Mechancs Physcs 5 Lecture 3 Contnuous Systems an Fels (Chapter 3) Where Are We Now? We ve fnshe all the essentals Fnal wll cover Lectures through Last two lectures: Classcal Fel Theory Start wth wave equatons smlar to Physcs 5c Do t wth Lagrangan an maybe wth Hamltonan Go nto relatvstc fel theory Not enough tme to scuss everythng Let s see how much we can o An take t easy! Longtunal Mechancal Waves An nfnte elastc ro s vbratng longtunally x Moel ths wth a chan of masses an sprngs m k x D ths n 5c -th mass s poston s η relatve to equlbrum T = mη Let s bul the V = k ( η+ η ) Lagrangan η
Lagrangan Lagrangan s L= ( ) m η k η+ η Rearrange m η+ η = η k x x a lttle x x m/ x s the lnear ensty (mass/unt length) k x s the elastc moulus K (force/fractonal elongaton) Thnk about Hooke s law It s not Young s moulus L F = k L = K How much the sprng s stretche L relatve to ts natural length an K reman constant as we shrnk x Contnuous Lmt η+ η Now we have L= η K x x Re-label η wth the equlbrum poston x η η( x) η( x+ x) η( x) L= η ( x) K x x x Shrnk! η K Lagrangan per unt length Lagrangan Densty We can wrte the Lagrangan as L = K L L s the Lagrangan ensty n -menson We may generally exten ths to 3-mensons L = L yz η η where L = ρ Y ρ s the volume ensty /A (A s the ro s cross secton) Y s Young s moulus K/A
Lagrange s Equatons η+ η Frst start from L= η K x x Do the usual Lagrange s equatons K η+ η K η η = η x η η + = x x x x Shrnk x η η K = That s wave equaton wth velocty v = K We want to get ths from the contnuous Lagrangan Lagrange s Equatons In the screte case we ha = η η η became η(x) Smple analogy gves = η( x) η( x) But ths oesn t work We must go back to Hamlton s Prncple δi = δ L = δ = L for each Hamlton s Prncple Our Lagrangan ensty s L = K Let s get general L may epen on η L= L η x t ( ) We nee the path of η an ts varaton η( x t; α) = η( x t;) + αζ ( x t) Wll make α Nomnal path Varaton Set varaton to zero at the bounares ζ( xt ) = ζ( xt ) = ζ( x t) = ζ( x t) = Intal Fnal Eges OK let s work Don t really matter for the nfnte ro 3
Hamlton s Prncple I t x = ( η xt ) α α L t x t x = + + t x η η η α α α t x = t x η η η α Hamlton s Prncple gves I t x = ζ ( x t) t x η η α = α = η =! Lagrange s Equaton Lagrange s equaton for the -m problem s + = η Let s try t wth L = K K = η η = K Yes the rght wave equaton 3-D Verson Easy to guess how t shoul look lke n 3-m. η η η η L= L η y z x yzt ( ) t x y z = L ( η y z ) I t x y z x y z t yz + + + = y y z z η Symmetrc between tme an space Hope for relatvstc formalsm Wll look nto ths n the next lecture 4
Mult-Component Fel I efne η as the splacement along x axs General 3-m. vbraton may be n any recton η η = ( ηx ηy ηz) We are now ealng wth 3 functons of space an tme x x x x η x y z y y y y ηy y z L= L z z z z ηz y z xyzt Ths s gettng really teous Shorthan Notaton Let s use nces (3) nstea of t x y z Smlar to what we n relatvty η We nee quanttes lke η Let s get lazy ρ ηρ ηρ ηρ an η etc. We can wrte e.g. L= L ( ηρ ηρ x) = η ρ η ρ Conservaton Laws Let s try what we wth the energy functon Conser the total ervatve of the Lagrangan ensty L ( ηη x ) L = L η + L η + L η η x Usng Lagrange s equatons: = η η L = η + η + x η η = η + η x Ths s 5
Stress-Energy Tensor We got η L δ = η x T Stress-energy tensor NB: T s not a tensor n the relatvstc sense Suppose L oes not epen explctly on x For = 3 that means no external force For = that means no source/snk of energy Free fel T = What oes ths conservaton conton mean? Dvergence of S-E Tensor T The conton = has a form of vergence T T T T = + = + T = Integrate over a fxe volume V an use Gauss s Law T V = V = T T S Total T n the volume Now we nee to know what T an T are What escapes from the surface Ths vector represents the flow Energy Densty Frst conser T = η L η T η L δ η Looks just lke the energy functon oesn t t? Thnk about the -m. elastc ro example L = K T = + K T shoul be the energy flow Knetc Potental energy energy T = η = K η s t? 6
Energy Current Densty Conser a small pece It s stretche by η( x + ) η( x) = Ths gves the Hooke s law force F = K η ( x) η ( x + ) The work one by ths pece to the next pece s F η = K η equals to T = K η Momentum Densty Frst conser T = η Agan wth the -m. elastc ro example = K Ths sn t so obvous L T T η L δ η = Momentum Densty T = How much mass s there between x an x +? to the zeroth orer To the frst orer η It s velocty s η η ( x) η ( x + ) so the momentum s η η Densty of excess momentum s = T T may be consere as the momentum ensty 7
Stress-Energy Tensor We can nterpret the stress-energy tensor T as T = energy ensty T = energy current ensty T = momentum ensty T j = momentum current ensty T The vergence conton = represents conservaton of energy an momentum Summary Bult Lagrangan formalsm for contnuous system Lagrangan L = L yz Lagrange s equaton Derve smple wave equaton = η η Energy an momentum conservaton gven by the energy-stress tensor T T η L δ = Conservaton laws η take the form of (tme ervatve) = (flux nto volume) 8