Mechanics Continuous Expansion, Lecture Notes - Physics, Study notes of Mechanics

Download Mechanics Continuous Expansion, Lecture Notes - Physics and more Study notes Mechanics in PDF only on Docsity! Mechanics Physics 151 Lecture 23 Continuous Systems and Fields (Chapter 13) Where Are We Now? We’ve finished all the essentials Final will cover Lectures 1 through 22 Last two lectures: Classical Field Theory Start with wave equations, similar to Physics 15c Do it with Lagrangian, and maybe with Hamiltonian Go into relativistic field theory Not enough time to discuss everything Let’s see how much we can do And take it easy! Continuous Limit Now we have Re-label ηi with the equilibrium position x 2 2 11 2 i i i i L K x x η ηµη + ⎡ ⎤−⎛ ⎞= − ∆⎢ ⎥⎜ ⎟∆⎝ ⎠⎢ ⎥⎣ ⎦ ∑ ( )i xη η→ 2 2 2 0 2 1 ( ) ( )( ) 2 1 2 i x x x xL x K x x dK dx dx η ηµη ηµη∆ → ⎡ ⎤+ ∆ −⎛ ⎞= − ∆⎢ ⎥⎜ ⎟∆⎝ ⎠⎢ ⎥⎣ ⎦ ⎡ ⎤⎛ ⎞⎯⎯⎯→ −⎢ ⎥⎜ ⎟ ⎝ ⎠⎢ ⎥⎣ ⎦ ∑ ∫Shrink! Lagrangian per unit length Lagrangian Density We can write the Lagrangian as L is the Lagrangian density in 1-dimension We may generally extend this to 3-dimensions ρ is the volume density µ/A (A is the rod’s cross section) Y is Young’s modulus K/A 2 21 2 d dL K dx dx dt dx η ηµ ⎡ ⎤⎛ ⎞ ⎛ ⎞= − ≡⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦ ∫ ∫L L dxdydz= ∫∫∫L 2 21 2 d dY dt dx η ηρ ⎡ ⎤⎛ ⎞ ⎛ ⎞= −⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦ Lwhere Lagrange’s Equations First, start from Do the usual Lagrange’s equations That’s wave equation with velocity We want to get this from the continuous Lagrangian 2 2 11 2 i i i i L K x x η ηµη + ⎡ ⎤−⎛ ⎞= − ∆⎢ ⎥⎜ ⎟∆⎝ ⎠⎢ ⎥⎣ ⎦ ∑ 1 1 0i i i i i i d L L K K x dt x x x x η η η ηµη η η + −⎛ ⎞ ⎡ − − ⎤∂ ∂ ⎛ ⎞ ⎛ ⎞− = − + ∆ =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥∂ ∂ ∆ ∆ ∆ ∆⎝ ⎠ ⎝ ⎠⎣ ⎦⎝ ⎠ 2 2 0 dK dx ηµη − =Shrink∆x Kv µ = Hamilton’s Principle Hamilton’s Principle gives ( )2 2 1 1 2 2 1 1 2 2 1 1 , , , , t x d d dx dtt x d dt x dx dt d dt x dx dt t x d dt x dx dt dI d x t dxdt d d d dd dxdt d d d d d d dxdt dx dt d η η η η η η η η η α α η η α α α η η α = ⎧ ⎫∂ ∂ ∂ = + +⎨ ⎬∂ ∂ ∂⎩ ⎭ ⎧ ⎫⎛ ⎞ ⎛ ⎞∂ ∂ ∂⎪ ⎪= − −⎨ ⎬⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎪ ⎪⎝ ⎠ ⎝ ⎠⎩ ⎭ ∫ ∫ ∫ ∫ ∫ ∫ L L L L L L L 2 2 1 10 ( , ) 0 t x d dt x dx dt dI d d x t dxdt d dx dtη ηα ζ α η= ⎧ ⎫⎛ ⎞ ⎛ ⎞∂ ∂ ∂⎪ ⎪⎛ ⎞ = − − =⎨ ⎬⎜ ⎟ ⎜ ⎟⎜ ⎟ ∂ ∂ ∂⎝ ⎠ ⎪ ⎪⎝ ⎠ ⎝ ⎠⎩ ⎭ ∫ ∫ L L L = 0! Lagrange’s Equation Lagrange’s equation for the 1-dim problem is Let’s try it with 0d d dt dx d d dt dxη η η ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ + − =⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠ L L L 2 21 2 d dK dt dx η ηµ ⎡ ⎤⎛ ⎞ ⎛ ⎞= −⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦ L 2 2 2 2 0 d d d d d dK K dt dt dx dx dt dx η η η ηµ µ⎛ ⎞ ⎛ ⎞− = − =⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ Yes, the right wave equation 3-D Version Easy to guess how it should look like in 3-dim. Symmetric between time and space Hope for relativistic formalism Will look into this in the next lecture 0d d d d dt dx dy dz d d d d dt dx dy dzη η η η η ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ + + + − =⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ L L L L L ( ), , , , , , , ,d d d ddx dy dz dt x y z tη η η ηη=L L ( )2 2 2 2 1 1 1 1 , , , , , , , , t x y z d d d d dx dy dz dtt x y z I x y z t dxdydzdtη η η ηη= ∫ ∫ ∫ ∫ L Conservation Laws Let’s try what we did with the energy function Consider the total derivative of the Lagrangian density Using Lagrange’s equations: ( ),, , xµ µη ηL , , , d dx xµ µνµ ν µ η η η η ∂ ∂ ∂ = + + ∂ ∂ ∂ L L L L , 0d dxµ µη η ⎛ ⎞∂ ∂ − =⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ L L , , , , , , d d dx dx x d dx x µ µν µ ν ν ν µ µ ν ν µ η η η η η η ⎛ ⎞∂ ∂ ∂ = + +⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎛ ⎞∂ ∂ = +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ L L L L L L This is , d dx µ ν η Stress-Energy Tensor We got NB: Tµν is not a tensor in the relativistic sense Suppose L does not depend explicitly on xµ For µ = 1, 2, 3, that means no external force For µ = 0, that means no source/sink of energy , , d dx xµ µνν ν µ η δ η ⎛ ⎞∂ ∂ − = −⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ L L L Free field 0 dT dx µν ν = Stress-energy tensorTµν≡ What does this “conservation” condition mean? Mechanics Physics 151 Lecture 23 Continuous Systems and Fields (Chapter 13) Where Are We Now? We’ve finished all the essentials Final will cover Lectures 1 through 22 Last two lectures: Classical Field Theory Start with wave equations, similar to Physics 15c Do it with Lagrangian, and maybe with Hamiltonian Go into relativistic field theory Not enough time to discuss everything Let’s see how much we can do And take it easy! Continuous Limit Now we have Re-label ηi with the equilibrium position x 2 2 11 2 i i i i L K x x η ηµη + ⎡ ⎤−⎛ ⎞= − ∆⎢ ⎥⎜ ⎟∆⎝ ⎠⎢ ⎥⎣ ⎦ ∑ ( )i xη η→ 2 2 2 0 2 1 ( ) ( )( ) 2 1 2 i x x x xL x K x x dK dx dx η ηµη ηµη∆ → ⎡ ⎤+ ∆ −⎛ ⎞= − ∆⎢ ⎥⎜ ⎟∆⎝ ⎠⎢ ⎥⎣ ⎦ ⎡ ⎤⎛ ⎞⎯⎯⎯→ −⎢ ⎥⎜ ⎟ ⎝ ⎠⎢ ⎥⎣ ⎦ ∑ ∫Shrink! Lagrangian per unit length Lagrangian Density We can write the Lagrangian as L is the Lagrangian density in 1-dimension We may generally extend this to 3-dimensions ρ is the volume density µ/A (A is the rod’s cross section) Y is Young’s modulus K/A 2 21 2 d dL K dx dx dt dx η ηµ ⎡ ⎤⎛ ⎞ ⎛ ⎞= − ≡⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦ ∫ ∫L L dxdydz= ∫∫∫L 2 21 2 d dY dt dx η ηρ ⎡ ⎤⎛ ⎞ ⎛ ⎞= −⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦ Lwhere Lagrange’s Equations First, start from Do the usual Lagrange’s equations That’s wave equation with velocity We want to get this from the continuous Lagrangian 2 2 11 2 i i i i L K x x η ηµη + ⎡ ⎤−⎛ ⎞= − ∆⎢ ⎥⎜ ⎟∆⎝ ⎠⎢ ⎥⎣ ⎦ ∑ 1 1 0i i i i i i d L L K K x dt x x x x η η η ηµη η η + −⎛ ⎞ ⎡ − − ⎤∂ ∂ ⎛ ⎞ ⎛ ⎞− = − + ∆ =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥∂ ∂ ∆ ∆ ∆ ∆⎝ ⎠ ⎝ ⎠⎣ ⎦⎝ ⎠ 2 2 0 dK dx ηµη − =Shrink∆x Kv µ = Hamilton’s Principle Hamilton’s Principle gives ( )2 2 1 1 2 2 1 1 2 2 1 1 , , , , t x d d dx dtt x d dt x dx dt d dt x dx dt t x d dt x dx dt dI d x t dxdt d d d dd dxdt d d d d d d dxdt dx dt d η η η η η η η η η α α η η α α α η η α = ⎧ ⎫∂ ∂ ∂ = + +⎨ ⎬∂ ∂ ∂⎩ ⎭ ⎧ ⎫⎛ ⎞ ⎛ ⎞∂ ∂ ∂⎪ ⎪= − −⎨ ⎬⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎪ ⎪⎝ ⎠ ⎝ ⎠⎩ ⎭ ∫ ∫ ∫ ∫ ∫ ∫ L L L L L L L 2 2 1 10 ( , ) 0 t x d dt x dx dt dI d d x t dxdt d dx dtη ηα ζ α η= ⎧ ⎫⎛ ⎞ ⎛ ⎞∂ ∂ ∂⎪ ⎪⎛ ⎞ = − − =⎨ ⎬⎜ ⎟ ⎜ ⎟⎜ ⎟ ∂ ∂ ∂⎝ ⎠ ⎪ ⎪⎝ ⎠ ⎝ ⎠⎩ ⎭ ∫ ∫ L L L = 0! Lagrange’s Equation Lagrange’s equation for the 1-dim problem is Let’s try it with 0d d dt dx d d dt dxη η η ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ + − =⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠ L L L 2 21 2 d dK dt dx η ηµ ⎡ ⎤⎛ ⎞ ⎛ ⎞= −⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦ L 2 2 2 2 0 d d d d d dK K dt dt dx dx dt dx η η η ηµ µ⎛ ⎞ ⎛ ⎞− = − =⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ Yes, the right wave equation 3-D Version Easy to guess how it should look like in 3-dim. Symmetric between time and space Hope for relativistic formalism Will look into this in the next lecture 0d d d d dt dx dy dz d d d d dt dx dy dzη η η η η ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ + + + − =⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ L L L L L ( ), , , , , , , ,d d d ddx dy dz dt x y z tη η η ηη=L L ( )2 2 2 2 1 1 1 1 , , , , , , , , t x y z d d d d dx dy dz dtt x y z I x y z t dxdydzdtη η η ηη= ∫ ∫ ∫ ∫ L Conservation Laws Let’s try what we did with the energy function Consider the total derivative of the Lagrangian density Using Lagrange’s equations: ( ),, , xµ µη ηL , , , d dx xµ µνµ ν µ η η η η ∂ ∂ ∂ = + + ∂ ∂ ∂ L L L L , 0d dxµ µη η ⎛ ⎞∂ ∂ − =⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ L L , , , , , , d d dx dx x d dx x µ µν µ ν ν ν µ µ ν ν µ η η η η η η ⎛ ⎞∂ ∂ ∂ = + +⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎛ ⎞∂ ∂ = +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ L L L L L L This is , d dx µ ν η Stress-Energy Tensor We got NB: Tµν is not a tensor in the relativistic sense Suppose L does not depend explicitly on xµ For µ = 1, 2, 3, that means no external force For µ = 0, that means no source/sink of energy , , d dx xµ µνν ν µ η δ η ⎛ ⎞∂ ∂ − = −⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ L L L Free field 0 dT dx µν ν = Stress-energy tensorTµν≡ What does this “conservation” condition mean? Divergence of S-E Tensor The condition has a form of divergence Integrate over a fixed volume V and use Gauss’s Law Now we need to know what Tµ0 and Tµ are 0 dT dx µν ν = 0 0 0i i dT dT dT dT dx dt dx dt µν µ µ µ µ ν = + = +∇ ⋅ =T 0 d T dV dV d dt µ µ µ = − ∇ ⋅ = − ⋅∫ ∫ ∫T T S This vector represents the “flow”Total Tµ0 in the volume What escapes from the surface Momentum Density First consider Again with the 1-dim. elastic rod example This isn’t so obvious… , , Tµν µ µν ν η δ η ∂ ≡ − ∂ L L 0i i dT dx η η ∂ = ∂ L 2 21 2 d dK dt dx η ηµ ⎡ ⎤⎛ ⎞ ⎛ ⎞= −⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦ L 10 d dT dt dx η ηµ= Momentum Density How much mass is there between x and x + dx? µdx to the zeroth order To the first order It’s velocity is , so the momentum is Density of excess momentum is –T10 may be considered as the momentum density 10 d dT dt dx η ηµ= dx ( )xη ( )x dxη + 1 d dx dx ηµ ⎛ ⎞−⎜ ⎟ ⎝ ⎠ η 1 d d dx dx dt η ηµ ⎛ ⎞−⎜ ⎟ ⎝ ⎠ 10 d d T dt dx η ηµ− = − Stress-Energy Tensor We can interpret the stress-energy tensor Tµν as T00 = energy density T0i = energy current density Ti0 = momentum density Tij = momentum current density The divergence condition represents conservation of energy and momentum 0 dT dx µν ν =