How Classical Mechanics Can Be Applied To A Triatomic Molecule

  In an earlier post I showed howLagrangian dynamics applies to simple mechanical systems, e.g.

• Lagrangian Dynamics Revisited (1)
• We now examine a more advanced application, namely in terms of modeling a linear triatomic molecule.  We sketch the basic model system below:

We see the linear molecule is composed of three components: m1, m2, and M,  with the smaller masses m1 = m2 = m.  We also apply the condition they can only move along the linear dimension with atoms specified at positions: x1, x2 and x3.  We designate the equilibrium position as b.

We recall here the Lagrangian L = T - V.

And obtain the kinetic energy. T and potential energy V:

Kinetic energy: 

T   =   ½  ( m x1’ ²   -   M x2’ ²    + m x3’ ² )

Potential energy:

V =   ½  k[ (x2 – x1) – b] ²   +  ½  k[ (x3 – x2) – b] ²      

T   =

½  [(m….0…..0)

       [(0….M…..0)

       [(0….0… ..m)

And make use here of the refined position parameter in terms of displacements h _i:

h _i        =  x _i  – x o _i  

So:   h ₁           =  x ₁  – x o ₁

x ₀₂  – x o ₁   =  x ₀₃  – x o₂ =   b

 Then the potential energy is:

V =   ½  k[ (h 2 – h 1) ²   +  ½  k[ (h 3 – h 2)  ² ] 

 And the kinetic energy:

T   =   ½  ( m h 1’ ²   -   M h 2’ ²    + m h 3’ ² )

Rewriting V:

V   =   ½  k[ (2 h 2 ²    + h 1 ²   +  h 3 ²  -  2  h 1 h 2  - 2  h 3 h 2 )]  

In tensor form:

 V   =

½  [(k… .-k…..0)

       [(-k….2k…. -k)

       [(0….-k… .. k)

 

T  =  h  ×  T ×   h

 

By Lagrange’s eqns.(Tensor form)

T ×   h   +     V ×   h   =  0  

Assume a solution of form:  exp (iw t)

w ²    T ×   h   +     V ×   h   =  0  

And the determinant of coefficients must = 0 for non-trivial soln.

è   | w ² T  +   V  |  =  0

è 

Secular determinant:

  

(k - w ² m    …-k…   . .0)

(-k …..2 k - w ² m  ..  -k)

(0…. ….-k….. k - w ² m)

 From here and working through some messy algebra (solving the secular equations from the determinant), we can arrive at three solutions for the angular frequency:

w1  =  0,     w2  = Ö(k/ m),   w3  = Ö{ k/ m (1 + 2m/ M)}

For amplitudes, three solutions are obtained:

h 1  =   A1 exp (iw t)

 h 2  =   A2 exp (iw t)

h 3  =   A3 exp (iw t)

Suggested Problem:

For each of the frequency solutions (h 1  etc. )applied to the linear tri-atomic molecule, give the relation between the normalized amplitudes: A1, A2 and A3 and the angular frequency.

A Media Disgrace: Craven WSJ Editors Blame Dems For U.S.S. Lincoln's Deplorable Conditions Instead Of Trump

 

"If he is no longer responsible for what he says or does, we should move beyond shock and outrage to outright fear. He could blow the world up. We must call on our elected representatives to remove him from office via the 25th Amendment as soon as possible, by reason of insanity."-  Robert Reich, 'Beyond Shock and Outrage', Substack

Again, we have a diseased, bloated madman at the helm of this country, trying to steer it into permanent wreckage - while a supplicant congress is simply content to follow his every order and suck his butt. The latest outrage is how the carrier U.S.S. Abraham Lincoln has been allowed to be deployed for over 250 days, as living conditions have  deteriorated along with morale. The latter leading to six sailors already reaching such desperation they tried to jump off the ship, see e.g. the Kyle Kulinski video podcast:

6 US Soldiers Tried To K*LL THEMSELVES By Jumping Off The USS Lincoln | The Kyle Kulinski Show

And what has been the response of the rotting orange fungal Turd? Well only to insist the besieged U.S.S. Lincoln tour "wasn't nearly long enough".  (WSJ, 'Trump Downplays Concerns About Carrier, Aug. 15-16, p. A7).  

Not nearly long enough with guys so stir crazy  they're jumping off the ship for escape?  What mushrooms or drugs is this fat fucker on?

This was the response of the demented pig who passes for 'commander in chief- though he couldn't command emptying a porto potty.  Why would he, as a draft dodger, traitor and grifting crypto criminal who cares more about completing his gold ballroom than the distressed sailors on the U.S.S. Lincoln?

 As NY Times' Maggie Haberman recently noted in an Ezra Klein interview: 

"70 percent of his mind share is on completing his ego projects like his ballroom and Arch de Trump."

This is the slime ball who the craven, conservo-cuck WSJ editors (in the same Journal issue) could not bring themselves to blame for the Carrier's problems in their editorial ('A Crisis in U.S. Naval Readiness Revealed', p. A 14).  And the Editor's asinine response? It's the Democrats to blame!  Writing: 

"Democrats this week are expressing outrage over press reports of low morale and tough conditions aboard the U.S.S. Abraham Lincoln, the aircraft carrier deployed for 250 days. This is a new level of chutzpah from the party that won't pass a bill to finance military operations."

Woah! 'A new level of chutzpah'? Back up there Dotard asslickers and bald-faced liars passing for 'editors'. The sole blame is on your vaunted Orange hero who launched the attacks on Iran with Israel. Iran then responded with an initial all out targeted assault on the U.S. supply depot in Bahrain - which forced the Navy to move supply operations 2,2oo miles away to the island of Diego Garcia -with a Brit base in the Indian Ocean. See e.g.

USS Lincoln wasn't resupplied because Hegseth 'hid' that Iran destroyed main US base: Rohde | Watch

 Check your damn facts and you will find the same moron that launched this war of choice also LOST it on Day one, see e.g., 

 "He lost the war on Day 1" — Paul Krugman Exposes Iran War Failure

But no, the mesmerized Journal editors are dug too deeply into Donnie Dotard's rear end. What with believing Trump's lies to the extent they will swallow anything he or keg master Kegsbreath bellows out of their pieholes. I.e. the latter gaslighting the nation with this horse shit:  "The brave Sailors of that Strike Group are very proud of their tough and historic mission!"

Er no, they are not, Petey. They are barely hanging on for dear life having "not made a port call in 200 days, setting a record for consecutive days at sea".

And all that time dealing with half rotten food, moldy showers, no hot water and a ship basically falling apart as Kyle Kulinski's podcast notes.  No wonder these guys are ready to jump off it for relief - and yeah troops are lashing out, e.g.

Trump has CATASTROPHIC SUNDAY as TROOPS LASH OUT!!

But the craven  WSJ editors go on to bark out this twaddle:

"If Democrats are so concerned perhaps they could expedite passage of the $17 billion for U.S. military operations that the Administration requested."

Oh you mean this "administration"? Actually a regime of criminals, clowns and congenital liars who can't even get their stories straight two days in a row? How about instead, Captain Bonespurs finally take an 'L' and admit he lost this war and that Iran will never cease making his life miserable'?  At least Hormuz would open again and the desperate people of the globe who need energy - including for fertilizer - would breathe easier.

But no, "Bonespurs" Trump ain't about to do that. He'd rather spit out tons more horse manure like asking voters to accept $4 gas prices for a war they are not behind.  (See top graphic). Do the WSJ editors mention any of this? Hell no! They prefer to piss all over the Dems in their editorial, but then in the next paragraph bleat: 

"Yet the GOP also seems ready to abandon the President's $350 billion defense request."

Well, hell's bells, maybe they've also seen the light, but more likely are trying a reconciliation bill end around. Personally, I don't care so long as Dotard Donnie and Kegsbreath don't get any more taxpayer money to blow on their futile war that's already lost. And clearly we're never going to get the truth from the WSJ Pooh Bahs anyway, so outside sources are essential. Below some of the videos, podcasts that tell it like it is:

Trump Just Lost Control of His Own Military — And It's Getting Worse | Col Douglas Macgregor

&

The Misery Will Continue Until Morale Improves

&

The Military’s SHAME: Inside the USS Lincoln Crisis

&

Deep Intel on Carrier LINCOLN's Morale Crisis - YouTube

&

BREAKING: Trump Says USS Lincoln Deployment Hasn’t Gone on “Nearly Long Enough” | Command Post

&

We need to accept that we elected an actual madman to be president

&

Why Trump's Ballroom Bunker Proves He's Terrified | The Daily Beast Podcast

The only WSJ Op-ed contributor who (for now) seems to possess a full deck and some grasp of reality appears to be Peggy Noonan ('Trump is Starting To Look Precarious', Aug. 15-16, p. A15) who writes this regarding Bonespurs:

"His political problem with Iran is not only that the conflict is dangerous and unresolved. or that his attempts at diplomatic resolution seem half-baked, but that almost six months in he seems utterly out of his depth.  Like someone who wouldn't have gone in there had he understood history or the nature of Iran's leadership."

To which all I can retort is a big "DUH!"  Of course this deranged nitwit doesn't understand history because he's never studied it. Hell, the dolt doesn't read and can't even handle a vastly reduced half page daily brief - as Maggie Haberman and Jonathan Swan note in their book Regime Change,

 'Regime Change' - Perhaps The Best Political Expose in 20 years (A Review)

Noonan herself is also wise enough to grasp Trump is to blame for whatever morass or lack of military readiness or weapons the country finds itself in. Especially given the idiots defended U.S. bases around the Gulf with thousands of Patriot missiles at $4 m each, while Iran took out its targets with $400k drones. Of course the U.S. -guided by certified morons- would run through its whole inventory!

Near the end of her piece Noonan writes that Trump spoke to Tucker Carlson "by phone in the weeks before the Iran war commenced"  Adding:"Mr. Carlson was trying to warn him against military action" But Trump replied: "I know you're worried about it. But it's going to be OK."

Again proving severe detachment from reality and military facts. Namely that Iran would shut down the Strait of Hormuz when the first missiles struck. But the Orangutan believed he'd have a cakewalk like in Venezuela.  Noonan again:

"It's hard to understand such a breezy gloss on history, or on life. Presidents don't say that kind of thing because presidents are precisely the people that learn each day that many things don't work out for the best."

Well, especially if that president is deluded, deranged or both - like Trump. Or which Pegs attributes to "boredom", i.e.

"Mr. Trump seems to have a real fear of boredom. Even at his age he is restless, and when there's peace and quiet he seems to become anxious as if he can't tolerate normality."

Exactly, Peggy! Because he's not normal though too much of the reactionary press treats him as such (even YOU at other times!)

But what Noonan interprets as aversion to normality I and many others - including his former lawyer Ty Cobb - trace to mounting insanity, i.e.

Trump Lawyer Calls For 25th Amendment Over PSYCHOTIC MELTDOWN

So it's past time this senile fool needs to be replaced under the 25th amendment.  If this traitor loses it with Iran and launches tactical nukes at some point (say at Pickaxe Mtn.), e.g.

MTG Gives URGENT WARNING on Trump USING NUKES!!!

 the nation's 'leaders' have only themselves to blame for refusing to act expeditiously.

See Also:

US Military Morale in the Middle East Is in the Overflowing Toilet

by Ann Wright | August 15, 2026 - 4:50am | permalink

Rotten food, overflowing toilets into bunking areas, no room in the incoming missile shelters, confiscated cellphones that are the notification devices for incoming missiles, infrequent mail, social media closed down, were some of the issues that military families identified at the national conference of Veterans For Peace.

Spouses and parents of Air Force, Army, Marine, and Navy members told of plummeting morale, lack of basic supplies, and attacks on US military bases that are not reported in the US media.

The frustration of the military families at the VFP conference echoed the descriptions of life on US military bases in the Middle East under attack by the Iranian response to the brutal US-Israeli five-month attack on Iran made by military families who met with the acting assistant secretary of the Navy and the commander of the US Pacific Fleet in San Diego on August 6, 2026.

» article continues...

And:

Opinion | Why the USS Abraham Lincoln crew has reached a breaking point - The Washington Post

Excerpt:

When missions expand while food, showers and sleep dwindle — and when mail and port calls are canceled, and homecoming dates repeatedly rescheduled — time seems to stop, making life in the middle of the ocean turn from hard to intolerable.

That breakdown matches recent reports of conditions on the aircraft carrier USS Abraham Lincoln, which is on a historic deployment with more than 260 consecutive days at sea. The crew of roughly 5,000 was supposed to be home several weeks ago, but the prolonged war with Iran has extended its operations. For months, worried family members have raised concerns about the well-being of their sailors and the status of the ship, sharing accounts of broken systems, supply shortages and extreme fatigue.

Rep. Mike Levin (D-California), who represents the congressional district neighboring the Lincoln’s home port in San Diego, chronicled the issues on social media: “Moldy showers, broken toilets, laundry down for weeks, long stretches with no hot water, a meal that came down to half a cup of rice and two tortillas. No soap, deodorant, or toothpaste.” 

And:

Donald Trump

Trump’s ‘Obvious Neurological Decay’ Poses Historic Danger: Diplomat

by Nick Hilden | August 18, 2026 - 5:17am | permalink

— from Alternet

On Monday, amid news that President Donald Trump is destroying longstanding U.S. alliances, coddling adversaries and considering the use of nuclear weapons, a prominent American diplomat called for bipartisan recognition of the historic danger presented by the Commander in Chief’s "unchecked narcissism" and “absolute cognitive failure.” This is according to attorney Andrew Weinstein, former delegate to the United Nations, who took to social media to raise the alarm.

“We are past the point of political disagreement,” posted Weinstein. “We are witnessing a catastrophic collision between unchecked narcissism and obvious neurological decay. It is the most dangerous intersection in American history: a man with the absolute power of the Presidency and the absolute cognitive failure of a person no longer tethered to reality.” He continued, “Turns out U.S. military readiness becomes strangely flexible when the priority is not upsetting a dictator who runs forced labor camps, starves his own people as punishment, and executes citizens for watching foreign media.”

» article continues...

And:

Professor Robert Pape: Iran War is a Trap and America Has No Way Out!

And:

Lawrence: Donald Trump has lost control – even his choice of airplane

And:

Chris Hedges - How Trump’s Iran War Is Bringing Down the American Empire

And:

🚨 Trump's FINAL NAIL in the COFFIN as Approval Ratings Hit ROCK BOTTOM! Iran DESTROYS HIM!

Solutions To Fresnel Diffraction Problems (Part 1)

 1) The innermost zone of a zone plate has a diameter 0.425mm.  

(a)Find the focal length of the plate when it is used with parallel incident light of wavelength 4471 Å from a helium lamp.

(b) Find the first subsidiary focal length for the zone plate.

Solutions.

a)  The focal length f =  s _m ² / m l

Where in this case: s _m  = s ₁   =  0. 0425 cm   (so m = 1)

l   =   4471 Å  = 4. 471   x  10 ^-5  cm

Then:  f =  (0. 0425 cm) 2 / (4. 471   x  10 ^-5  cm)  =  40. 39 cm

b) The first subsidiary focal length will correspond to f/ 3,  so:

f /3  =  40. 39 cm/  3   =  13. 46 cm

2) A zone plate is to be set up on an optical bench where it will be used as an enlarging lens.  Its innermost zone will have a diameter of 0.2250 mm with monochromatic green light of wavelength 4800 Å  to be used.  If all enlargements are to be eightfold in diameter find the focal length f of the zone plate.

Solution.

An eight-fold enlargement means we need to find f/ 16 for the 8th subsidiary maximum of the plate.  Then:  

 f =  (0.0225 cm)/  (4. 8   x  10 ^-5  cm) =  10. 54 cm

And: f/ 16 = 10. 54 cm/  16  =  0.66 cm

Why Population Decline Is The Way To Reduce Inequality - As Opposed To Dependence On Higher Fertility

 

               Roadside market in Nigeria: 80% of  population growth until 2100 will be in Africa

In the most recent Population Connection issue we learn some important facts especially given the barrage of hysteria concerning the need to expand fertility – especially in the U.S. e.g.

           Unreal - Another Pro-Natalist Buffoon Thinks We Need More Than 1 Billion Americans

 – mostly pumped by clueless politicos. Or lamebrains like Elon Musk who has predicted “catastrophic population collapse.”   But all such hand wringing focuses on the wrong issue or index. Typically, warnings about falling fertility in the U.S. case focus on the old-age dependency ratio — the number of retirees relative to the economically active population. In countries where fertility decline started earliest the McKinsey Global Institute (MGI) estimates that the ratio of people over 65 to working-age individuals (aged 15–64) will rise from around one-quarter today to one-half by 2050. In the U.S. the worry is that there will be insufficient active workers to support the elders receiving Social Security and Medicare.

But this assumes that retirement ages remain fixed despite rising life expectancy, thereby conflating two distinct demographic trends: falling fertility and increasing longevity. You'd think a billionaire genius like Elon Musk would get this but they don't.

Second, the standard fertility template ignores the fact that children are also dependents, which means lower fertility reduces child dependency even as old-age dependency rises. Politicians - especially on the Right - forget or dismiss that side of the equation. They call for “more babies” but fail to recognize that if birth rates do increase, the total dependency ratio will rise even faster than before until those children enter the workforce two decades later.

The major bugbear and fear throughout is spreading inequality, which will surely increase among the elderly worldwide if social support systems - like Medicare and Social Security in the U.S.-  are cut. (Which the GOOPs are already planning to do to extend and enhance their abominable 'big beautiful bill' which has already taken a wood chipper to Medicaid and Obamacare.

So the GOP and Trump are quite content to increase the misery index as Paul Krugman recently stated in a podcast.

Misery Business

And will do more damage to social support programs if they manage to get control of government again. But let's get back to whether and how inequality relates to fertility - population increase. Let's assume the GOP degrading factor to add economic misery is eliminated by 2028, then we ask: Can population decline reduce inequality?

Recall Thomas Piketty’s 2013 book, Capital in the Twenty-First Century, showed that wealth-to-income (W/Y) ratios have increased sharply across most of the world’s developed economies over the past 70 years,

Piketty, Capital, Soaring Stock Markets and Why Americans Aren't Rioting in the Streets

 Piketty noted the innate tendency for capital to increase  encapsulated by the shorthand,  r > g  with r the long run returns on capital and g is the rate of economic growth. Basically, it shows the "past devours the future". This is highlighted by the fact that the heirs of the fatcats  pay zero capital gains taxes on their stocks or other assets. Americans aren't yet into torches and riots because they firmly believe one day they will join the 'upper crust'.

But here's the kicker: Because wealth is distributed far more unequally than income, and because it is inherited, this trend undermines social mobility. Without higher taxes on wealth or inheritance, Piketty argues, W/Y ratios will continue to rise. But the chances of higher taxes on wealth or inheritance are slim to none so long as the Reeps are in power- so how can the equation be made more favorable for those caught in the economic undertow fed by Trump's follies? 

It turns out Pikettey's analysis assumes that the annual increase in total wealth equals the savings rate multiplied by total income. In other words, wealth comes from household savings, which implies that the W/Y ratio depends on the savings rate relative to the economy’s growth rate. By reducing GDP growth, then, declining fertility would boost the W/Y ratio.

But hold strain. Piketty’s own data also show that most of the increase in the W/Y ratio is the result of housing prices rising faster than average earnings, driven by competition for limited supply in desirable locations. This implies that, contrary to Piketty’s prediction, population decline would actually slow the rise in W/Y ratios.  In other words, inequality could be reduced.

Not only would a slower or falling W/Y ratio benefit those without access to inheritance or the “bank of mom and dad”, i.e.  as a path to homeownership. But population decline will also reduce income inequality, benefiting lower-income groups.   Let's also bear in mind, as the recent issue of Population Connection notes(p.22)

"Historically, slow population growth — or outright decline — has tended to benefit workers and disadvantage capital owners. In his 2017 book, The Great Leveler, the Stanford University economic historian Walter Scheidel notes that the collapse in Europe’s population after the Black Death of 1348 drove up real wages and cut landowners’ rents. A 2020 study suggests that these wage increases also spurred innovation, putting northwest Europe on a path to sustained improvements in living standards."

And maybe the best part of all? A shrinking population would also enhance human welfare by reducing the burden on natural systems and resources, especially water.  What’s not to like?

See Also:

Clutter, Capitalism and Inequality: Why So Many Uncaring 'Muricans ?

And:

Economic Policy

The House’s ‘One Big Beautiful Bill’ Is Only Beautiful If You’re Rich

by Robert Reich | May 23, 2025 - 5:35am | permalink

— from Robert Reich's Substack

The old professor in me thinks the best way to convey to you how utterly awful the so-called “one big beautiful bill” passed by the House last night actually is would be to give you this short 10-question exam. (Answers are in parenthesis but first try to answer without looking at them.)

1. Does the House’s “one big beautiful bill” cut Medicare? (Answer: Yes, by an estimated $500 billion.)

2. Because the bill cuts Medicaid, how many Americans are expected to lose Medicaid coverage? (At least 8.6 million.)

3. Will the tax cut in the bill benefit the rich or the poor or everyone?(Overwhelmingly, the rich.)

4. How much will the top 0.1% of earners stand to gain from it? (Nearly $390,000 per year).

» article continues...

And:

Passage Of Trump's Gargantuan Tax Cuts Bill Means Potential Medicare Cuts For The First Time (Can You Say 'Sequestration'?)

And:

A Skewed Economic System (Weighted for the Wealthy) Explains Why Gen Z Is Embracing "Financial Nihilism"

And:

Corporate America

Nothing Is Ours Anymore: Is the "Rental Economy" the Most Destructive Scam in Modern American Capitalism?

by Thom Hartmann | December 30, 2025 - 6:18am | permalink

— from The Hartmann Report

And:

Lower Living Standards For Generations To Come? Thank Global Monetary Policy

Fermat's Principle Of Least Time: Feynman's Intro & A Proof From Classical Mechanics

Fermat's principle is an intriguing one in physics, first pointed out by Feynman in his lectures.  Therein he noted Fermat discovered this principle in 1650, basically given all possible paths that light might take to get from one point to another, the one it actually takes requires the shortest time.  

In Sec. 26-3 of his Feynman Lectures, Richard Feynman uses a plane mirror and states Fermat's law includes both 'the law of straight line propagation' and 'the law for the mirror'.  He uses the diagram shown below for reference with MM' the mirror surface and asks for the way to get from A to B  - in the shortest time - but the ray striking the mirror first.

The evident point needed for the shortest time is C, but how to get it? Feynman proposes a 'geometrical trick' - to wit, constructing an 'artificial point' B' on the other side of MM' which is the same distance below MM' as point B is above it. Since BFM is a right angle and BF = FB', then EB is equal to EB'.

Then the sum of the two distances: AE + EB is proportional to the time it will take, assuming a constant value c (speed of light). Which is also the sum of the two lengths AE + EB'.  The problem is then easily re-phrased as: When is the sum of the two lengths the least? The answer, by inspection, is when the reflection line goes through point C as a straight line from A to B'.  C works because it is the best choice to get to the artificial point B' in the most direct manner from A.  To break it down: if ACB' is a straight line then the angle BCF is equal to the angle B'CF and thence to angle ACM. So saying the angle of incidence is equal to the angle of reflection is equivalent to saying the light travels to the mirror in such a way that it comes back to point B' in the least possible time.

 

In a more modern context, given Fermat's least time principle also applies in classical mechanics, it can be proven using a more quantitative method from calculus. We use the diagram below:

                                  Sketch for mathematical analysis of situation

The purpose here is to demonstrate that for a ray going from a to b  the angle of incidence will equal the angle of reflection and hence be in accordance with the principle of least time.  Consider then, from the diagram, a path such that the time from a to b is a minimum (i.e. x1,a to xo,o back to x2,b), then we have;

t₁ =  ò _{x1, a} ^xo,0     ds/  C     (1st integral for path 1) 

where:   ds   =   Ö (dx² + dy ² )  =    dx Ö [ 1  +  (dy/dx) ² ]

Let:   f  =   1  +  (dy/dx) ²     

Þ

f  -  y'   ¶ f / ¶ y'    =  const.

t₁ =   1/ c ò _{x1, a} ^xo,0     dx f

Therefore, 

Ö  1  -  y ²    =   y'   (½ )   [y' / Ö (1  +  y'  ²  ) ]  =  const.  = A

Þ

1  +  y'  ²    - 2 y' ²  =   A ((1  +  y'  ²  )  (½ )

1  +  y' ²    =   1/  A²  =  B

But:   y'   =  dy/dx  =   Ö B  -1 

 Þ

y   =   (Ö B  -1 )  x  + C      (straight line)

Then:   

 For x 1,a  to   x 0,0  to  x1, b  :

t₁ +   t₂    =   

1/ C  [(x 1   - x 0 )²  +   a ²] ^1/2  +  [(x 0   - x 2 )²  +   b ²] - ^1/2  (2 (x 0   - x 2 )

For minimum:  

¶ ( t₁ + t₂ ) / ¶ x 0    =  0

Þ

 ½ [(x 1   - x 0 )²  +   a ²] ^-1/2   2 (x 1   - x 0) (-1) 

+ ½ [(x 0   - x 2 )²  +   b ²] ^{- 1/2}  (2 (x 0   - x 2 )   =  0   

Þ

(x 1   - x 0 )²/ (x 1   - x 0 )²  +   a ²   =  (x 0   - x 2 )² / (x 0   - x 2 )²  +   b ²

Therefore:   cos² q 1   =    cos² q 2 

q 1   =    q 2      I.e.

q i   =    q r 

The statement of the law of reflection that the angle of incidence ( q i) is equal to the angle of reflection (q r ) is equivalent to the statement that light travels (via the mirror)  from point  x 1,a  to  x1, b    in  the least possible time.   Hence, the principle of least time or Fermat's principle.