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# Special Relativity | Lecture 4

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I'm Steiger University today
Oilers Lagrange equations led you work aren't of course this object and he would be called or grungy Granzien em all over to fight back squared my history of fighting and the equations of given by d by DT of the derivative of a Lagrangian with respect to fight not equals derivative 0 grungy with respect to 5 this is the 4th arrived and signed and this will be may at times acceleration acceleration double 2nd Prime derivative of we worked out you just get Newton's equations here OK tho only 2 or write it down and double docked equals minus DVD fight it's no use equations and that's the net upshot albeit just 1 more . 1 more point that's worth pointing out armed action principle or the order Lagrange equations of the solution to the problem of finding the trajectory with the minimum action on so I'm going to use the term minimum but you know very well that I mean stationery could be maximum as code the minimum action on a trajectory from year to year holding these . 6 of a point a N . be fixed find the trajectory of least action that connects them that's a course the order Lagrangian equations and the way that I like to think about it but we worked this out last time is divided climaxes while pieces and a set of thinking of the Lagrangian or instead of thinking of the action as an integral just think of it as a somewhat there a those terms depend on they depend on fly at each time another toddler action is simply a function of a lot of values of 5 if some functions of all whole bunch of values of 5 How do you minimize a functional fight you do you differentiate with respect the fact that the order Lagrange equations are no other way to say it is the solution of the problem of molding these points around until you find the trajectory which which minimize action think so that's nonrelativistic misses 902 because of the relativistic particles theory the nonrelativistic description of particles In a force field now let's come back over a year he advised the a field which is a function of time and base space Kamoke Sunday when our world there are 3 but we can consider world and dimensions of space 9 dimensions is based its even interesting to think a world tour dimensions of space that would be used stable things move around the tabletop at a world of 2 dimensions you could even think of a world With 1 dimension of space and time of course one-dimensional space could be be moving up and down on a wire particles I'd be like particles moving up and down our a wire but you could think of Zero dimensional space is 0 dimensional space means a point but Zero but what will the corresponding space be nor have time and no space a magical world now with time and Nos base camp at a field in each case the field was a function of time coordinator and what every space coordinates that happened the beat if we don't they don't do this limit this strange limit with space has no dimensions of all but only time there were talking about a theory of a function love the photoconductor This is a fury of a function but only depends on current Suzie you could think of you could think of ordinarily particle mechanics as a field theory in a world which has no nobody mentions of space only a dimension of time instead of thinking of the feel instead of thinking of fine now mathematics but a different interpretation instead of thinking of fly as the position of a particle think of it as a field which depend only on time calling field just the name you could call it the coordinator something you could cause a field in quote whatever you want but it's just a function of time around the call on the to think of it now as we feel that in a world with 0 space dimension so it is essential that it cases or not it could still depend not that would be of turned me energy which would depend on the value of fine you could imagine a field and you couldn't engine that the potential energy in the field depends on fire the cost energy to change life and that remind you could remember that flies stood for the will be positioned the particle of light but the sale is a potential energy just means that if you move this curve up and down the energy changes absorb With where of course are not going to think of funny as the position of the particle we are going to think of it as a field then will go any other direction with generalized from this pattern here this was a field theory In a world with no spaces and one-time River generalized field theory time direction and a collection of space caught the field that was not just a function of time it's a function of space and time 5 x unauthorized again areas conceptually editor of pungent state all are regional add overstates the you could it wouldn't matter it's very hard to visualize a world without coordinates so think about as a world without coordinates in which nothing depends on position only on part far right of quite so now let's try to guess what the principles Of mechanics the principles of mechanics but the principles of field theory are knowing what the principles of field theory are In a world with only 1 type a best guess my best guess somebody's best guess is that the world should be governed by a principle of least action least action was a very powerful principle and all of ordinary mechanics not just of 1 particle could step 2 particles and particles of lead particles and gas whatever it was a very powerful principle which encoded in Sunrise huge numbers of laws of physics and we're going to take it has a mouse and man of giving so that feel theories that the whole world is described by an action principle I know what is action principle look like now the world world is
really kiddie we've learned a sort of think of the 4 coordinates are common plane has been all parts a space time and just give them a name or for coordinates call them X knew where you runs over 4 values kind 3 components space so it's frequently but frequently standard right integral all all 4 quarter myths of space is just will be 4 x Of course if we were dealing with a fury in Firebrick tool 1 they mentioned we would replace this body to XT for XT Rex while XP it was only 1 dimension would be done right so before Forex includes DT but also includes the suspects you could just think of it as DXT Desi and then some look around here of something as a shared which that is just to be integrated over time be integrated office space any time so often called the Lagrange density but stressed the analog of Lagrangian he and what does it depend on well of course exposed to build a grand jury in the field funny to tell you how far be so 1 of the things it certainly do depend on the side that's a catalog of saying Lagrangian wouldn't say over a year but that's the order on a lot of saying ground geology of depends on depends on position it may mean that the composition of course but it also depends on the derivatives of flying in this case with respect to time if time and space on the same footing in relativity we might expect that the Lagrangian depends not only on the kind derivative of fighting which put in never be fired by duty a lot of it's going to depend on defy by D. C and with dealing in the a theory of relativity with time and space are pretty much the same or very closely related footing we might expect it also depends on the derivatives of 5 with respect 2 X with respect to Why for harsh but Tuesday and however many more coordinates the Art Basel transient programs is a function of the fields 580 maybe more than 1 field incidentally just as In this situation over here a particle may have more than 1 coordinate locating its position the XYZ so for God and maybe more than 1 field but for the moment let's just concentrate our problem with 1 field called 5 programs in depends on the field and its various derivatives the effects are not just as the Lagrangian Mike depend on T over here it might now our throw closed system ordinarily system which has an energy conservation it means that the Lagrangian doesn't depend explicitly on time and the same will be true here would not depend explicitly on time or space but it got a voted out here 5 or a general at the moment at the moment that have so far I haven't said anything that makes a scalar field other than the fact that I only wrote 1 component vector feel there would have to have some more components but it's a scalar field and would only have 1 component a yes I certainly have the back of my mind here a scalar field but up till now it does not mean it could be anything I'd so it's so let's some let's condensed notation let's just write this as a function refined and the collection of derivatives D 5 IDX mural of which there are several time derivatives and space we could even convince the notation a little bit more time derivative we would use 5 . 0 used the space derivatives well remains convinced locations not to use this the ice condenser patient Mr. Right 5 he Pfizer X Y Y Y Z and these things mean by definition derivative of fire with respect to the river of fire service would be 55 IDT should be defined by the X and so forth as moral standard notation party but will tear an extra orders each year we could afford light right find some new meaning derivative of fighting with respect text but I will leave the waiters 5 IPX mail but such a problem problem is the to solve the problem of fixing the field on the boundary and wiggling around your we're going around until we find the configuration and minimizes this action if we replace based in time by agreed like there's obsessing about how we do it practice we would replace the space time by the a grid of points we would replace the Lagrangian it would depend on fire fired each point but what will really replaced 5 by IDX with good different those differences and the process we would actually write this action as a function on the values of society at each grid point who just iPhone to be a function of a lot of variables and many of them attend this way in this way BAT variables there are some who have a lot of variables finally make the lattice moreover would just be the action would just be a function of some collection of variables and how we minimize it we would minimize it by differentiating the function with respect to the variables and setting it equal to 0 exactly the same operational we do know here doing that operations and its senior notes in your classical mechanical in the classical mechanics notes we went from that principle the Maori grunge equation he or I will grant equations written in this form year minus theirs equal 0 is basically the equation which says differentiate the action with respect finally Ed each point considered equal to 0 this was going took a few steps work it out the question is How would that change every have multi-dimensional analog of exactly the same thing multi-dimensional iron ore on the same thing in a answer is it would change tho change in a rather recognizable way and I'm not going to work it out With right down you what the equations are which are the equivalent of saying that the the river the action with the valve with respect to fly in each good point is equal to 0 so we would do the corresponding or Lagrange equations all right the largest show you with the correspondence is well let's let's let's look outlets what's rewrite the ordinary Iran's equations over here and no changes originally was deemed IDT off partial of El with respect 2 0 0 before I buy DT with different Schuett the French Lagrangian with respect if I D T minus the director Grand and with respect
over squared of one-liners squared Prime because y pride Jose things which transformed in this way complexes of 4 quantities of time component of X component a component is component things which transformed this way of called for vectors so this it is also true for for the differential displacement With the differential displacement might be a differential displacement along a trajectory also the transformation property for the Australia you divide by deep How it tells you how the 4 velocity components transform anything which transforms this way is a format that I didn't mean a generic name for records on I will use vie for vector but these also already velocity used a but a stands for erode the vector potential used but stands for the magnetic field seized speed of light these derivatives he is energy have force I've run medical letters some will stick with age 80 80 a Arab how good Arab and later on they will have a dubious back so a floor has 4 components call at the time component but used to calling it a naught a Texas a wide easy and the transformation properties of these things are in my coordinate system I call on Prime that equal to the values in your coordinate system X my nest vii 18 naught divided by squares of 1 might be squared notice a not the same thing as the kind components a X minus V 8 wherever this 1 but that correctly not be did not a not minus X and this 1 is a X mines vii on my squared this 1 is right variable primed primed uh but but but an example of field which obviously transforms we can give you an example 1 that that that makes clear why the transforms gap imagine a fluid filling space time emerge a fluid filling space times every point in the fluid there was a need for velocity ordinary velocity for the fluid points but also for velocity and the 4 velocity could be called you knew Willoughby be a position of a position dependent thing if the flow was flowing the velocity might be different in different places before a lot this is you let before velocity could be thought of as a field before velocity of the fluid in space time could be thought of as a field instances of poor velocity it's a fought back there it would transform exactly this way except we recall you wins over and we got you would have a value of you'll in your frame would be different than the value of you in my framed the components and they would be related in this manner here but there are lots of other things which could be for vectors I want to tell you and your right now broke of 4 vectors is a possible kind of fields of 4 that there's are aII new also of and take a transformer in this manner here Twitter you could make galas out of full vector if you take the form of components of a force that you could make a scalar but if you take any tool for vectors to think In tool for vectors you can make a scalar out of them but let's just stick with 1 for you could make a scalar rather at the same exact way that we made a scalar of DX mu because now is equal to cow squared it is equal to D square above or below minus the exclamations 2 white women square in the same way every 4 vector could be used to make a scalar and it's just a naught squared minus 80 x squared say why square soulful that's a scalar exactly the same Rees wants to know that the eighties transformed saying ways X X and take them by the same little bit of algebra you could see that the difference of the squares kind component of space component will not change under Lorentz transformation St. little bit of algebra the see squared he squared minus X square my national minus square half wants to know that little bit of algebra the to riders How squares squared minus 6 square miles west square Manzi square for the same reason that this doesn't change in the Lorentz transformation neither so we have now constructed a scalar difference of the squares of recurrent components in space components last point was point is that given a scalar you could construct a 4th sector and not by just looking at it but by differentiating by differentiating it with respect for the 4 components of space and time goes for derivatives form of for I saw special case over 4 vector given the scale of collection of components the derivative of flying With respect X mule the review of a fire with respect to the river 5 with respect X through the fires but for wine and so forth these form of 4 vector not surprisingly not surprisingly these forms a before vector and so now we have a set of tools for creating Lagrangian awoke yes we know how things transformed and we know how to build a scalar and vectors our other objects What's the rules for making a Lagrangian on action invariant is very simple we required the goal of grungy itself is the same in every quarter but
Groningen the thing that we cannot always little cells is the same in every quarter that means a scalar but as means the scale of the Lagrangian should be a scalar so that's oral the Lagrangian should be a scalar under Lorentz transformations of squalor gala and that's all there is to it you take your field which happens to be fine in this case you consider old possible Steelers but you couldn't make out of it and they are possible candidates to put into a grungy and so let's what start with somebody running out a blackboard space Igor only discerning will work to make well fire itself as a scalar but now his fire scalar but any function of fires a scalar Everybody agrees on the value of 5 they will also agree on the value of squint fight fueled the hyperbolic co-sign the no functions of flight will also be scale so any function functional of is also scalar and therefore it's a candidate for something that you could around Jim well we've already done that here that next Wells could you put it well we certainly want to put in derivatives of the field if we don't put rivers of fewer fuel theories to be very very trivial it would be very interesting where to put the rivers and but we have to put the derivatives Ian in a way which constitutes a scale so what can we do what's easy we just thought with vectors build a 5 X X meal that's a erect and we simply construct the invariant scalar Out of the components of the vector sewers at me in practice it means 5 IDT squared but my AST 5 IDX squared but dynasty fiber you don't get too tired of writing a column I wish I had us shorthand notation for now an example passing law passed in 2nd derivative this is the 1st to apply RCN is like an acceleration of passing and uses a different thing this is the sum of the squares on the difference of the squares so there's something Yukon grew from around and you can be sure that when you do the equations that you get out of the same in every reference frame just because quantity is is the same in every reference in other words it equal to corresponding things in another Frank served years other other things you go right there we were messes none of the church
message the parameter it's not a dynamical thing you could say we put the Nets in their birth their mess of what would mess it will put numbers yarn numbers a perfectly good things political of we did that ready reserve booklets they via fight to be new squared over 2 5 square because the number so yes to corporate numbers and just numerical constants but something else you could do this would stop nickel complicated story but became this which is a scalar and you could multiply by another scalar multiplying Scalia's together gives galas multiply 2 things each of which is the same in every reference frame multiply still scale so it can multiply this by some function if we like a finite as a legal thing to do it starts the make rather complicated Lagrangian and when a considerate of the moment but certainly a reasonable thing Lorenzen variance thing to do we could do something even uglier this would this we give very ugly we could take this scalar and squared and have high you do really have a higher powers of derivatives that begins very ugly in fact it's not the
things which appears and burn off Chris higher were driven decision-makers allegedly was still but but you will but not within the can't find Our rules of classical mechanics rules of classical mechanics you could have functions of Oakwood nets and kind derivatives so if you're staying within the boundaries Our standard classical physics on you want to stay we just 1st derivatives and higher powers of 1st reviews but only for France or once we do that a citywide this was water while I chose it the way I did the restore the units output oversee squared here and purely for notational reasons meanings purely on 4 conventions with poor 1 half from the whole thing is same 1 half goes from 1 here FMV squared you don't have to do it warned he FMV squares you could have taken the liberty in the B and B and squared Of course the man's would differed from masked by a factor of tool that's OK you could say it's a convention to conventions that f equals and not articles twice and that's a convention so this 1 half as purely a conventions no content in it will want to fix it you fixed it so our examples example that that will focus on it will be the story and a over again over here From a believer I would see a Leivasy there for the moment minors but let's take New Square or 0 why squared that's simplest field theory fact even simpler if we don't have Mr. Freeling it's a very simple feel theory that a wave equation gives rise to wave equation written down with right the equation we directly equation but lost the are it's right down for the wave equation derives from this is it is 1 overseas squared d 2nd IDT Square the site acceleration and then as analogous terms to each 1 of the the X Y and Z with a minus sign he said confined by X squared plus Y squared Posey squared In the from here we get plus new squared 580 equals 0 analysis said as a wave equation it's a particularly simple wave equation because linear ruin the means it means that the field only if the field
just do your of branch equation motion with just 2 they're ugly and pretty yes I'm sorry but should be in need absolutely was at a message functional reacts that's correct muscle have to be just a function of X this represents the position of the particle at yes it should be it should be an integral best we need to go because even know fight Mike did not depend on T TV X depends on of fire x depends on of next to him to decide the integral that the 1st all notice something interesting America's Sapulpa for some reason the field like Tom migrate some value for some reason the field like toast seek some staple a stable value which wasn't 0 I like to get frozen for some reason away from 0 constant completely constant firewood just be a constant approximately a constant the motion of a particle would be absolutely identical to a particle whose and they asked I was In the minus 5 who would be exactly the same as if the particle man was in the minors flight when miners worked miners and past 5 of romance would be and minors why this is the simplest example of how a scalar field can give rise to a shift of the article what I said those meetings and began what fielders most likely to nature Higgs field is a thing which is most likely scalar field nature and you could see how what possible that if the Higgs field did shifted by some mechanism its value shifted in turn it will shift the value of the massive particles in fact if the mess of some particle naturally began as 0 but it was coupled this field and this feel fine naturally sought some particular value beat particle would effectively have amassed which would be related to the so this is not exactly the Higgs mechanism but its closely connected with it and it's what the roughly what made when people say the Higgs field gives the particle maps Higgs field enters into the equations as if it were part of the a man but that starts but just the equation of motion would be and then maybe will try to take the nonrelativistic limit in order get done this for years and now but still it but strife OK so here is the Lagrangian and find that was just thought of as a function of X X function of the position had a pot of tea but so the 1st thing to calculate is a derivative Lola grungy partial the river Legrand June with respect to X . 4 X . right here was that that is In my mind this Friday Rexon tee times X Dr. although the square root of 1 my X . squared I'm doing this as if there was only 1 direction space will make it simple will make it a problem movie in which a particle moves of only 1 direction then really X direction which give I get this from well if you remember the last time we calculated the momentum by differentiating Lagrangian with respect next what we get we got we got the Mass cancel velocity over square of what might be squared nonetheless that was a momentum was equal to the derivative around June with respect of velocity differentiate the Lagrangian with respect the velocity will get exactly the same kind of thing except every place she saw and it will not be in this fight we did do it explicitly to differentiate this with respect to a component of the velocity we differentiate the square root we we multiply by this this is what you get is not in times of velocity but in minus 5 sets a 1st step was the next step we differentiate this with respect to of next EPA's differentiated suspected time is deemed IDT Of and minus 5 8 X . over square root of 1 minus X . squared but still it's 1 then what's the other terminal grand equations Our Lagrange equations he is a son of your after differentiated Lewin right what see general my dynasty OBX those X appear in the yes it does it's in fiery fright depends on Exon take sorrow at equal to 0 direct minus derivative of 580 With respect to X the Red Dress poses a partial live my partial derivative of firewood acts it find of Suez Lagrangian yacht would differentiating Lagrangian with respect acts he is the only place that X appears but this thing multiplies Square Square restore Hey square root 1 minus X . square but such that's the equation of motion of differential equation the left-hand side has something which is more complicated than ordinary acceleration the Debye IDT has all kinds of terms are a nuisance to work out it's not traders but this is it but there our hopes worker arts that's worked through a little bit of it OK so 1st of all we get I wonder if we figure out where the speed of light go now From didn't do that can do a lot of thought only a small us that Bush that were a little Granzien have finally Kansas square root of minus textile square that was in around right was with my assigned there was the plus sign was minus any costs and where differentiated fire with respect to x still contain the factor at that's their arms but this is not pretty this is not a it's nice the wind-up electoral lecture with a nice beautiful elegant result this has not happened in this case I know that word are but let's just she works here 1st of all the Debye he
hit this thing over here now this thing over here is what would have been in the canonical momentum had this turned well away it is what it is His work is term which is Debye dt on X . all of the square root of 1 mind up squared itself was not very pretty somebody by DT hits this 3rd times and minus 5 a derivative acting on this factor fire and then another terms which is plus we have to take a derivative of T of minors fight and is constant and as a constant derivative of the specter of minus 5 batters minors but 1st right was 5 . times X . over square of 1 mind 6 . glared from differentiating Miers and that's still equal to arrive he inside 5 IDX would get a minus sign square 1 minus 6 . square I bought off I got what I do if I got exercise because of space is also the 1st partial success as a simple case were finally depends on X just be simple finally depends on justice simplified if then it was 5 . equal to 0 now is now 0 because it depends on X an X is not constant price August 5 . fi . and physical to derivative fire with respect to x times X . bird so this is going going to be from here defined IDX and the annexed up square here among make it any simple vendors so says that river monstrosity on I'll leave you the exercise of putting the speed of light into it was a point but to speed of light into it and then study it when the velocity is much slower than the speed of light and see if you can recognize India many familiar anything familiar about she recognize anything familiar about are it will be it should be familiar in that case is to put the speed of light and and then take the limit in which the velocity is much lower than the speed of light the authority of the Arab world reaffirmed for more please visit us at Stanford .
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Klassische Elektronentheorie
Magnetisches Dipolmoment
Masse <Physik>
Mechanik
Sturm
Staubmessung
Dreidimensionale Integration
Bildqualität
Begrenzerschaltung
Minute
Fahrgeschwindigkeit
Windgeschwindigkeit
Array
Eisenkern
Zugangsnetz
Elektronisches Bauelement
Weltall
Kardieren
Elementarteilchenphysik
Schiffsklassifikation
Elementarteilchen
Weiß
Gleichstrom
Elektromagnetische Welle
Scheinbare Helligkeit
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Magnetspule
Ringgeflecht
Ersatzteil
Telefax
Temperatur
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Schaft <Waffe>
Intervall
Mechanikerin
Treiberschaltung
A6M Zero-Sen
Satz <Drucktechnik>
Abformung
Leitungstheorie
Zelle <Mikroelektronik>
Sonntag
Abwrackwerft
Kaltumformen
Licht
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Elektronenschale
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Angeregtes Atom
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Motor
Eisendraht
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Jahr
Lineal
Analogsignal
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Schreibware
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Schnittmuster
Hobel
Begrenzerschaltung
Blei-209
Gasturbine
Speckle-Interferometrie
Energielücke
Tagesanbruch
Waffentechnik
Entfernung
Tag
Photowiderstand
Rauschzahl
Strangeness
Gas
Druckkraft
Trajektorie <Meteorologie>
Elementarteilchen
Gleichstrom
Knopf
Ersatzteil
Unterwasserfahrzeug
Erder
Magnetisches Dipolmoment
Mechanik
Hobel
Dienstag
Kondensor
Vakuumröhre
Feldeffekttransistor
Computeranimation
Sternkatalog
Prozessleittechnik
Trenntechnik
Eis
Gasturbine
Fuß <Maßeinheit>
Klangeffekt
Array
Tastverhältnis
Kaltumformen
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Elektronisches Bauelement
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Wirtsgitter
Energieeinsparung
Elektronenschale
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Steckdose
Werkzeug
Elementarteilchen
Jahr
Ersatzteil
Ringgeflecht
Analogsignal
Drehen
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Direkte Messung
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Lichtgeschwindigkeit
Besprechung/Interview
Auslenkung
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Minute
Zelle <Mikroelektronik>
Kapazität
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Magnetfeld
Schall
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Flüssiger Brennstoff
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Material
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Greiffinger
Anzeige <Technik>
Erwärmung <Meteorologie>
Mechanikerin
Lichtgeschwindigkeit
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Satz <Drucktechnik>
Computeranimation
Juni
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Erdefunkstelle
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Kompendium <Photographie>
Umlaufzeit
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Unterbrechungsfreie Stromversorgung
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Trajektorie <Meteorologie>
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Walken <Textilveredelung>
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Minute
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Steckverbinder
Potentiometer
Windrose
Waffentechnik
D.H. 98
Rotverschiebung
Trajektorie <Meteorologie>
Schiffsklassifikation
Elementarteilchen
Gleichstrom
Magnetspule
Ersatzteil
Kleiderstoff
Homogene Turbulenz
Kaltumformen
Licht
Lichtgeschwindigkeit
Gasturbine
Fahrgeschwindigkeit
Begrenzerschaltung
August
Rootsgebläse
Übungsmunition
Computeranimation