Use Of Generalized Coordinates, Momenta and Force In Theoretical Mechanics

 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).

*

Revisiting Laplace Transforms Applied To Differential Equations

The Laplace transform is perhaps one of the most useful devices for solving differential equations. In this post I look at some of the basics underlying its use, especially in solving  DEs.

Definition : Let  F be a function defined for t > 0. Then define a new function f by:

f(s) =  ò ^¥_o    exp(-st) t dt

For all s such that the integral exists, then f is called the Laplace transform of F and is written as:

f =  £ {F} or f(s)  =    £ {F(t) }

Example:  Compute £ {t}   where F(t)  =   t

£ {t}  =   ò^¥_o  exp(-st) t dt  =    lim _R ® 0  ò ^R_o exp(-st) F(t) dt   

=     lim _R ® 0    [ - t/s  exp (-st)]  ^R_o   + 1/s   ò ^R_o  t exp(-st) dt

=    lim _R ® 0    - R/s  exp (-Rs)  +  (-1/ s 2  exp(-Rs)  + 1/ s 2  )

 General properties of Laplace Transforms:

1)Let F1 and F2 both have Laplace transforms on some common interval. Let c1 and c2 be constants. Then:

£ {c1 F1 + c2 F2} =  c1 £ {F1}  + c2 £ {F2} 

Let F1 = 1, and F2 = cos t

Then:  £ {1 – cos t} =   £ {1} - £ {cos t} = 1/s  - s /1 +  s ² 

2)Let F be continuous for t > 0 and of exponential order exp (a t).  Assume F’ is piecewise continuous on every interval of the form [0, b], and 0 <  b <  ¥  .   Then,

£ {F’}exists and:

£ {F’(t) }  =  s  £ {F(t) }  -  F (0)

Problem example:

Solve:   dY / dt  +   2Y = cos t

Using Laplace transforms:

Write:

£ {Y’(t) }  + 2 £ {Y (t) }  =  £ {cos (t) } 

And:

£ {Y’(t) }  + 2 £ {Y (t) }  =    s / s ² +  1

Further:   £ {Y’(t) }   =  y(s)

s £ {Y’(t) }  -  Y(0)  + 2 £ {Y (t) }  =    s / s ² +  1

s y(s) + 1  + 2 y(s)   =  s / s ² +  1

y(s)  [s + 2}  =   s -   s ² +  1  / s ² +  1

  

Whence:  y(s)  =    -  s ² + s  -  1  / ( s ² +  2) ( s  +  2)

Separate using partial fractions:

As + B/ s ² +  1  +   C/ s + 2   =

(As + B)  (s + 2) + Cs² +  C/  ( s ² +  2) ( s  +  1)

So:

(C  + A) s²   +  (2A + B) s  + 2B + C  =  =    -  s ² + s  -  1  

From which we see by inspection:

A + C = -1,   2A +  B  = 1,  2B  + C  = -1

Add:

-2A – 2C  = 2

 2A   + B = 1

-----------------

B – 2C   =   3

Add:

B  - 2C   =  3

4B  + 2C =  -2

----------------

5B        =  1     Therefore:  B = 1/5  

2A + B = 1 and 2A =   1 - 1/5   =   4/5

A =   ½ (4/5)   =   2/5  so:   C = -1 – 2B = -1 – 2(1/5) = -7/5

The inverse transform is therefore:

£ ^-1 {y(s)}  =  2 cos t/ 5 -  sin t/5 – 7/5 exp (-2t)  = Y(1)

                                            Table of common Laplace transforms:

Suggested Problems:

1)  Solve:    

d ³ Y/ dt ³ -    d Y /dt   =   0 

Using the Laplace transform and the conditions:

Y(0) =  1,  Y’(0) =  0  and  Y’ (0) = 1

2)   Solve:    

  d² y/ dt ²  +  4y    =   3 sin t 

 Using the appropriate Laplace transform, given: 

y= F(t),   F(0)= 1,  F'(t) = 0

Results Of Latest ABC-WaPo Poll Again Shows A Population Detached From Political Reality & Critical Thinking

                Results of key question in ABC-Washington Post poll

                                "I don' believe Biden dun' nuthin' since he got in."

"If this poll is valid then most Americans don't seem to have the  slightest idea about how well or badly their government is run."  WaPo comment

“What are they, certified imbeciles, or maybe idiots?” Janice huffed on seeing the news this a.m. of a  Washington Post-ABC News poll. The poll found that 62 percent of Americans think Biden has accomplished “not very much” or “little or nothing” during his presidency, while 36 percent say he has accomplished “a great deal” or “a good amount.” Janice agreed that at least the latter groups – just over a third polled- showed a “workable intelligence and higher IQ” than the first set. 

 To refresh readers' memories, Janice has held a jaundiced view of my countrymen's intelligence since we moved to the U.S. (from Barbados) in 1992. From the foolish consumer obsessions to social media addictions (Tik Tok), to talk radio and FOX News, not to mention the political conspiracy theories such as 'Pizzagate'  e.g.

Crackpot "Pizzagate" Tale Promoted By Michael G. Flynn Almost Led To D.C. Tragedy 

And then the misguided 2016 election of the hare -brained demagogue Trump followed by the belief Biden really lost in 2020, she has not been enamored of Americans' IQ demonstrations. This has not been helped now by the poll finding "A majority of Americans do not believe Biden has accomplished much of anything."

What are these polling respondents taking, magic shrooms?   High dose MJ candies?  At the very least Biden has shown he's high functioning and more than capable - something Trump never did apart from his endless batshit nuts tweets.  

Oh, wait. Did I forget Biden's masterful Ukraine policy? See e.g. this WaPo article:

Biden deserves props for his masterful Ukraine policy

Or as Janice put it: Where have these twits been the past two years? Do they ever read or is watching dancing monkeys on Tik Tok all they do?”  Good questions!   Is my better half being hyperbolic? I don’t believe so.

The poll found that 62 percent of Americans think Biden has accomplished “not very much” or “little or nothing” during his presidency, while 36 percent say he has accomplished “a great deal” or “a good amount.” Janice agreed that at least the latter groups – just over a third polled- showed a “workable intelligence and higher IQ” than the first set.

 

Here's the clincher for Janice on the low IQ aspect: On many of Biden’s signature initiatives — from improving the country’s infrastructure to making electric vehicles more affordable to creating jobs — majorities of Americans say they do not believe he has made progress, the poll finds.

 

“In other words, they are all moron FOX News watchers and brainless conspiracy toads!”, according to her.  In other words, they are paying more attention to the blather and propaganda on FOX and its related conservo shit shows than to sober reporting of actual Biden accomplishments, such as at MSNBC.  She blames much of this on the recurring “Woke” bunkum which she insists too many are too dense to parse and see through.  I agree and have skewered this crap and its derogative, degenerate origins from the start e.g.

Enough Of The Anti- Woke Bunkum - The Culture Warriors Remain On The Right

The sad fact is that since the pandemic erupted  three years ago too many American brains have been flushed into a rabbit hole of unreality.  This has been reflected in what the late Steve Allen used to call 'Dumbth' (in his eponymous book) and meant an absence of rationality, critical thinking, common sense and possession of basic facts.  In the case of the ABC-WaPo poll basic facts of  how our system of government works and what exactly defines governance success.  It is clear as numerous sensible observers (including several WaPo commenters) have pointed out, that these Americans dinging Biden in the poll have no clue.  As Janice joked, "They probably can't even name the three branches of government."  

Much of this can be traced to too much distraction, excessive social media use and immersion in half-assed conspiracy bunkum via FOX News troglodytes such as Tucker Carlson.  It is a sad commentary that Biden will now have to factor a majority of the nation into account as distracted (or conspiracy-deluded) ignorant morons when he announces his achievements in the SOTU tomorrow night.   Or to quote Janice: "What did you expect?"

See Also:

Biden's Age - A Problem for the Media - But Not Sensible Citizens Uninfected by Ageism 

With Chinese 'Spy Balloon' Hysteria In US of A Maybe It's Good The Aliens Keep Their Craft Mostly Hidden

                     The Chinese spy balloon before being shot down after national freakout              

                            Headline from Washington Daily New about D.C. Sky Ghosts in July, 1952

                     More recent take on 'frenzy' over the 1952 D.C. UFO flap 

The Chinese spy balloon has  'Muricans freaking out from coast to coast, even those who aren't directly under it or for whom it's not visible.  One sheriff quoted in the Kansas City Star Friday swore if Biden didn't have jets shoot it down then he would. He didn't mention what specific weapon he'd use to reach a target 60,000 feet up.  The hysteria reached a crescendo after GOOP  Rep. James Comer appeared on FOX wailing about the thing dropping "lab hatched viruses" from Wuhan labs.  Then Saturday with all the GOOPs in an uproar Biden had little choice but to have an F-22 waste money and fuel to shoot it down.

But the utter lunacy and hysteria reminded me of another manifested hysteria 60 years ago: the massive UFO flap over Washington, D.C. - captured in the clippings shown at top of post.  According to the Wikipedia account of the events:

The sightings of July 19–20, 1952, made front-page headlines in newspapers around the nation. A typical example was the headline from the Cedar Rapids Gazette in Iowa. It read "SAUCERS SWARM OVER CAPITAL" in large black type.^[10] By coincidence, USAF Captain Edward J. Ruppelt, the supervisor of the Air Force's Project Blue Book investigation into UFO sightings, was in Washington at the time. However, he did not learn about the sightings until Monday, July 21, when he read the headlines in a Washington-

At 8:15 p.m. on Saturday, July 26, 1952, a pilot and stewardess on a National Airlines flight into Washington observed some lights above their plane. Within minutes, both radar centers at National Airport, and the radar at Andrews AFB, were tracking more unknown objects.^[14] USAF master sergeant Charles E. Cummings visually observed the objects at Andrews, he later said that "these lights did not have the characteristics of shooting stars. There was [sic] no trails . . . they traveled faster than any shooting star I have ever seen."^[7]

Meanwhile, Albert M. Chop, the press spokesman for Project Blue Book, arrived at National Airport and, due to security concerns, denied several reporters' requests to photograph the radar screens. He then joined the radar center personnel.^[15] By this time (9:30 p.m.) the radar center was detecting unknown objects in every sector. At times the objects traveled slowly; at other times they reversed direction and moved across the radarscope at speeds calculated at up to 7,000 mph (11,250 km/h).^[   

In the Robertson Panel’s controversial estimate, the Air Force and Project Blue Book needed to spend less time analyzing and studying UFO reports and more time publicly debunking them. The panel recommended that the Air Force and Project Blue Book should take steps to "strip the Unidentified Flying Objects of the special status they have been given and the aura of mystery they have unfortunately acquired

A more recent WaPo account from July 20, 2012 tried to punk readers with this:

The month that E.T. came to D.C.

Writing:

The Washington Daily News had written at the time:

 “Recent attempts to explain ‘saucers’ as optical illusions have been shaken by recent radar sightings. Illusions don’t show up on a radar screen.”

Illusions don’t, but temperature inversions do. A temperature inversion occurs when a layer of cold air is trapped under a layer of warm air. 

However, temperature inversions aren’t radar clocked at 7,000 m/h.  And temperature inversions, or mere ‘lights’ from optical effects, don’t result in reporters being denied requests to photograph radar screens.  Let’s also please note the Air Force’s own Director of Project Blue Book, Capt. Edward Ruppelt, was not consulted prior to the press conference in which the USAF tried the debunking routine. Ruppelt later publicly denounced the Air Force’s explanation, saying that temperature inversions were already well-documented radar anomalies, and what happened in Washington, D.C., was not related to an inversion.

Again, we behold the pathetic efforts of small minds at work to reach for utterly ridiculous prosaic nonsense to try to deny the obvious: actual craft from another world having their way in our airspace. With the resident alleged "intelligent" inhabitants too dense to admit their reality.

“Oh, oh but it’s impossible ‘cause other worlds is too far and they’d have to travel at light speed or more to get here!” 

Which babble I have already addressed in the post below taking the irrepressible WSJ Troll Holman Jenkins Jr. to task:

 Holman Jenkins Jr. In Latest WSJ Piece Shows UFO Topics Are Way Beyond His Pay Grade 

Of course, Jenkins still can't get over his fixation that UFO-UAP reports are really being used to "hide" the truth. As he wrote in a recent piece:

For those who couldn’t figure out why I devoted four columns to the Pentagon UFO debate, this is why. It became clear that, whether from serendipity or design, national security agencies were using UFOs to hide something they didn’t want us to see. That something, it has slowly dribbled out since last May, was Chinese surveillance in U.S. airspace. 

But does this twit really believe the Chinese spy balloon so many 'Muricans are freaking out over is anything remotely like the craft reported by the Nimitz pilots, e.g.

 https://youtu.be/Ce6ZevfbIK0 

 Newsflash Chinese spy balloons don't move at 7,000 mph.

What Jenkins Jr. still doesn't get - likely never will because of an underwhelming IQ --  is that the military and media deflection pranks are all about the loss of “sovereignty” so must desperately try to rationalize what can’t be rationalized.  

But Holman probably saw yet another breaking Jan. 13 WSJ story (p. A6): 

Reported UFO Sighting Up Sharply Over 2 Years

As more reinforcement for his nonsense about "hiding" the truth of Chinese sy balloon surveillance.  In any case that piece likely fell off most folks' 'radar' - what with all the media focus on Biden's missing "classified" files and the nutso furor resulting. The WSJ story by Warren P. Strobel was important as it focused on increased UFO sightings since 2020, noting:

"Reported sightings of Unidentified Aerial Phenomena, known popularly as UFOs, have climbed significantly in the past two years, and almost half the new sightings remain unexplained, U.S. spy agencies and the Pentagon said in a report released Thursday.

The study led by the Office of the Director of National Intelligence said the number of UAP sightings—often by Navy and Air Force pilots—stands at 510, with 366 of those reports coming in since March 2021...'Some of these uncharacterized UAP appear to have demonstrated unusual flight characteristics or performance capabilities, and require further analysis,” it said."

But from where I sit, any such analysis now needs to be detached from the 'sovereignty' bunkum which is holding back any objective assessment.  Once again, that malformed trope tends to reject the entire UFO-UAP phenomenon as real because the very existence of such collides with the human concept of state sovereignty. This based on a superb 2008 paper in the journal Political Theory by Alexander Wendt and Raymond Duvall, e.g.

Sovereignty and the UFO - Alexander Wendt, Raymond Duvall, 2008

In which the authors make the compelling case the real reason UFOs have been dismissed or treated as unworthy of serious human attention is because of the existential challenge they pose to a species that fancies itself the lords of creation - as well as the most technologically advanced life form. 

 As physicist Jack Sarfatti makes clear in his Youtube talk (link below) it is time this baloney cease and the powers-that-be and their media allies face reality. We are being visited and it isn't by the Russians (who can barely sustain a war with Ukraine), or the Chinese - apart from their spy balloons.  

At the same time, until a firm human reality consciousness emerges, I'd strongly advise any aliens piloting the UAP to avoid any prominent appearances.  They wouldn't want to get attacked by F-22s like the Chinese  "spy" balloon. That shoot down done to satiate the hysterics among us - to "preserve sovereignty"- was successful and the pieces fell harmlessly in the Atlantic Ocean. However, a similar attempt against a benign alien craft may not end with a beneficial outcome. For us.

See Also:

Republicans

Eek, a Balloon! How China Easily Got Republicans to Beclown Themselves

by Amanda Marcotte | February 8, 2023 - 8:32am | permalink

— from Salon

Excerpt:

It's a truth that should be self-evident: If you are emasculated by a balloon, you weren't so tough to begin with. Last week, a balloon that U.S. officials believe to be a Chinese surveillance device was spotted floating over Montana, likely to take pictures of military assets. In the grown-up world where President Joe Biden and Democrats live, this was a situation that called for a thoughtful, careful response. While displeased with the predicament, adults recognized that freaking out was not the answer. Instead, they waited until the balloon wasn't a direct threat to people and property underneath and shot it down over the ocean.

Meanwhile, Republicans — many in elected office who could be doing something useful with their time — saw the Chinese balloon as an opportunity for cringeworthy tough guy cosplay. Ohio Sen. J.D. Vance was just one of many, many Republicans who spent the Days of the Chinese Balloon ping-ponging between performative hysterics and childish games of dress-up where they pretended they were somehow going to shoot the balloon down.

And:

China

APB: The Balloon is Up

by Mel Gurtov | February 4, 2023 - 8:43am | permalink

Excerpt:

 The detection yesterday of a Chinese balloon hovering over Montana, where the US houses ICBMs, probably falls into the category of military surveillance, though the fact of the matter remains to be determined.

To my mind, the US has overreacted to the discovery, postponing an important visit to Beijing by the secretary of state. Granted, a Chinese high-altitude balloon should not have been floating over US territory; as Secretary Blinken said, it violated sovereignty and international law. 

And:

Republicans

House Oversight Chairman Mocked for Pushing Unfounded Balloon Conspiracy Theories

by David Badash | February 4, 2023 - 7:24am | permalink

— from The New Civil Rights Movement

James Comer is pushing baseless conspiracy theories about the Chinese high-altitude surveillance balloon floating over the United States – currently, over Montana – that the Pentagon is tracking, and he’s being widely mocked for his unfounded fear-mongering.

And:

• Are Laser Plasma Holographs Created By "Foreign Adveraries" Really The UFOs Reported By Navy Pilots?...

And:

Anthropocentric Parochialism Defines SETI Scientism.

And:

Jack Sarfatti - UAPs, Warp-Drives, Time-Travel & Consciousness (Part 1) - YouTube