I dreamed last night of speaking to Feynman, and he told me to save what I had just learned. Seeing that I had just finished working through a problem in Path Integrals, it seemed appropriate. On the other hand, it's one of the bunny problems from chapter 2 - I doubt Dick would have been impressed.
The problem is to calculate the classical action of a harmonic oscillator.
L = T - V = (m/2)(x'²) - (m/2)ω²x² = (m/2)(x'² - ω²x²)
The classical path is that in which the Euler-Lagrange equation is satisfied.
d/dt( ∂L/∂x' ) - ∂L/∂x = 0 d/dt( mx' ) = - mω²x mx'' = - mω²x x'' = - ω²xx = A cos(ωt) + B sin(ωt)
x'= ω [B cos(ωt) - A sin(ωt)]
To find the action, we just plug in x and x' and start working.
S = ∫ L(x',x,t) dt = ∫ (m/2)(x'² - ω²x²) dt = (m/2) ∫ (x'² - ω²x²) dt = (m/2) ∫ (ω² [B cos(ωt) - A sin(ωt)]² - ω²[A cos(ωt) + B sin(ωt)]²) dt = (m/2) ω² ∫ ( [B cos(ωt) - A sin(ωt)]² - [A cos(ωt) + B sin(ωt)]²) dt = (m/2) ω² ∫ ( [B² cos²(ωt) - 2AB cos(ωt)sin(ωt) + A²sin²(ωt)] - [A²cos²(ωt) + 2ABcos(ωt)sin(ωt)+ B²sin²(ωt)]) dt = (m/2) ω² ∫ B² cos²(ωt) - 2AB cos(ωt)sin(ωt) + A²sin²(ωt) - A²cos²(ωt) - 2ABcos(ωt)sin(ωt) - B²sin²(ωt) dt = (m/2) ω² ∫ B²cos²(ωt) - B²sin²(ωt) - 2ABcos(ωt)sin(ωt) - 2ABcos(ωt)sin(ωt) + A²sin²(ωt) - A²cos²(ωt) dt = (m/2) ω² ∫ [B²-A²][cos²(ωt) - sin²(ωt)] - 2AB[cos(ωt)sin(ωt) + cos(ωt)sin(ωt)] dt = (m/2) (ω/2) ∫ [B²-A²]cos(2ωt) - 2ABsin(2ωt) 2ωdt = (m/2) (ω/2) ∫ [B²-A²]cos(u) - 2ABsin(u) du = (m/2) (ω/2) ( [B²-A²] ∫ cos(u) du - 2AB ∫ sin(u) du ) = (m/2) (ω/2) ( [B²-A²] [sin(u₂) - sin(u₁)] + 2AB [cos(u₂) - cos(u₁)]) = (m/2) (ω/2) ( [B²-A²] [sin(2ωt₂) - sin(2ωt₁)] + 2AB [cos(2ωt₂) - cos(2ωt₁)]) = (m/2) (ω/2) ( [B²-A²] [2cos(ωt₂)sin(ωt₂) - 2cos(ωt₁)sin(ωt₁)] + 2AB [cos(2ωt₂) - cos(2ωt₁)]) = (m/2) (ω/2) 2 ( [B²-A²] [cos(ωt₂)sin(ωt₂) - cos(ωt₁)sin(ωt₁)] + AB [cos(2ωt₂) - cos(2ωt₁)]) = (mω/2) ( [B²-A²] [cos(ωt₂)sin(ωt₂) - cos(ωt₁)sin(ωt₁)] + AB [cos(2ωt₂) - cos(2ωt₁)]) = (mω/2) ( [B²-A²] [cos(ωt₂)sin(ωt₂) - cos(ωt₁)sin(ωt₁)] + AB [cos²(ωt₂) - sin²(ωt₂) + sin²(ωt₁) - cos²(ωt₁)])
Upon reaching this point, it wasn't at all obvious to me how to proceed. In fact, I jumped to the answer and began fighting my way backwards. Having bridged the two halves together in the middle, I think I can continue from here. It will be useful to introduce a few definitions.
T = t₂ - t₁ x₁ = A cos(ωt₁) + B sin(ωt₁) x₂ = A cos(ωt₂) + B sin(ωt₂) S = (mω/2) ( [B²-A²] [cos(ωt₂)sin(ωt₂) - cos(ωt₁)sin(ωt₁)] + AB [cos²(ωt₂) - sin²(ωt₂) + sin²(ωt₁) - cos²(ωt₁)]) = (mω/2) (sin(ωT)/sin(ωT))( [B²-A²] [cos(ωt₂)sin(ωt₂) - cos(ωt₁)sin(ωt₁)] + AB [cos²(ωt₂) - sin²(ωt₂) + sin²(ωt₁) - cos²(ωt₁)]) = (mω/2sin(ωT)) (sin(ωT))( [B²-A²] [cos(ωt₂)sin(ωt₂) - cos(ωt₁)sin(ωt₁)] + AB [cos²(ωt₂) - sin²(ωt₂) + sin²(ωt₁) - cos²(ωt₁)]) = (mω/2sin(ωT)) ( (sin(ωT))[B²-A²] [cos(ωt₂)sin(ωt₂) - cos(ωt₁)sin(ωt₁)] + AB (sin(ωT))[cos²(ωt₂) - sin²(ωt₂) + sin²(ωt₁) - cos²(ωt₁)]) = (mω/2sin(ωT)) ( (sin(ωT))[B²-A²] [cos(ωt₂)sin(ωt₂) - cos(ωt₁)sin(ωt₁)] + AB (sin(ωt₂)cos(ωt₁)-sin(ωt₁)cos(ωt₂))[cos²(ωt₂) - sin²(ωt₂) + sin²(ωt₁) - cos²(ωt₁)]) = (mω/2sin(ωT)) ( (sin(ωT))[B²-A²] [cos(ωt₂)sin(ωt₂) - cos(ωt₁)sin(ωt₁)] + AB (sin(ωt₂)cos(ωt₁)[cos²(ωt₂) - sin²(ωt₂) + sin²(ωt₁) - cos²(ωt₁)]+sin(ωt₁)cos(ωt₂)[ sin²(ωt₂) - cos²(ωt₂) + cos²(ωt₁) - sin²(ωt₁)])) = (mω/2sin(ωT)) ( (sin(ωT))[B²-A²] [cos(ωt₂)sin(ωt₂) - cos(ωt₁)sin(ωt₁)] + AB (sin(ωt₂)cos(ωt₁)[cos²(ωt₂) + 1 - sin²(ωt₂) + sin²(ωt₁) + 1 - cos²(ωt₁) - 2]+sin(ωt₁)cos(ωt₂)[ sin²(ωt₂) + 1 - cos²(ωt₂) + cos²(ωt₁) + + 1 - sin²(ωt₁) - 2])) = (mω/2sin(ωT)) ( (sin(ωT))[B²-A²] [cos(ωt₂)sin(ωt₂) - cos(ωt₁)sin(ωt₁)] + AB (sin(ωt₂)cos(ωt₁)[cos²(ωt₂) + cos²(ωt₂) + sin²(ωt₁) + sin²(ωt₁) - 2]+sin(ωt₁)cos(ωt₂)[ sin²(ωt₂) + sin²(ωt₂) + cos²(ωt₁) + cos²(ωt₁) - 2])) = (mω/2sin(ωT)) ( (sin(ωT))[B²-A²] [cos(ωt₂)sin(ωt₂) - cos(ωt₁)sin(ωt₁)] + AB (sin(ωt₂)cos(ωt₁)[2cos²(ωt₂) + 2sin²(ωt₁) - 2]+sin(ωt₁)cos(ωt₂)[ 2sin²(ωt₂) + 2cos²(ωt₁) - 2])) = (mω/2sin(ωT)) ( (sin(ωT))[B²-A²] [cos(ωt₂)sin(ωt₂) - cos(ωt₁)sin(ωt₁)] + 2AB (sin(ωt₂)cos(ωt₁)[cos²(ωt₂) + sin²(ωt₁) - 1]+sin(ωt₁)cos(ωt₂)[ sin²(ωt₂) + cos²(ωt₁) - 1])) = (mω/2sin(ωT)) ( (sin(ωT))[B²-A²] [cos(ωt₂)sin(ωt₂) - cos(ωt₁)sin(ωt₁)] + 2AB (sin(ωt₂)cos(ωt₁)[cos²(ωt₂) + sin²(ωt₁)]+sin(ωt₁)cos(ωt₂)[ sin²(ωt₂) + cos²(ωt₁)])- (sin(ωt₂)cos(ωt₁) + sin(ωt₁)cos(ωt₂) ) ) = (mω/2sin(ωT)) ( (sin(ωT))[B²-A²] [cos(ωt₂)sin(ωt₂) - cos(ωt₁)sin(ωt₁)] + 2AB (sin(ωt₂)cos(ωt₁)[cos²(ωt₂) + sin²(ωt₁)]+sin(ωt₁)cos(ωt₂)[ sin²(ωt₂) + cos²(ωt₁)]))- 2AB(sin(ωt₂)cos(ωt₁) + sin(ωt₁)cos(ωt₂) ) = (mω/2sin(ωT)) ( (sin(ωT))[B²-A²] [cos(ωt₂)sin(ωt₂) - cos(ωt₁)sin(ωt₁)] + 2AB (sin(ωt₂)cos(ωt₁)[cos²(ωt₂) + sin²(ωt₁)]+sin(ωt₁)cos(ωt₂)[ sin²(ωt₂) + cos²(ωt₁)])) + 2A²cos(ωt₂)cos(ωt₁) + 2B²sin(ωt₂)sin(ωt₁) - 2A²cos(ωt₂)cos(ωt₁) - 2B²sin(ωt₂)sin(ωt₁) -2AB(sin(ωt₂)cos(ωt₁) + sin(ωt₁)cos(ωt₂) ) = (mω/2sin(ωT)) ( (sin(ωT))[B²-A²] [cos(ωt₂)sin(ωt₂) - cos(ωt₁)sin(ωt₁)] + 2AB (sin(ωt₂)cos(ωt₁)[cos²(ωt₂) + sin²(ωt₁)]+sin(ωt₁)cos(ωt₂)[ sin²(ωt₂) + cos²(ωt₁)])) + 2A²cos(ωt₂)cos(ωt₁) + 2B²sin(ωt₂)sin(ωt₁) - 2 [ A²cos(ωt₂)cos(ωt₁) + B²sin(ωt₂)sin(ωt₁) + ABsin(ωt₂)cos(ωt₁) + ABsin(ωt₁)cos(ωt₂) ] ) = (mω/2sin(ωT)) ( (sin(ωT))[B²-A²] [cos(ωt₂)sin(ωt₂) - cos(ωt₁)sin(ωt₁)] + 2AB (sin(ωt₂)cos(ωt₁)[cos²(ωt₂) + sin²(ωt₁)]+sin(ωt₁)cos(ωt₂)[ sin²(ωt₂) + cos²(ωt₁)])) + 2A²cos(ωt₂)cos(ωt₁) + 2B²sin(ωt₂)sin(ωt₁) - 2 [ Acos(ωt₂)Acos(ωt₁) + Bsin(ωt₂)Bsin(ωt₁) + Bsin(ωt₂)Acos(ωt₁) + Bsin(ωt₁)Acos(ωt₂) ] ) = (mω/2sin(ωT)) ( (sin(ωT))[B²-A²] [cos(ωt₂)sin(ωt₂) - cos(ωt₁)sin(ωt₁)] + 2AB (sin(ωt₂)cos(ωt₁)[cos²(ωt₂) + sin²(ωt₁)]+sin(ωt₁)cos(ωt₂)[ sin²(ωt₂) + cos²(ωt₁)])) + 2A²cos(ωt₂)cos(ωt₁) + 2B²sin(ωt₂)sin(ωt₁) - 2 [ Acos(ωt₂)Acos(ωt₁) + Bsin(ωt₂)Acos(ωt₁) + Bsin(ωt₂)Bsin(ωt₁) + Bsin(ωt₁)Acos(ωt₂) ] ) = (mω/2sin(ωT)) ( (sin(ωT))[B²-A²] [cos(ωt₂)sin(ωt₂) - cos(ωt₁)sin(ωt₁)] + 2AB (sin(ωt₂)cos(ωt₁)[cos²(ωt₂) + sin²(ωt₁)]+sin(ωt₁)cos(ωt₂)[ sin²(ωt₂) + cos²(ωt₁)])) + 2A²cos(ωt₂)cos(ωt₁) + 2B²sin(ωt₂)sin(ωt₁) - 2 [ (Acos(ωt₂)+ Bsin(ωt₂))Acos(ωt₁) + (Bsin(ωt₂)+Acos(ωt₂)) Bsin(ωt₁) ] ) = (mω/2sin(ωT)) ( (sin(ωT))[B²-A²] [cos(ωt₂)sin(ωt₂) - cos(ωt₁)sin(ωt₁)] + 2AB (sin(ωt₂)cos(ωt₁)[cos²(ωt₂) + sin²(ωt₁)]+sin(ωt₁)cos(ωt₂)[ sin²(ωt₂) + cos²(ωt₁)])) + 2A²cos(ωt₂)cos(ωt₁) + 2B²sin(ωt₂)sin(ωt₁) - 2 [ (Acos(ωt₂)+ Bsin(ωt₂))(Acos(ωt₁) + Bsin(ωt₁)) ] ) = (mω/2sin(ωT)) ( (sin(ωT))[B²-A²] [cos(ωt₂)sin(ωt₂) - cos(ωt₁)sin(ωt₁)] + 2AB (sin(ωt₂)cos(ωt₁)[cos²(ωt₂) + sin²(ωt₁)]+sin(ωt₁)cos(ωt₂)[ sin²(ωt₂) + cos²(ωt₁)])) + 2A²cos(ωt₂)cos(ωt₁) + 2B²sin(ωt₂)sin(ωt₁) - 2 [ x₂x₁ ] ) = (mω/2sin(ωT)) ( (sin(ωT))[B²-A²] [cos(ωt₂)sin(ωt₂) - cos(ωt₁)sin(ωt₁)] + 2AB (sin(ωt₂)cos(ωt₁)[cos²(ωt₂) + sin²(ωt₁)]+sin(ωt₁)cos(ωt₂)[ sin²(ωt₂) + cos²(ωt₁)])) + 2A²cos(ωt₂)cos(ωt₁) + 2B²sin(ωt₂)sin(ωt₁) - 2 x₁x₂ ) = (mω/2sin(ωT)) ( [B²-A²] (sin(ωT))[cos(ωt₂)sin(ωt₂) - cos(ωt₁)sin(ωt₁)] + 2A²cos(ωt₂)cos(ωt₁) + 2B²sin(ωt₂)sin(ωt₁) + 2AB (sin(ωt₂)cos(ωt₁)cos²(ωt₂) + sin²(ωt₁)sin(ωt₂)cos(ωt₁)+sin(ωt₁)sin²(ωt₂)cos(ωt₂) + sin(ωt₁)cos²(ωt₁)cos(ωt₂))) - 2 x₁x₂ ) = (mω/2sin(ωT)) ( [B²-A²] (sin(ωt₂)cos(ωt₁)-sin(ωt₁)cos(ωt₂))[cos(ωt₂)sin(ωt₂) - cos(ωt₁)sin(ωt₁)] + 2A²cos(ωt₂)cos(ωt₁) + 2B²sin(ωt₂)sin(ωt₁) + 2AB (sin(ωt₂)cos(ωt₁)cos²(ωt₂) + sin²(ωt₁)sin(ωt₂)cos(ωt₁)+sin(ωt₁)sin²(ωt₂)cos(ωt₂) + sin(ωt₁)cos²(ωt₁)cos(ωt₂))) - 2 x₁x₂ ) = (mω/2sin(ωT)) ( [B²-A²] [sin²(ωt₂)cos(ωt₁)cos(ωt₂) - sin(ωt₁)sin(ωt₂)cos²(ωt₂) - sin(ωt₁)sin(ωt₂)cos²(ωt₁) + sin²(ωt₁)cos(ωt₁)cos(ωt₂)] + 2A²cos(ωt₂)cos(ωt₁) + 2B²sin(ωt₂)sin(ωt₁) + 2AB (sin(ωt₂)cos(ωt₁)cos²(ωt₂) + sin²(ωt₁)sin(ωt₂)cos(ωt₁)+sin(ωt₁)sin²(ωt₂)cos(ωt₂) + sin(ωt₁)cos²(ωt₁)cos(ωt₂))) - 2 x₁x₂ ) = (mω/2sin(ωT)) ( [B²-A²] [(sin²(ωt₂)+sin²(ωt₁))cos(ωt₁)cos(ωt₂) - sin(ωt₁)sin(ωt₂)(cos²(ωt₁)+cos²(ωt₂)) ] + 2A²cos(ωt₂)cos(ωt₁) + 2B²sin(ωt₂)sin(ωt₁) + 2AB (sin(ωt₂)cos(ωt₁)cos²(ωt₂) + sin²(ωt₁)sin(ωt₂)cos(ωt₁)+sin(ωt₁)sin²(ωt₂)cos(ωt₂) + sin(ωt₁)cos²(ωt₁)cos(ωt₂))) - 2 x₁x₂ ) = (mω/2sin(ωT)) ( B²[(sin²(ωt₂)+sin²(ωt₁))cos(ωt₁)cos(ωt₂) - sin(ωt₁)sin(ωt₂)(cos²(ωt₁)+cos²(ωt₂)) ]-A²[(sin²(ωt₂)+sin²(ωt₁))cos(ωt₁)cos(ωt₂) - sin(ωt₁)sin(ωt₂)(cos²(ωt₁)+cos²(ωt₂)) ] + 2A²cos(ωt₂)cos(ωt₁) + 2B²sin(ωt₂)sin(ωt₁) + 2AB (sin(ωt₂)cos(ωt₁)cos²(ωt₂) + sin²(ωt₁)sin(ωt₂)cos(ωt₁)+sin(ωt₁)sin²(ωt₂)cos(ωt₂) + sin(ωt₁)cos²(ωt₁)cos(ωt₂))) - 2 x₁x₂ ) = (mω/2sin(ωT)) ( 2B²sin(ωt₂)sin(ωt₁) + B²[(sin²(ωt₂)+sin²(ωt₁))cos(ωt₁)cos(ωt₂) - sin(ωt₁)sin(ωt₂)(cos²(ωt₁)+cos²(ωt₂)) ] + 2A²cos(ωt₂)cos(ωt₁) - A²[(sin²(ωt₂)+sin²(ωt₁))cos(ωt₁)cos(ωt₂) - sin(ωt₁)sin(ωt₂)(cos²(ωt₁)+cos²(ωt₂)) ] + 2AB (sin(ωt₂)cos(ωt₁)cos²(ωt₂) + sin²(ωt₁)sin(ωt₂)cos(ωt₁)+sin(ωt₁)sin²(ωt₂)cos(ωt₂) + sin(ωt₁)cos²(ωt₁)cos(ωt₂))) - 2 x₁x₂ ) = (mω/2sin(ωT)) ( B²[(sin²(ωt₂)+sin²(ωt₁))cos(ωt₁)cos(ωt₂) + sin(ωt₁)sin(ωt₂)(sin²(ωt₁)+sin²(ωt₂)) ] - A²[-(cos²(ωt₂)+cos²(ωt₁))cos(ωt₁)cos(ωt₂) - sin(ωt₁)sin(ωt₂)(cos²(ωt₁)+cos²(ωt₂)) ] + 2AB (sin(ωt₂)cos(ωt₁)cos²(ωt₂) + sin²(ωt₁)sin(ωt₂)cos(ωt₁)+sin(ωt₁)sin²(ωt₂)cos(ωt₂) + sin(ωt₁)cos²(ωt₁)cos(ωt₂))) - 2 x₁x₂ ) = (mω/2sin(ωT)) ( B²[(sin²(ωt₂)+sin²(ωt₁))cos(ωt₁)cos(ωt₂) + sin(ωt₁)sin(ωt₂)(sin²(ωt₁)+sin²(ωt₂)) ] + A²[(cos²(ωt₂)+cos²(ωt₁))cos(ωt₁)cos(ωt₂) + sin(ωt₁)sin(ωt₂)(cos²(ωt₁)+cos²(ωt₂)) ] + 2AB (sin(ωt₂)cos(ωt₁)cos²(ωt₂) + sin²(ωt₁)sin(ωt₂)cos(ωt₁)+sin(ωt₁)sin²(ωt₂)cos(ωt₂) + sin(ωt₁)cos²(ωt₁)cos(ωt₂))) - 2 x₁x₂ ) = (mω/2sin(ωT)) ( B²[(sin²(ωt₂)+sin²(ωt₁))(cos(ωt₁)cos(ωt₂) + sin(ωt₁)sin(ωt₂)) ] + A²[(cos²(ωt₂)+cos²(ωt₁))(cos(ωt₁)cos(ωt₂) + sin(ωt₁)sin(ωt₂)) ] + 2AB (sin(ωt₂)cos(ωt₁)cos²(ωt₂) + sin²(ωt₁)sin(ωt₂)cos(ωt₁)+sin(ωt₁)sin²(ωt₂)cos(ωt₂) + sin(ωt₁)cos²(ωt₁)cos(ωt₂))) - 2 x₁x₂ ) = (mω/2sin(ωT)) ( B²[(sin²(ωt₂)+sin²(ωt₁))cos(ωT) ] + A²[(cos²(ωt₂)+cos²(ωt₁))cos(ωT) ] + 2AB (sin(ωt₂)cos(ωt₂)[cos(ωt₁)cos(ωt₂)] + [sin(ωt₁)sin(ωt₂)]sin(ωt₁)cos(ωt₁)+[sin(ωt₁)sin(ωt₂)]sin(ωt₂)cos(ωt₂) + sin(ωt₁)cos(ωt₁)[cos(ωt₁)cos(ωt₂)])) - 2 x₁x₂ ) = (mω/2sin(ωT)) ( B²[(sin²(ωt₂)+sin²(ωt₁))cos(ωT) ] + A²[(cos²(ωt₂)+cos²(ωt₁))cos(ωT) ] + 2AB (sin(ωt₂)cos(ωt₂)[cos(ωt₁)cos(ωt₂)] + [sin(ωt₁)sin(ωt₂)]sin(ωt₂)cos(ωt₂) + sin(ωt₁)cos(ωt₁)[cos(ωt₁)cos(ωt₂)]+ sin(ωt₁)cos(ωt₁)[sin(ωt₁)sin(ωt₂)])) - 2 x₁x₂ ) = (mω/2sin(ωT)) ( B²[(sin²(ωt₂)+sin²(ωt₁))cos(ωT) ] + A²[(cos²(ωt₂)+cos²(ωt₁))cos(ωT) ] + 2AB (sin(ωt₂)cos(ωt₂)[cos(ωt₁)cos(ωt₂) + sin(ωt₁)sin(ωt₂)] + sin(ωt₁)cos(ωt₁)[cos(ωt₁)cos(ωt₂) + sin(ωt₁)sin(ωt₂)])) - 2 x₁x₂ ) = (mω/2sin(ωT)) ( B²[(sin²(ωt₂)+sin²(ωt₁))cos(ωT) ] + A²[(cos²(ωt₂)+cos²(ωt₁))cos(ωT) ] + 2AB (sin(ωt₂)cos(ωt₂)cos(ωT) + sin(ωt₁)cos(ωt₁)cos(ωT)) - 2 x₁x₂ ) = (mω/2sin(ωT)) ( B²cos(ωT)[(sin²(ωt₂)+sin²(ωt₁)) ] + A²cos(ωT)[(cos²(ωt₂)+cos²(ωt₁)) ] + 2AB cos(ωT)(sin(ωt₂)cos(ωt₂) + sin(ωt₁)cos(ωt₁)) - 2 x₁x₂ ) = (mω/2sin(ωT)) ( cos(ωT) [B²(sin²(ωt₂)+sin²(ωt₁)) + A²(cos²(ωt₂)+cos²(ωt₁)) + 2AB(sin(ωt₂)cos(ωt₂) + sin(ωt₁)cos(ωt₁)) - 2 x₁x₂ ] = (mω/2sin(ωT)) ( cos(ωT) [A²cos²(ωt₁) + 2ABsin(ωt₁)cos(ωt₁) + B²sin²(ωt₁) + A²cos²(ωt₂) + 2ABsin(ωt₂)cos(ωt₂) + B²sin²(ωt₂)] - 2 x₁x₂ ) = (mω/2sin(ωT)) [ cos(ωT) ( x₁² + x₂²) - 2 x₁x₂ ]
December 20, 2003 7:13 PM
| TrackBack