Mah 225-4 Fri Apr 4 7 Sysems of differenial equaions - o model muli-componen sysems via comparmenal analysis: hp://enwikipediaorg/wiki/muli-comparmen_model Here's a relaively simple 2-ank problem o illusrae he ideas: Exercise ) Find differenial equaions for solue amouns, above, using inpu-oupu modeling Assume solue concenraion is uniform in each ank If = b, = b 2, wrie down he iniial value problem ha you expec would have a unique soluion answer (in marix-vecor form): 4 2 = 4 2 = b b 2
Geomeric inerpreaion of firs order sysems of differenial equaions The example on page is a special case of he general iniial value problem for a firs order sysem of differenial equaions: x = F, x x = x We will see how any single differenial equaion (of any order), or any sysem of differenial equaions (of any order) is equivalen o a larger firs order sysem of differenial equaions And we will discuss how he naural iniial value problems correspond Why we expec IVP's for firs order sysems of DE's o have unique soluions x : From eiher a mulivariable calculus course, or from physics, recall he geomeric/physical inerpreaion of x as he angen/velociy vecor o he parameric curve of poins wih posiion vecor x, as varies This picure should remind you of he discussion, bu ask quesions if his is new o you: Analyically, he reason ha he vecor of derivaives x compued componen by componen is acually a limi of scaled secan vecors (and herefore a angen/velociy vecor) is: x x x lim : : x n x n = lim : x n x n provided each componen funcion is differeniable Therefore, he reason you expec a unique soluion o he IVP for a firs order sysem is ha you know where you sar (x = x ), and you know your "velociy" vecor (depending on ime and curren locaion) you expec a unique soluion! (Plus, you could use somehing like a vecor version of Euler's mehod or he Runge-Kua mehod o approximae i! You jus conver he scalar quaniies in he code ino vecor quaniies And his is wha numerical solvers do) = x n :,
Exercise 2) Reurn o he page ank example = 4 2 = 4 2 = 9 = 2a) Inerpre he parameric soluion curve, T o his IVP, as indicaed in he pplane screen sho below ("pplane" is he siser program o "dfield", ha we were using in Chapers -2) Noice how i follows he "velociy" vecor field (which is ime-independen in his example), and how he "paricle moion" locaion, T is acually he vecor of solue amouns in each ank, a ime If your sysem involved en coupled anks raher han wo, hen his "paricle" is moving around in 2b) Wha are he apparen limiing solue amouns in each ank? 2c) How could your smar-alec younger sibling have old you he answer o 2b wihou considering any differenial equaions or "velociy vecor fields" a all?
Firs order sysems of differenial equaions of he form x = A x are called linear homogeneous sysems of DE's (Think of rewriing he sysem as x A x = in analogy wih how we wroe linear scalar differenial equaions) Then he inhomogeneous sysem of firs order DE's would be wrien as x A x = f or x = A x f Noice ha he operaor on vecor-valued funcions x defined by L x x A x is linear, ie L x y = L x L y L c x = c L x SO! The space of soluions o he homogeneous firs order sysem of differenial equaions x A x = is a subspace AND he general soluion o he inhomogeneous sysem x A x = f will be of he form x = x P x H where x P is any single paricular soluion and x H is he general homogeneous soluion Exercise 3) In he case ha A is a consan marix (ie enries don' depend on ), consider he homogeneous problem x = A x Look for soluions of he form x = e v, where v is a consan vecor Show ha x = e v solves he homogeneous DE sysem if and only if v is an eigenvecor of A, wih eigenvalue, ie A v = v Hin: In order for such an x o solve he DE i mus be rue ha and Se hese wo expressions equal x = e v A x = A e v = e A v
Exercise 4) Use he idea of Exercise 3 o solve he iniial value problem of Exercise 2!! Compare your soluion x o he parameric curve drawn by pplane, ha we looked a a couple of pages back Exercise 5) Lessons learned from ank example: Wha condiion on he marix A n n will allow you o uniquely solve every iniial value problem x = A x x = x n using he mehod in Exercise 3-4? Hin: Chaper 6 (If ha condiion fails here are oher ways o find he unique soluions)
Mah 225-4 Week 4 April 7-2: secions 7-73 firs order sysems of linear differenial equaions; 74 mass-spring sysems Mon Apr 7 Coninue discussing sysems of differenial equaions, 7-73 Review exercises 3-5 in las Friday's noes, which foreshadow he eigenvalue-eigenvecor mehod for solving homogeneous firs order sysems of differenial equaion IVPs x = A x x = x ha he book covers in deail in secion 73 Then discuss he imporan fac ha every n h order differenial equaion or sysem of differenial equaions is acually equivalen o a (possibly quie large) sysem of firs order differenial equaions This discussion is in oday's noes and is he conen of secion 7 The exisence-uniqueness heorem for firs order sysems of differenial equaions will hen explain why he naural iniial value problems for higher order differenial equaions, ha we sudied in Chaper 5, also always have unique soluions Anoher mysery ha his correspondence will solve is why we used "characerisic polynomial" when looking for basis soluions e r o n h order consan coefficien homogeneous differenial equaions and hen used exacly he same erminology, "characerisic polynomial", when we were looking for marix eigenvalues (see secion 73) Along he way and omorrow as well, we will also be discussing how he heory and emplae for finding soluions o firs order sysems of linear differenial equaions precisely mirrors he emplae we developed for single n h -order linear differenial equaions in Chaper 5 This is he conen of secion 72
Convering higher order DE's or sysems of DE's o equivalen firs order sysems of DEs: Example: Consider his configuraion of wo coupled masses and springs: Exercise ) Use Newon's second law o derive a sysem of wo second order differenial equaions for,, he displacemens of he respecive masses from he equilibrium configuraion Wha iniial value problem do you expec yields unique soluions in his case?
Exercise 2) Consider he IVP from Exercise, wih he special values m = 2, m 2 = ; k = 4, k 2 = 2; F = 4 sin 3 : = 3 = 2 2 4 sin 3 = b, = b 2 = c, = c 2 2a) Show ha if, solve he IVP above, and if we define v v 2 hen,, v, v 2 solve he firs order sysem IVP = v = v 2 v = 3 v 2 = 2 2 4 sin 3 = b v = b 2 = c v 2 = c 2 2b) Conversely, show ha if,, v, v 2 solve he IVP of four firs order DE's, hen, solve he original IVP for wo second order DE's
I is always he case ha he naural iniial value problems for single or sysems of differenial equaions are equivalen o an iniial value problem for a larger sysem of firs order differenial equaions, as in he previous example A special case of his fac is ha if you have an IVP for single n h order DE for x, i is equivalen o an IVP for a sysem of n firs-order DE's, in which he funcions are = x, x, x 3 x,, x n x n For example, consider his second order underdamped IVP for x : x 5 x = x = 4 x = 4 Exercise 3) 3a) Conver his single second order IVP ino an equivalen firs order sysem IVP for x and x 3b) Solve he second order IVP in order o deduce a soluion o he firs order IVP Use Chaper 5 mehods even hough you love Laplace ransform more 3c) How does he Chaper 5 "characerisic polynomial" in 3b compare wih he Chaper 6 (eigenvalue) "characerisic polynomial" for he firs order sysem marix in 3a? hmmm 3d) Is your analyic soluion x, v in 3b consisen wih he parameric curve shown on he nex page, in a "pplane" screensho? The picure is called a "phase porrai" for posiion and velociy
If you've been using Wolfram alpha o solve second order differenial equaions you migh have noiced picures ha look jus like he one above, even hough you migh no have hough abou hem look a he plo a he lower righ corner of his screensho!
Theorem For he IVP x = F, x x = x If F, x is coninuous in he variable and differeniable in is x variable, hen here exiss a unique soluion o he IVP, a leas on some (possibly shor) ime inerval Theorem 2 For he special case of he firs order linear sysem of differenial equaions IVP x = A x f x = x If he marix A and he vecor funcion f are coninuous on an open inerval I conaining hen a soluion x exiss and is unique, on he enire inerval Remark: The soluions o hese sysems of DE's may be approximaed numerically using vecorized versions of Euler's mehod and he Runge Kua mehod The ideas are exacly he same as hey were for scalar equaions, excep ha hey now use vecors This is how commerical numerical DE solvers work For example, wih ime-sep h he Euler loop would incremen as follows: j = h j x j = x j h F j, x j Remark: These heorems are he rue explanaion for why he n h -order linear DE IVPs in Chaper 5 always have unique soluions - because each n h order linear DE IVP is equivalen o an IVP for a firs order sysem of n linear DE's In fac, when sofware finds numerical approximaions for soluions o higher order (linear or non-linear) DE IVPs ha can' be found by he echniques of Chaper 5 or oher mahemaical formulas, i works by convering hese IVPs o he equivalen firs order sysem IVPs, and uses algorihms like Euler and Runge-Kua o approximae he soluions
Theorem 3) Vecor space heory for firs order sysems of linear DEs (Noice he familiar hemeswe can compleely undersand hese facs if we ake he inuiively reasonable exisence-uniqueness Theorem 2 as fac) 3) For vecor funcions x differeniable on an inerval, he operaor L x x A x is linear, ie L x z = L x L z L c x = c L x check! 32) Thus, by he fundamenal heorem for linear ransformaions, he general soluion o he nonhomogeneous linear problem x A x = f I is x = x p x H where x p is any single paricular soluion and x H is he general soluion o he homogeneous problem x A x = We frequenly wrie he homogeneous linear sysem of DE's as x = A x
33) For A n n and x n he soluion space on he inerval I o he homogeneous problem x = A x is n-dimensional Here's why: Le X, X 2,X n be any n soluions o he homogeneous problem chosen so ha he Wronskian marix a I W X, X 2,, X n X X 2 X n is inverible (By he exisence heorem we can choose soluions for any collecion of iniial vecors - so for example, in heory we could pick he marix above o acually equal he ideniy marix In pracice we'll be happy wih any inverible marix ) Then for any b n he IVP x = A x x = b has soluion x = c X c 2 X 2 c n X n where he linear combinaion coefficiens are he soluion o he Wronskian marix equaion c b X X 2 X n c 2 : = b 2 : c n b n Thus, because he Wronskian marix a is inverible, every IVP can be solved wih a linear combinaion of X, X 2,X n, and since each IVP has only one soluion, X, X 2,X n span he soluion space The same marix equaion shows ha he only linear combinaion ha yields he zero funcion (which has iniial vecor b = ) is he one wih c = Thus X, X 2,X n are also linearly independen Therefore hey are a basis for he soluion space, and heir number n is he dimension of he soluion space
73 Eigenvecor mehod for finding he general soluion o he homogeneous consan marix firs order sysem of differenial equaions x = A x Here's how: We look for a basis of soluions x = e v, where v is a consan vecor Subsiuing his form of poenial soluion ino he sysem of DE's above yields he equaion e v = A e v = e A v Dividing boh sides of his equaion by he scalar funcion e gives he condiion v = A v We ge a soluion every ime v is an eigenvecor of A wih eigenvalue! If A is diagonalizable hen here is an n basis of eigenvecors v, v 2,v n and soluions X = e v, X 2 = e 2 v 2,, X n = e n v n which are a basis for he soluion space on he inerval I =, because he Wronskian marix a = is he inverible diagonalizing marix P = v v 2 v n ha we considered in Chaper 6 If A has complex number eigenvalues and eigenvecors i may sill be diagonalizable over n, and we will sill be able o exrac a basis of real vecor funcion soluions If A is no diagonalizable over n or over n he siuaion is more complicaed Exercise 4a) Use he mehod above o find he general homogeneous soluion o = 6 7 4b) Solve he IVP wih = 4
Exercise 5a) Wha second order overdamped iniial value problem is equivalen o he firs order sysem IVP on he previous page And wha is he soluion funcion o his IVP? 5b) Wha do you noice abou he Chaper 5 "Wronskian marix" for he second order DE in 4a, and he Chaper 7 "Wronskian marix" for he soluion o he equivalen firs order sysem? 5c) Since in he correspondence above, equals he mass velociy x = v, I've creaed he pplane phase porrai below using he leering x, v T raher han x, T Inepre he behavior of he overdamped mass-spring moion in erms of he pplane phase porrai 5d) How do he eigenvecors show up in he phase porrai, in erms of he direcion he origin is approached from as, and he direcion soluions came from (as )? hp://mahriceedu/~dfield/dfpphml
Mah 225-4 Tues Apr 8 7-73 Summary of wha is covered in Monday's and Tuesday's noes: Any iniial value problem for a differenial equaion or sysem of differenial equaions can be convered ino an equivalen iniial value problem for a sysem of firs order differenial equaions There is a "shor-ime" exisence-uniqueness heorem for firs order DE iniial value problems x = F, x x = x and soluions can be approximaed using Euler or Runge-Kua ype algorihms A special case of he IVP above is he one for a firs order linear sysem of differenial equaions : x = A x f x = x If he marix A and he vecor funcion f are coninuous on an open inerval I conaining hen a soluion x exiss and is unique, on he enire inerval The general soluion o an inhomogeneous linear sysem of DEs x = A x f ie x A x = f will be of he form x = x P homogeneous sysem x H where x P is a paricular soluion, and x H is he general soluion o he x = A x For A n n and x n he soluion space on he inerval I o he homogeneous problem is n-dimensional x = A x ie x A x = For A n n a consan marix, we ry o find a basis for he soluion space o x = A x consising of soluions of he form x = e v, where v is an eigenvecor of A wih eigenvalue We will succeed as long as A is diagonalizable Today: We will coninue using he eigenvalue-eigenvecor mehod for finding he general soluion o he homogeneous consan marix firs order sysem of differenial equaions x = A x ha we discussed yeserday, and is in Monday's noes Today we'll consider examples where he eigenvalues and eigenvecors are complex There is such an example in he homework due Wednesday problem w33
So far we've no considered he possibiliy of complex eigenvalues and eigenvecors Linear algebra heory works he same wih complex number scalars and vecors - one can alk abou complex vecor spaces, linear combinaions, span, linear independence, reduced row echelon form, deerminan, dimension, basis, ec Then he model space is n raher han n Definiion: v n (v ) is a complex eigenvecor of he marix A, wih eigenvalue if A v = v Jus as before, you find he possibly complex eigenvalues by finding he roos of he characerisic polynomial A I Then find he eigenspace bases by reducing he corresponding marix (using complex scalars in he elemenary row operaions) The bes way o see how o proceed in he case of complex eigenvalues/eigenvecors is o work an example There is a general discussion on he page afer his example ha we will refer o along he way: Glucose-insulin model (adaped from a discussion on page 339 of he ex "Linear Algebra wih Applicaions," by Oo Brescher) Le G be he excess glucose concenraion (mg of G per ml of blood, say) in someone's blood, a ime hours Excess means we are keeping rack of he difference beween curren and equilibrium ("fasing") concenraions Similarly, Le H be he excess insulin concenraion a ime hours When blood levels of glucose rise, say as food is digesed, he pancreas reacs by secreing insulin in order o uilize he glucose Researchers have developed mahemaical models for he glucose regulaory sysem Here is a simplified (linearized) version of one such model, wih paricular represenaive marix coefficiens I would be mean o apply beween meals, when no addiional glucose is being added o he sysem: G 4 G = H H Exercise 3a) Undersand why he signs of he marix enries are reasonable Now le's solve he iniial value problem, say righ afer a big meal, when G = H 3b) The firs sep is o ge he eigendaa of he marix Do his, and compare wih he Maple check on he nex page
> wih LinearAlgebra : 2 > A 5 ; Eigenvecors A ; 2 A := 5 5 I 5 I, 2 I 2 I () Noice ha Maple wries a capial I = 3c) Exrac a basis for he soluion space o his homogeneous sysem of differenial equaions from he eigenvecor informaion above: 3d) Solve he iniial value problem
Here are some picures o help undersand wha he model is predicing you could also consruc hese graphs using pplane () Plos of glucose vs insulin, a ime hours laer: > wih plos : > G exp cos 2 : H 5 exp sin 2 : plo plo G, = 3, color = green : plo2 plo H, = 3, color = brown : display plo, plo2, ile = `underdamped glucose-insulin ineracions` ; 6 underdamped glucose-insulin ineracions 2 3 2) A phase porrai of he glucose-insulin sysem: > pic fieldplo G 4 H, G H, G = 4, H = 5 4 : soln plo G, H, = 3, color = black : display pic, soln, ile = `Glucose vs Insulin phase porrai` ; H Glucose vs Insulin phase porrai 4 3 2 4 2 4 6 8 G The example we jus worked is linear, and is vasly simplified Bu mahemaicians, docors, bioengineers, pharmaciss, are very ineresed in (especially more realisic) problems like hese
Soluions o homogeneous linear sysems of DE's when marix has complex eigenvalues: x = A x Le A be a real number marix Le = a b i C v = i n saisfy A v = v, wih a, b,, n Then z = e v is a complex soluion o x = A x because z = e v and his is equal o A z = A e v = e A v = e v Bu if we wrie z in erms of is real and imaginary pars, z = x i y hen he equaliy z = A z x i y = A x i y = A x i A y Equaing he real and imaginary pars on each side yields x = A x y = A y ie he real and imaginary pars of he complex soluion are each real soluions If A i = a b i i hen i is sraighforward o check ha A i = a b i i Thus he complex conjugae eigenvalue yields he complex conjugae eigenvecor The corresponding complex soluion o he sysem of DEs e a i b i = x i y so yields he same wo real soluions (excep wih a sign change on he second one) Anoher way o undersand how we ge he wo real soluions is o ake he wo complex soluions z = x i y w = x i y and recover x, y as linear combinaions of hese homogeneous soluions: x = 2 z w y = 2 i z w
Mah 225-4 Wed Apr 9 Summary of Chaper 7 so far: Any iniial value problem for a differenial equaion or sysem of differenial equaions can be convered ino an equivalen iniial value problem for a sysem of firs order differenial equaions There is a "shor-ime" exisence-uniqueness heorem for firs order DE iniial value problems x = F, x x = x and soluions can be approximaed using Euler or Runge-Kua ype algorihms A special case of he IVP above is he one for a firs order linear sysem of differenial equaions : x = A x f x = x If he marix A and he vecor funcion f are coninuous on an open inerval I conaining hen a soluion x exiss and is unique, on he enire inerval The general soluion o an inhomogeneous linear sysem of DEs x = A x f ie x A x = f will be of he form x = x P homogeneous sysem x H where x P is a paricular soluion, and x H is he general soluion o he x = A x For A n n and x n he soluion space on he inerval I o he homogeneous problem x = A x ie x A x = is n-dimensional (This is also he "real" reason why he soluion spaces o nh order homogeneous linear DE's are n-dimensional, since hose DE's are equivalen o homogeneous syems of n firs order linear DE s) For A n n a consan marix, we ry o find a basis for he soluion space o x = A x consising of soluions of he form x = e v, where v is an eigenvecor of A wih eigenvalue We will succeed as long as A is diagonalizable Finish 73 Applicaions of firs order sysems of differenial equaions
Here are some of he deails from Tuesdday's discussion of he Glucose-insulin model G H = 4 Characerisic polynomial A I = 2 4 has roos = 2 i For he eigenvalue = 2 i we wish o solve he eigenvecor sysem A I v = : R 2 R, 5 R R 2 : 2 i 4 2 i 2 i i 2 Alhough i doesn' look like i, he second row is a muliple of he firs row, as i mus be: ir R 2 R 2 : 2 i So we may choose he eigenvecor v = 2 i, T This yields a complex funcion soluion z = e v = e 2 i 2 i If we rewrie z = x i y wih x, y real vecor funcions, hen each of x, y will be real soluions o he sysem of differenial equaions, and will in fac be a basis (See general discussion a end of Wednesday's noes) A he end of class we did his decomposiion of z ino real and imaginary pars: e 2 i 2 i = e 2 i cos 2 i sin 2 = e cos 2 i sin 2 2 i cos 2 i sin 2 This gives real soluions G H = e 2 sin 2 cos 2 i e x = e 2 sin 2, y = e 2 cos 2 cos 2 sin 2 and general homogeneous soluion using real funcions: x H = c x c 2 y = c e 2 sin 2 c cos 2 2 e 2 cos 2 sin 2 2 cos 2 cos 2
The Glucose-insulin example is linearized, and is vasly simplified Bu mahemaicians, docors, bioengineers, pharmaciss, are very ineresed in (especially more realisic) problems like hese Prof Fred Adler and a recen graduae suden Chris Remien in he Mah Deparmen, and collaboraing wih he Universiy Hospial recenly modeled liver poisoning by aceominophen (brand name Tyleonol), by sudying a non-linear sysem of 8 firs order differenial equaions They came up wih a sae of he ar and very useful diagnosic es: hp://unewsuahedu/news_releases/mah-can-save-ylenol-overdose-paiens-2/ Here's a link o heir published paper For fun, I copied and pased he non-linear sysem of firs order differenial equaions from a preprin of heir paper, below: hp://onlinelibrarywileycom/doi/2/hep25656/full hp://wwwmahuahedu/~korevaar/225spring2/adler-remien-preprinpdf
Example) consider he hree componen inpu-oupu model below: Exercise a) Derive he firs order sysem for he ank cascade above b) In case he ank volumes (in gallons) are V = 2, V 2 = 4, V 3 = 5, he flow rae r = gal min, and pure waer (wih no solue) is flowing ino he firs (op) ank, show ha your sysem in (a) can be wrien as x 3 = 5 5 25 25 2 (This sysem is acually worked ou in he ex, page 422-434bu we'll modify he IVP, and hen consider a second case as well) x 3
c) Here's he eigenvecor daa for he marix in b You may wan o check or derive pars of i by hand, especially if you're sill no exper a finding eigenvalues and eigenvecors I enered he marix enries as raional numbers raher han decimals, alhough in oher problems you'd wan o use decimals Use he eigendaa o wrie down he general soluion o he sysem in b d) Solve he IVP for his ank cascade, assuming ha here are iniially 5 lb of sal in he firs ank, and no sal in he second and hird anks
Your answer o d should be 3 = 5 e 5 6 3 e 25 25 e 2 x 3 5 5 e) We can plo he amouns of sal in each ank o figure ou wha's going on Make sure you undersand how he formulas below are relaed o he vecor equaion above, and inerpre he graphical resuls > wih plos : > x 5 exp 5 : plo plo x, = 3, color = black : x2 3 exp 5 3 exp 25 : plo2 plo x2, = 3, color = brown : x3 25 exp 5 5 exp 25 25 exp 2 : plo3 plo x3, = 3, color = green : display plo, plo2, plo3, ile = `polluan flushing in ank cascade` ; > 5 5 polluan flushing in ank cascade 2 3 Exercise 2) Use he same ank cascade Only now, assume ha here is iniially 3 lb sal in he firs ank, none in he ohers, and ha when he waer sars flowing he inpu pipe conains saly waer, wih lb concenraion 5 gal 2a) Explain why his yields an IVP for an inhomogeneous sysem of linear DE's, namely x 3 = 5 5 25 25 2 x 3 5 x 3 = 3
2b) Use a vecor analog of "undeermined coefficiens" o guess ha here migh be a paricular soluion ha is a consan vecor, ie x P = d Plug his guess ino he inhomgeneous sysem o deduce d In oher words, if we wrie he sysem wih he "linear operaor" par of he lef as in Chaper 5, we'd wrie x A x = b and we'd guess ha since he righ side was a consan vecor here migh be a paricular soluion ha was also a consan vecor (wih undeermined enries) 2c) Use x = x P x H o solve he IVP Compare your soluion o he plos below > x 3 exp 5 : plo plo x, = 5, color = black : x2 2 6 exp 5 4 exp 25 : plo2 plo x2, = 5, color = brown : x3 25 5 exp 5 7 exp 25 exp 2 : plo3 plo x3, = 5, color = green : display plo, plo2, plo3, ile = `polluan level sabilizing in ank cascade` ; 2 polluan level sabilizing in ank cascade > 2 3 4 5
Fri Apr 2 74 Mass-spring sysems and unehered mass-spring rains In your homework and lab for his week you sudy special cases of he spring sysems below, wih no damping Alhough we draw he picures horizonally, hey would also hold in verical configuraion if we measure displacemens from equilibrium in he underlying graviaional field Le's make sure we undersand why he naural sysem of DEs and IVP for his sysem is m = k k 2 m 2 = k 2 k 3 = a, = a 2 = b, = b 2 Exercise a) Wha is he dimension of he soluion space o his homogeneous linear sysem of differenial equaions? Why? b) Wha if one had a configuraion of n masses in series, raher han jus 2 masses? Wha would he dimension of he homogeneous soluion space be in his case? Why? Examples:
We can wrie he sysem of DEs for he sysem a he op of page in marix-vecor form: m x k k k 2 2 = m x k k k 2 2 2 3 2 We denoe he diagonal marix on he lef as he "mass marix" M, and he marix on he righ as he spring consan marix K (alhough o be compleely in sync wih Chaper 5 i would be beer o call he spring marix K) All of hese configuraions of masses in series wih springs can be wrien as = M x = K x If we divide each equaion by he reciprocal of he corresponding mass, we can solve for he vecor of acceleraions: k k k 2 2 x m m k 2 m 2 k 2 k 3 m 2, which we wrie as x = A x (You can hink of A as he "acceleraion" marix) Noice ha he simplificaion above is mahemaically idenical o he algebraic operaion of muliplying he firs marix equaion by he (diagonal) inverse of he diagonal mass marix M In all cases: M x = K x x = A x, wih A = M K
How o find a basis for he soluion space o conserved-energy mass-spring sysems of DEs x = A x (*) Based on our previous experiences, he naural hing for his homogeneous sysem of linear differenial equaions is o ry and find a basis of soluions of he form x = f v (**) You migh guess ha f = e bu ha urns ou o no be he bes way o go Le's see wha f should equal by subsiuing in our guess! (We would maybe also hink abou firs convering he second order sysem o an equivalen firs order sysem of wice as many DE's, one for for each posiion funcion and one for each velociy funcion, and hen he exponenial guess would work, bu hey'd end up being complex exponenials) Subsiuing (**) ino (*) yields f v = A f v = f A v Since for each, he lef side is a scalar muliple of he consan vecor v, so mus be he righ side So v mus be an eigenvecor of A, A v = v, and if f is a real funcion and if v is a real (as opposed o complex) vecor, hen is also real Then f v = A f v = f v So we mus have So possible f 's are Case ) Case 2) Case 3) f f = f = c c 2 if = f = c cos c 2 sin if, = f = c e c 2 e 2 = if Case 3 will never happen for our mass-spring configuraions, because of conservaion of energy!
This leads o he Soluion space algorihm: Consider a very special case of a homogeneous sysem of linear differenial equaions, x = A x If A n n is a diagonalizable marix and if all of is eigenvalues are non-posiive hen for each eigenpair j, v wih here are wo linearly independen sinusoidal soluions o x = A x given by j j x j = cos j v y j j = sin j v j wih And for an eigenpair funcions j, v j wih j j = j = here are wo indpenden soluions given by consan and linear x j = v j y j = v j This procedure consrucs 2 n independen soluions o he sysem x soluion space = A x, ie a basis for he Remark: Wha's amazing is ha he fac ha if he sysem is conservaive, he acceleraion marix will always be diagonalizable, and all of is eigenvalues will be non-posiive In fac, if he sysem is ehered o a leas one wall (as in he firs wo diagrams on page ), all of he eigenvalues will be sricly negaive, and he algorihm above will always yield a basis for he soluion space (If he sysem is no ehered and is free o move as a rain, like he hird diagram on page, hen = will be one of he eigenvalues, and will yield he consan velociy and displacemen conribuion o he soluion space, c c 2 v, where v is he corresponding eigenvecor Togeher wih he soluions from sricly negaive eigenvalues his will sill lead o he general homogeneous soluion)
Exercise 2) Consider he special case of he configuraion on page one for which m = m 2 = m and k = k 2 = k 3 = k In his case, he equaion for he vecor of he wo mass acceleraions reduces o 2 k m k m = k m 2 k m a) Find he eigendaa for he marix = k m 2 2 2 2 b) Deduce he eigendaa for he acceleraion marix A which is k imes his marix m c) Find he 4 dimensional soluion space o his wo-mass, hree-spring sysem
soluion The general soluion is a superposiion of wo "fundamenal modes" In he slower mode boh masses oscillae "in phase", wih equal ampliudes, and wih angular frequency = k In he faser m mode, boh masses oscillae "ou of phase" wih equal ampliudes, and wih angular frequency 2 = 3 k m The general soluion can be wrien as = C cos C 2 cos 2 2 = c cos c 2 sin c 3 cos 2 c 4 sin 2 Exercise 3) Show ha he general soluion above les you uniquely solve each IVP uniquely This should reinforce he idea ha he soluion space o hese wo second order linear homgeneous DE's is four dimensional 2 k k = m k m m 2 k m = a, = a 2 = b, = b 2