Mechanics Physics 151

Transcription

1 Mechanics Physics 151 Lecture 3 Continuous Systems an Fiels (Chapter 13)

2 Where Are We Now? We ve finishe all the essentials Final will cover Lectures 1 through Last two lectures: Classical Fiel Theory Start with wave equations, similar to Physics 15c Do it with Lagrangian, an maybe with Hamiltonian Go into relativistic fiel theory Not enough time to iscuss everything Let s see how much we can o An take it easy!

3 Longituinal Mechanical Waves An infinite elastic ro is vibrating longituinally x Moel this with a chain of masses an springs m k x Di this in 15c i-th mass s position is η i relative to equilibrium T η i 1 = mη i 1 V = k ( ηi+ 1 ηi ) i i Let s buil the Lagrangian

4 Lagrangian Lagrangian is Rearrange a little 1 L= mη k( η η ) m/ x is the linear ensity µ (mass/unit length) k x is the elastic moulus K (force/fractional elongation) Think about Hooke s law L F k L = K L i i i+ 1 i 1 m ηi+ 1 ηi = ηi k x x i x x µ an K remain constant as we shrink x 0 It s not Young s moulus = How much the spring is stretche relative to its natural length

5 Continuous Limit 1 ηi+ 1 ηi L= µη i K x i x Re-label η i with the equilibrium position x η η( x) Now we have Shrink! = i x 1 η( x+ x) η( x) L µη ( x) K x 1 x 0 µη K i Lagrangian per unit length

6 Lagrangian Density We can write the Lagrangian as 1 L = µ K t L L is the Lagrangian ensity in 1-imension We may generally exten this to 3-imensions L = L 1 yz η η where L = ρ Y t ρ is the volume ensity µ/a (A is the ro s cross section) Y is Young s moulus K/A

7 Lagrange s Equations 1 ηi+ 1 ηi L= µη i K x i x Do the usual Lagrange s equations L L K ηi+ ηi K ηi ηi = µη x t ηi η + = i x x x x First, start from Shrink x µη = η K 0 That s wave equation with velocity v = K µ We want to get this from the continuous Lagrangian

8 Lagrange s Equations In the iscrete case, we ha η i became η(x) Simple analogy gives But this oesn t work L L = ηi t ηi L L = 0 η( x) t η( x) We must go back to Hamilton s Principle δi = δ Lt = δ L t = for each i

9 Hamilton s Principle Our Lagrangian ensity is Let s get general L may epen on L 1 L = µ K t (, η η,,, ) t xt We nee the path of η an its variation = L η( xt, ; α) = η( xt, ;0) + αζ ( xt, ) Will make α 0 Nominal path Variation Set variation to zero at the bounaries ζ( xt, ) = ζ( xt, ) = ζ( x, t) = ζ( x, t) = Initial Final Eges OK, let s work Don t really matter for the infinite ro

10 Hamilton s Principle I = α α t x t t x x 1 1 ( η,,,, ) = + + t 1 x η 1 t x L = t 1 x η 1 Hamilton s Principle gives t x t t L L L t t η α α t α L L L t η t t α L L L = ζ ( x, t) t = 0 η I t x t 1 x η 1 t α α = 0 t = 0!

11 Lagrange s Equation Lagrange s equation for the 1-im problem is t Let s try it with L L L + = t η 1 L = µ K t 0 K µ = µ η η = t t t K 0 Yes, the right wave equation

12 3-D Version Easy to guess how it shoul look like in 3-im. L I = L (, η, η, η, η η,,,, ) t x y z = L t1 x1 y1 z1 y z t xyzt ( η,,,,,,,, ) y z t L L L L L = t t y y z z η Symmetric between time an space Hope for relativistic formalism x y z t yzt Will look into this in the next lecture 0

13 Multi-Component Fiel I efine η as the isplacement along x axis General 3-im. vibration may be in any irection η η = ( η, η, η ) x y z We are now ealing with 3 functions of space an time x x x x η,,,,, x y z t y y y y ηy,, y, z, t, L= L z z z z ηz,, y, z, t, xyzt,,, This is getting really teious

14 Shorthan Notation Let s use inices (0,1,,3) instea of t, x, y, z Similar to what we i in relativity η i i We nee quantities like ηi µ µ Let s get lazy ρ ηρ ηρµ, ηρµ, an η, µ etc. We can write, e.g. L = L µ ( η, η,x ) ρ ρµ, µ µ µ η ρ, µ η ρ µ L L = 0

15 Conservation Laws Let s try what we i with the energy function Consier the total erivative of the Lagrangian ensity L ( ηη, ), µ,x L µ = L η, µ + L η, µ + L η η x Using Lagrange s equations: µ, µ L L = η µ η, µ L L L L = η + η + x, µ, µ µ η, η, µ L = η + L, µ η, xµ This is, µ 0

16 Stress-Energy Tensor We got L η L δ = L, µ µ η, xµ T µ Stress-energy tensor NB: T µ is not a tensor in the relativistic sense Suppose L oes not epen explicitly on x µ For µ = 1,, 3, that means no external force For µ = 0, that means no source/sink of energy Free fiel T µ = 0 What oes this conservation conition mean?

17 Divergence of S-E Tensor T µ The conition = 0 has a form of ivergence Tµ Tµ 0 Tµ i Tµ 0 = + = + Tµ = 0 t t i Integrate over a fixe volume V an use Gauss s Law T µ 0 V = Tµ V = Tµ S t Total T µ0 in the volume What escapes from the surface This vector represents the flow Now we nee to know what T µ0 an T µ are

18 Energy Density First consier T00 L = η L η T 0 shoul be the energy flow L T µ η, µ L δ µ η, Looks just like the energy function, oesn t it? Think about the 1-im. elastic ro example 1 1 L = µ K T00 = µ + K t t T L = η = K η is it? 01 Kinetic energy Potential energy

19 Energy Current Density Consier a small piece It s stretche by η( x+ ) η( x) = This gives the Hooke s law force F = K η ( x) η ( x+ ) The work one by this piece to the next piece is F η = K η equals to T 01 = K η

20 Momentum Density First consier T Again with the 1-im. elastic ro example This isn t so obvious i0 L = η 1 = µ K t L T10 i L T µ η, µ L δ µ η, = µ t

21 Momentum Density How much mass is there between x an x +? µ to the zeroth orer To the first orer 1 η µ It s velocity is η, η ( x) η ( x+ ) so the momentum is 1 η η µ t Density of excess momentum is µ = T10 t T 10 may be consiere as the momentum ensity T 10 = µ t

22 Stress-Energy Tensor We can interpret the stress-energy tensor T µ as T 00 = energy ensity T 0i = energy current ensity T i0 = momentum ensity T ij = momentum current ensity = 0 µ The ivergence conition represents T conservation of energy an momentum

23 Summary Built Lagrangian formalism for continuous system Lagrangian L Lagrange s equation Derive simple wave equation Energy an momentum conservation given by the energy-stress tensor L T Conservation laws = L yz µ η, µ take the form of (time erivative) = (flux into volume) T µ η, µ δ µ η, L L = η 0 µ L = 0