Mechanics Canonical Transformation 3, Lecture Notes - Physics, Study notes of Mechanics

Download Mechanics Canonical Transformation 3, Lecture Notes - Physics and more Study notes Mechanics in PDF only on Docsity! Mechanics Physics 151 Lecture 22 Canonical Transformations (Chapter 9) What We Did Last Time Direct Conditions Necessary and sufficient for Canonical Transf. Infinitesimal CT Poisson Bracket Canonical invariant Fundamental PB ICT expressed by Infinitesimal time transf. generated by Hamiltonian Hamilton’s equations ,, ji j i Q Pq p pQ q P ⎛ ⎞ ∂⎛ ⎞∂ =⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠ ,, ji j i Q Pq p qQ p P ⎛ ⎞ ∂⎛ ⎞∂ = −⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠ ,, ji j i Q Pq p pP q Q ⎛ ⎞ ∂⎛ ⎞∂ = −⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠ ,, ji j i Q Pq p qP p Q ⎛ ⎞ ∂⎛ ⎞∂ =⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠ [ ], i i i i u v u vu v q p p q ∂ ∂ ∂ ∂ ≡ − ∂ ∂ ∂ ∂ [ , ] [ , ] 0i j i jq q p p= = [ , ] [ , ]i j i j ijq p p q δ= − = [ , ] uu u G t t δ ε δ∂= + ∂ Infinitesimal Time CT Infinitesimal CT We know that the generator = Hamiltonian Integrating it with time should give us the “finite” CT that turns the initial conditions q(t0), p(t0) into the configuration q(t), p(t) of the system at arbitrary time That’s a new definition of “solving” the problem ( ), ( )q t p t ( ), ( )q t dt p t dt+ + [ , ] udu dt u H dt t ∂ = + ∂ [ , ]q q H= [ , ]p p H= Hamiltonian is the generator of the system’s motion with time a iltonian is the generator of the syste ’s otion ith ti e Static vs. Dynamic Two ways of looking at the same thing System is moving in a fixed phase space Hamilton’s equations Integrate to get q(t), p(t) System is fixed and the phase space is transforming ICT given by the PB Integrate to get CT for finite t Equations are identical You’ll find yourself integrating exactly the same equations Did we gain anything? Conservation Consider an ICT generated by G Suppose G is conserved and has no explicit t-dependence How is H (without t-dependence) changed by the ICT? A transformation that does not affect H Symmetry of the system Generator of the transformation is conserved [ , ] uu u G t t δ ε δ∂= + ∂ [ , ] 0G H = [ , ] 0HH H G t t δ ε δ∂= + = ∂ If an ICT does not affect Hamiltonian, its generator is conserved Angular Momentum The generator is obviously i.e. the z-component of the total momentum Generator for rotation about an axis given by a unit vector n should be We now know generators of 3 important ICTs Hamiltonian generates displacement in time Linear momentum generates displacement in space Angular momentum generates rotation in space i iy i ixG x p y p= − ( )z i i zL = ×r p G = ⋅L n Integrating ICT I said we can “integrate” ICT to get finite CT How do we integrate ? First, let’s rewrite it as We want the solution u(α) as a function of α, with the initial condition u(0) = u0 Taylor expand u(α) from α = 0 [ , ]u u Gδ ε= [ , ]du d u Gα= [ , ]du u G dα = 2 2 3 3 0 2 3 0 0 0 ( ) 2! 3! du d u d uu u d d d α αα α α α α = + + + + This is [u,G]0 What can I do with these? Integrating ICT Since is true for any u, we can say Now apply this operator repeatedly Going back to the Taylor expansion, Now we have a formal solution – But does it work? [ , ]du u G dα = [, ]d G dα = 2 2 [ , ] [[ , ], ] d u d u G u G G d dα α = = [ [[ , ], ], , ] j j d u u G G G dα = 2 2 3 3 0 2 3 0 0 0 2 3 0 0 0 0 ( ) 2! 3! [ , ] [[ , ], ] [[[ , ], ], ] 2! 3! du d u d uu u d d d u u G u G G u G G G α αα α α α α α αα = + + + + = + + + + Free Fall An object is falling under gravity Hamiltonian is Integrate the time ICT 2 2 pH mgz m = + z 2 3 0 0 0 0( ) [ , ] [[ , ], ] [[[ , ], ], ]2! 3! t tz t z t z H z H H z H H H= + + + + [ , ] pz H m = [[ , ], ]z H H g= − [[[ , ], ], ] 0z H H H = 20 0( ) 2 p gz t z t t m = + − It’s easier than it looked Infinitesimal Rotation ICT for rotation is generated by We’ve studied infinitesimal rotation in Lecture 8 Infinitesimal rotation of dθ about n moves a vector r as Compare the two expressions Equation holds for any r that rotates together with the system Several useful rules can be derived from this G = ⋅L n d dθ= ×r n r [ , ]d d dθ θ= ⋅ = ×r r L n n r [ , ]⋅ = ×r L n n r [ , ]⋅ = ×r L n n r Scalar Products Consider a scalar product of two vectors Try to rotate it Obvious: scalar product doesn’t change by rotation Also obvious: length of any vector is conserved ⋅a b [ , ] [ , ] [ , ] ( ) ( ) ( ) ( ) 0 ⋅ ⋅ = ⋅ ⋅ + ⋅ ⋅ = ⋅ × + ⋅ × = ⋅ × + ⋅ × = a b L n a b L n b a L n a n b b n a a n b a b n [ , ]⋅ = ×r L n n r Mechanics Physics 151 Lecture 22 Canonical Transformations (Chapter 9) What We Did Last Time Direct Conditions Necessary and sufficient for Canonical Transf. Infinitesimal CT Poisson Bracket Canonical invariant Fundamental PB ICT expressed by Infinitesimal time transf. generated by Hamiltonian Hamilton’s equations ,, ji j i Q Pq p pQ q P ⎛ ⎞ ∂⎛ ⎞∂ =⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠ ,, ji j i Q Pq p qQ p P ⎛ ⎞ ∂⎛ ⎞∂ = −⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠ ,, ji j i Q Pq p pP q Q ⎛ ⎞ ∂⎛ ⎞∂ = −⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠ ,, ji j i Q Pq p qP p Q ⎛ ⎞ ∂⎛ ⎞∂ =⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠ [ ], i i i i u v u vu v q p p q ∂ ∂ ∂ ∂ ≡ − ∂ ∂ ∂ ∂ [ , ] [ , ] 0i j i jq q p p= = [ , ] [ , ]i j i j ijq p p q δ= − = [ , ] uu u G t t δ ε δ∂= + ∂ Infinitesimal Time CT Infinitesimal CT We know that the generator = Hamiltonian Integrating it with time should give us the “finite” CT that turns the initial conditions q(t0), p(t0) into the configuration q(t), p(t) of the system at arbitrary time That’s a new definition of “solving” the problem ( ), ( )q t p t ( ), ( )q t dt p t dt+ + [ , ] udu dt u H dt t ∂ = + ∂ [ , ]q q H= [ , ]p p H= Hamiltonian is the generator of the system’s motion with time a iltonian is the generator of the syste ’s otion ith ti e Static vs. Dynamic Two ways of looking at the same thing System is moving in a fixed phase space Hamilton’s equations Integrate to get q(t), p(t) System is fixed and the phase space is transforming ICT given by the PB Integrate to get CT for finite t Equations are identical You’ll find yourself integrating exactly the same equations Did we gain anything? Conservation Consider an ICT generated by G Suppose G is conserved and has no explicit t-dependence How is H (without t-dependence) changed by the ICT? A transformation that does not affect H Symmetry of the system Generator of the transformation is conserved [ , ] uu u G t t δ ε δ∂= + ∂ [ , ] 0G H = [ , ] 0HH H G t t δ ε δ∂= + = ∂ If an ICT does not affect Hamiltonian, its generator is conserved Angular Momentum The generator is obviously i.e. the z-component of the total momentum Generator for rotation about an axis given by a unit vector n should be We now know generators of 3 important ICTs Hamiltonian generates displacement in time Linear momentum generates displacement in space Angular momentum generates rotation in space i iy i ixG x p y p= − ( )z i i zL = ×r p G = ⋅L n Integrating ICT I said we can “integrate” ICT to get finite CT How do we integrate ? First, let’s rewrite it as We want the solution u(α) as a function of α, with the initial condition u(0) = u0 Taylor expand u(α) from α = 0 [ , ]u u Gδ ε= [ , ]du d u Gα= [ , ]du u G dα = 2 2 3 3 0 2 3 0 0 0 ( ) 2! 3! du d u d uu u d d d α αα α α α α = + + + + This is [u,G]0 What can I do with these? Integrating ICT Since is true for any u, we can say Now apply this operator repeatedly Going back to the Taylor expansion, Now we have a formal solution – But does it work? [ , ]du u G dα = [, ]d G dα = 2 2 [ , ] [[ , ], ] d u d u G u G G d dα α = = [ [[ , ], ], , ] j j d u u G G G dα = 2 2 3 3 0 2 3 0 0 0 2 3 0 0 0 0 ( ) 2! 3! [ , ] [[ , ], ] [[[ , ], ], ] 2! 3! du d u d uu u d d d u u G u G G u G G G α αα α α α α α αα = + + + + = + + + + Free Fall An object is falling under gravity Hamiltonian is Integrate the time ICT 2 2 pH mgz m = + z 2 3 0 0 0 0( ) [ , ] [[ , ], ] [[[ , ], ], ]2! 3! t tz t z t z H z H H z H H H= + + + + [ , ] pz H m = [[ , ], ]z H H g= − [[[ , ], ], ] 0z H H H = 20 0( ) 2 p gz t z t t m = + − It’s easier than it looked Infinitesimal Rotation ICT for rotation is generated by We’ve studied infinitesimal rotation in Lecture 8 Infinitesimal rotation of dθ about n moves a vector r as Compare the two expressions Equation holds for any r that rotates together with the system Several useful rules can be derived from this G = ⋅L n d dθ= ×r n r [ , ]d d dθ θ= ⋅ = ×r r L n n r [ , ]⋅ = ×r L n n r [ , ]⋅ = ×r L n n r Scalar Products Consider a scalar product of two vectors Try to rotate it Obvious: scalar product doesn’t change by rotation Also obvious: length of any vector is conserved ⋅a b [ , ] [ , ] [ , ] ( ) ( ) ( ) ( ) 0 ⋅ ⋅ = ⋅ ⋅ + ⋅ ⋅ = ⋅ × + ⋅ × = ⋅ × + ⋅ × = a b L n a b L n b a L n a n b b n a a n b a b n [ , ]⋅ = ×r L n n r Angular Momentum Remember the Fundamental Poisson Brackets? Now we know Poisson brackets between Lx, Ly, Lz are non-zero On the other hand, , so |L| may be a canonical momentum QM: You may measure |L| and, e.g., Lz simultaneously, but not Lx and Ly, etc. [ , ] [ , ] 0i j i jq q p p= = [ , ] [ , ]i j i j ijq p p q δ= − = PB of two canonical momenta is 0 [ , ]i j ijk kL L Lε= 2[ , ] 0iL L = Only 1 of the 3 components of the angular momentum can be a canonical momentum Phase Volume Static view: CT moves a point in one phase space to a point in another phase space Dynamic view: CT moves a point in one phase space to another point in the same space If you consider a set of points, CT moves a volume to anther volume, e.g. q p Q P dq dp How does the area change? Phase Volume Easy to calculate the Jacobian for 1-dimension i.e., volume in 1-dim. phase space is invariant This is true for n-dimensions Goldstein proves it using simplectic approach dQdP dqdp= M Q q Q p P q P p ∂ ∂ ∂ ∂⎡ ⎤ = ⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦ Mwhere [ , ] 1Q P P Q Q P q p q p ∂ ∂ ∂ ∂ = − = = ∂ ∂ ∂ ∂ M dQdP dqdp= Volume in Phase Space is a Canonical Invariant Ideal Gas Dynamics Imagine ideal gas in a cylinder with movable piston Each molecule has its own position and momentum They fill up a certain volume in the phase space What happens when we compress it? q p q p Compress slowly Extra momenta Gas gets hotter! Liouville’s Theorem The phase volume occupied by a group of particles (ensemble in stat. mech.) is conserved Thus the density in phase space remains constant with time Known as Liouville’s theorem Theoretical basis of the 2nd law of thermodynamics This holds true when there are large enough number of particles so that the distribution may be considered continuous More about this in Physics 181 Summary Introduced dynamic view of Canonical Transf. Hamiltonian is the generator of the motion with time Symmetry of the system Hamiltonian unaffected by the generator Generator is conserved How to integrate infinitesimal transformations Discussed infinitesimal rotation Angular momentum QM Invariance of the phase volume Liouville’s theorem Stat. Mech. 2 3 0 0 0 0( ) [ , ] [[ , ], ] [[[ , ], ], ]2! 3! u u u G u G G u G G Gα αα α= + + + + [ , ]⋅ = ×r L n n r