's chapel (see image) just two blocks down from our hotel in the twin town of Garmisch-Partenkirchen, one whole wall is decorated with the image of the Reaper slaying the people of the town. The Black Death did take a huge toll and no one there seemed to know how many but some guessed "in the thousands".

We ventured into the small cloistered chapel and sat in its rear most row, taking in the fetid odor of rank age, and trying to imagine conditions ca. 1634-35 when it may have been packed with the faithful desperately seeking to make some kind of bargain with the Almighty to be spared. (We were most careful not to dally too long in our ruminations, lest we found ourselves caught in a time slip, e.g. http://www.brane-space.blogspot.com/2013/04/time-slips1-flights-of-fancy-or.html )


After ten minutes or so we exited the rear of the chapel and walked outside to find a green expanse of about an acre with a small notice that this ground (extending behind the chapel)  formed a "cemetery". Oddly, there were no grave markers. Evidently, at least 1, 100 plague victims were simply buried in unmarked graves. Likely these victims made up the faifthful of the small congregation, while others ....e.g. unbelievers, witches, or Jews who perished, simply had their bodies consigned to huge funeral pyres - which actually manifested throughout Europe during the Black Death. (Europe lost an estimated one-third of its then population.)

A few days later, our German friends Reinhardt and Elli drove us to Oberammagau.  There they pointed out  engraved iron works displaying other unsettling images of the Plague "Reaper".  These were often accompanied by prayers or other pleas to spare the town. Reinhardt related that the plague toll was fearsome with so many corpses piled up in the streets and few places to put them.  Most had to be carted away to be incinerated in open pits.

Then, as now, people often didn't get it and shrugged off warnings to stay away from certain areas - like the markets where the rats congregated- and so brought the plague home to infect the household.  "Schlimm!"  Reinhardt would exclaim, "Diese leute sind dumpkofen!"   

Recalling his words, "These folks are idiots!"  brings to mind the tens of thousands of blithering morons now clamoring for "escape" and pursuit of their "freedom" in the Trump era U.S, even as Covid-19 deaths soar past 670,000.  Also, projections now are that we'll double that by the end of August because of the total lack of leadership, and the idiocy of opening up the economy too soon. It is as if these daft dummies don't grasp the essential biology of the virus, or that it doesn't give a damn about their precious "freedom".

In the mid 1600s, the plague ("Black Death") ravaged large areas of Germany obliterating perhaps a quarter million lives. All snuffed out, and most under horrific conditions of suffering as well as social alienation, exclusion. How could it be otherwise when it was seen what the disease did to the human body, so no one in his or her right mind wanted to tempt fate and contract it.

According to Sean Martin, the author of 'The Black Death' (2007) what we now refer to as 'the Black Death' was actually the second pandemic of the plague - the first occurring over 527- 565, during the reign of the Roman Emperor Justinian, and hence called 'the Plague of Justinian'. Unlike the Black Death, Justinian's plague is believed to have come from East Africa.


Meanwhile, the Black Death in Europe is generally traced to the Mongol hordes who poured across the Steppes westward and raced toward the Black Sea eager to seize ever more territory and booty as they plundered their way along well known trade routes. Ultimately meeting resistance from the Europeans (near the city of Kaffa), the Mongols were at the time more beset by the Plague (Bubonic) in their midst. With their corpses mounting and collective will crumbling they made a last resort effort to use catapults to hurl the plague -ravaged corpses of their comrades into the middle of the city. (This scenario is also confirmed by Martin, op. cit., p. 25). This is perhaps the first known instance of human biological warfare, albeit crude!


According to Martin (ibid.):

"In the confined environment of a walled city, the plague quickly did its work. No matter how fast the Genoese merchants disposed of the bodies of the Khan's troops (by dumping them in the Black Sea), they couldn't out run the pace of the disease. They took to their boats and fled.  Within a few days they would have realized they had unwittingly brought the plague back with them."

According to Martin, the Black Death is "traditionally thought to be a mixture of bubonic, septicaemic, and pneumonic plague" and he speculates on some unnamed 'third factor' that may have made it even more virulent. In the bubonic plague, the lymphatic system is attacked generating boil-like black eruptions ('buboes') the size of oranges. The growths are hideous and painful to the touch and can appear anywhere from the neck, to the armpits and groin....but often most where the victim was first bitten.

As time goes on, the person vomits repeatedly, has a high fever and the buboes themselves make 'gurgling' sounds and the potential to erupt. In 60% of the time when they are untreated the person perishes, often with his or her body peppered by black buboes - hence the term, 'Black Death'. Vomiting itself, accompanied by high fever, is often the sign that the disease has become septicaemic ....non-stop watery diarrhea follows with black, bruise-like discolorations appearing on the skin, just before the hapless victim expires.

Pneumonic or pulmonary plague occurs when the plague bacillus misses the lymphatics and settles in the lungs instead. The most frequent symptom is the violent expectoration of blood. The problem? With each cough-released stream of bloody sputum, the plague bacilli are released as well - resulting in airborne transmission.

The target flea for the Yersinias pestis bacterium is Xenopsylla cheopsis. Once it infects this flea, Yersinias' tendency is to keep on multiplying until it eventually totally blocks the flea's oesophagus making the critter chronically thirsty. The flea must then seek blood sources. Since rats are the most prolific, and especially at the times of the plague bred relentlessly because of sewage and garbage strewn all about, the fleas seek out these beasties first. When one dies, they jump to another, and have no qualms about choosing humans as the next best source to quench their thirst.

Of course, none of this was known at the time the Plague ravaged Germany from 1633- 1670. Also, given the advance of scientific knowledge had not yet accounted for bacterial existence, blame was most often based on "supernatural" or "spiritual" attributes. I.e. blaming people who were likely not pious enough, or committed foul deeds, or were miscreants in some other fashion, e.g. "witches", atheists, heretics or ....Jews. In many regions, alas, Jewish populations were blamed for the plague's spread, part and parcel of the misbegotten German Völkisch tradition (which also gave rise to the Passion Play at Oberammergau).

It's difficult for most of us today to contemplate such grisly conditions as I described ever arising again,. but who knows?  What we do know is the coronavirus has the potential to mutate - though evidently not as fast as flu viruses - and these mutations may bring more severe symptoms than we've documented to date. (And that is as we're still learning about the disease's current progression and attendant current symptoms, including: "silent hypoxia",  "Covid toe" and severe acute respiratory distress syndrome in children.

The tragic irony now is that the U.S. doesn't appear to learn from past history - tracking back even just to the Spanish flu pandemic 100 years ago. Under the criminal administration of Trump, which has rushed economic reopening, many tens of thousands more Americans will die by the end of August. Despite that, Trump is prepared to  abandon his coronavirus task force by the fall, in order to "focus on the states reopening". Thereby giving a measure of the Trump cabal's callous dimensions and total lack of any moral compass, placing Trump's re-election bid over the rising daily casualties.

In Italy, meanwhile, we have learned (WSJ yesterday, p A8) the government is steadfast in that "it doesn't wish to be rushed" and this despite "much of corporate Italy chafing at the slow pace of Italy's reopening.".   Further, like in the U.S. (based on polls, 63% now concerned about too early reopening): "Most people agree that reopening too early is more dangerous than waiting too long."   But they are blessed with a real leader who places their welfare foremost, as opposed to his own ego and one track narcissist mind to get re-elected.  In the words of Economic Development Minister Stefano Patuanelli (Ibid.): 


"The Italian economy can't afford a second lockdown. The gradual approach and the possibility in the coming days for regions to expand the reopening based on epidemiological data is the only possible approach."

Sadly, this core lesson is one that the Trump reprobates have not learned and likely never will. Instead they have turned the nation over to their own corruption in the wake of Covid-19's depredations. This is given the virus can incubate for more than 2 weeks amidst millions of "silent carriers".   Projections now are for the equivalent of a 9/11 each day by June. Whereas Saudi terrorists were responsible for the original 9/11, Donald J. Trump will be responsible for the succession to follow, on account of doing nothing to halt the SARS- Cov-2 virus. From that point of view, Trump can be said to be an accomplice of the terrorist virus now running amuck among us.



Tuesday, September 21, 2021

Revisiting The Delaunay Variables of Celestial Mechanics

 

                             Perturbation diagram to for Delaunay variable analysis

We already saw the Delaunay variables play an important role in celestial mechanics, especially in perturbation theory.  This is concerned with how the presence of a third body (say m 3  in the accompanying diagram) perturbs or disturbs the motion of m 2.  Given the complexity, it is useful to revisit these concepts.  These Delaunay variables are defined by the equations:

ℓ  =   n(t  - T)

L =  Ö(m a)

G  =   L    Ö (1 -   e 2)

H =   G cos i

g   =  w

h  =   Ω


The symbols used are standard, e.g.  w   is the argument of the perihelion, W is the longitude of the ascending node, e is the eccentricity, a is the semi-major axis of the orbit,   and i is the inclination of the orbit.   Also, n is the mean motion (i.e. of the planet or other body per day),  T is the date of perihelion, t is any future date assigned,  and m  is the reduced mass, i.e.

=   1/ (1/m 2   + 1/ m 3)

As with all similar problems in Hamiltonian mechanics, the approach begins with the applicable Hamiltonian for the system (see diagram).  In its most general form we write:

 =   ½ å 3 i=1  P i 2    -  m / r  +  R

After modification and as a function of the Delaunay variables (L, ):

H  (L,  ℓ  )  =   -  m 2 / 2 L 2  -  2  3    {1/  D (L,  ℓ  )  +   r ·  r 3   /   r  }

A particular set of differential equations will always be associated with the new Hamiltonian, e.g.

L’ =    -  H  /  ℓ        ℓ    =     H  /  L

g ‘  =   H  /  G          h’ =    H   H

G’ =   -    H  /  g     H’  =  -    H  /  h

 A huge simplification arrives  if R = 0., which then reduces to the 2-body problem, which we've examined in previous blog posts.   Note that R in the second Hamiltonian is expressed:

R   =   2  3    {1/  D (L,  ℓ  )  +   r ·  r 3   /   r  }

Now, for the case at hand (consult diagram), we may write:

D 2     =  r  2   +   r  2    -   2 ·  r 3  cos S

 And:  r ·  r 3  =   r   r 3  cos S

Assume now that 3  is greater than r (perturbing an inner planet or body by an outer one):

1 /D      =  1/r [ 1 + (r / r 3)  2  -  2 (r / r 3)  cos S] 1/2

Then the above can be rewritten as the sum:

½ å ¥ r= 0  (r / r 3i  P i 

=   1/  r 3  [ o   +   (r / r 3)    + (r / r 3)  2     +  ...]

The P i   are functions of the angle S and are called Legendre polynomials.  In this case, the first three may be written:

o   =   1,      P 1     =   cos S,    P 2     =   ½ ( cos 2 S   -   1) 


Problem for ambitious readers:

Using the Legendre functions for the angle S just given rewrite the perturbation term R in terms of them.

Thence or otherwise, rewrite a Hamiltonian incorporating R and these functions.   From this, write out the form of at least one applicable differential equation incorporating Delaunay variables and explain your basic procedure (say for the Delaunay variables L, G,   and g.)

For those really wanting a challenge:  Estimate the magnitude of the error in the modified Hamiltonian H  in terms of k and   m 3.  (You can take  m =  mass of Jupiter,  m = mass  of Earth, and a =  semi-major axis of Jupiter).   Make sure your reference Hamiltonian conforms with these.