Hamiltonian Formalism in Classical Mechanics Lectures Summary
Explore the Hamiltonian formalism in classical mechanics through lectures covering topics such as phase space, Liouville's theorem, time evolution, Poisson brackets, and phase space diagrams for one-dimensional motion under different forces. Dive into the mathematical methods behind Hamiltonian mechanics for a deep understanding of this fundamental concept.
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PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 12: Continue reading Chapter 3 & 6 1. Hamiltonian formalism 2. Phase space 3. Liouville s theorem 9/22/2014 PHY 711 Fall 2014 -- Lecture 12 1
9/22/2014 PHY 711 Fall 2014 -- Lecture 12 2
Hamiltonian formalism ( , ) t ) = ( ( , ) t H H q p t Canonical dq = equations H of motion dt p dp H = dt q 9/22/2014 PHY 711 Fall 2014 -- Lecture 12 3
Hamiltonian formalism and time evolution: ( , ) ) = ( ( , ) H H q t p t t dq H = dt p dp H = dt q dH H H H H ( arbitrary an = + + = q p dt For q p t t , ) ( ) = function : ( , ) F F q t p t t dF F F F F H F H F = + + = + q p dt q p t q p p q t 9/22/2014 PHY 711 Fall 2014 -- Lecture 12 4
Hamiltonian formalism and time evolution: Poisson brackets: ( ) = arbitrary an For function : ( , ) ( , ) F F q t p t t dF F F F F H F H F = + + = + q p dt Define q p t q p p q t : F G F G = F,G G,F PB PB q p p q dF F = + So that : F,H PB dt t 9/22/2014 PHY 711 Fall 2014 -- Lecture 12 5
Poisson brackets -- continued: F G F G = F,G G,F PB PB q p p q Examples x,x : ,L = = = 0 = 1 0 x,p x,p PB x y PB PB L L x y z PB Liouville theorem density Let dD of particles D phase in space : D = + = 0 D,H PB dt t 9/22/2014 PHY 711 Fall 2014 -- Lecture 12 6
Phase space defined is space Phase at the set of all coordinate q momenta and s t p system a of : ( For dimensiona d ) ( , ) t ( ) a system l with particles, dN N correspond space phase the to s 2 degrees of freedom. 9/22/2014 PHY 711 Fall 2014 -- Lecture 12 7
Phase space diagram for one-dimensional motion due to constant force p x 2 p p m ( ) = = = , H x p F x p F x 0 0 2 m 1 2 p m = + = + + 2 ( ) ( ) p t p F t x t x t F t 0 i 0 0 0 0 i i i i 9/22/2014 PHY 711 Fall 2014 -- Lecture 12 8
Phase space diagram for one-dimensional motion due to spring force p x 2 1 2 p p m ( ) = + = = 2 2 2 , H x p m x p m x x 2 m p m ( ) ( ) = + = + ( ) cos ( ) sin p t p t x t t 0 i 0 0 0 i i i i i 9/22/2014 PHY 711 Fall 2014 -- Lecture 12 9
Liouvilles Theorem (1838) The density of representative points in phase space corresponding to the motion of a system of particles remains constant during the motion. ( , ) t ) = Denote density the q of particles phase in space : ( ( , ) t D D q p t dD D D D = + + q p dt p t dD = According Liouville' to theorem s : 0 dt 9/22/2014 PHY 711 Fall 2014 -- Lecture 12 10
Liouvilles theorem (x+ x,p+ p) (x,p+ p) p D x t (x+ x,p) (x,p) p x 9/22/2014 PHY 711 Fall 2014 -- Lecture 12 11
Liouvilles theorem -- continued (x+ x,p+ p) (x,p+ p) p D x t (x+ x,p) (x,p) p x D time rate of change of particles within vo lume t = time rate of particle entering minus particles leaving D D = x p x p 9/22/2014 PHY 711 Fall 2014 -- Lecture 12 12
Liouvilles theorem -- continued (x+ x,p+ p) (x,p+ p) p D x t (x+ x,p) (x,p) p x D D D = x p t x p D D D dD + + = = 0 x p t x p dt 9/22/2014 PHY 711 Fall 2014 -- Lecture 12 13
Review: Liouville s theorem: Imagine a collection of particles obeying the Canonical equations of motion in phase space. distributi " the denote Let p q q D D N = on" of particles phase in space : D ( , ) : , p t 1 3 1 3 N Liouville' dD = theorm s shows that 0 constant is D in time dt 9/25/2013 PHY 711 Fall 2013 -- Lecture 13 14
Proof of Liouvillee theorem: v D Continuity equation : t D ( ) = v D t v Note v = velocity the case, in this : p r r r N the is 6 dimensiona vector l : N ( also = ) p p , , , , , 1 2 1 2 N N We have 6 a dimensiona gradient l : ( ) , , , , , r r r p p p 1 2 N 1 2 N 9/25/2013 PHY 711 Fall 2013 -- Lecture 13 15
D ( ) = v D t q p ( ) ( ) 3 N = j = + q D p D j j 1 j j q p 3 3 N N D D = j = j j j = + + q p D j j q p q p 1 1 j j j j q p q p 2 2 H H j j + = + = 0 q p p q j j j j j j 9/25/2013 PHY 711 Fall 2013 -- Lecture 13 16
0 q p 3 3 N N D D D = j = j j j = + + q p D j j t q p q p 1 1 j j j j 3 = j N D D D = + q p j j t q p 1 j j 3 = j N D D D dD + + = = 0 q p j j t q p dt 1 j j 9/25/2013 PHY 711 Fall 2013 -- Lecture 13 17
dD = 0 dt Importance of Liouville s theorem to statistical mechanical analysis: In statistical mechanics, we need to evaluate the probability of various configurations of particles. The fact that the density of particles in phase space is constant in time, implies that each point in phase space is equally probable and that the time average of the evolution of a system can be determined by an average of the system over phase space volume. 9/22/2014 PHY 711 Fall 2014 -- Lecture 12 18
9/22/2014 PHY 711 Fall 2014 -- Lecture 12 19
Virial theorem (Clausius ~ 1860) = r F 2 T Proof : p r Define : A dA ( ) = + = + p r p r F r 2 T dt dA = + F r 2 T dt ( ) t ( ) ( ) 0 1 dA dA A A = = 0 dt dt dt 0 9/22/2014 PHY 711 Fall 2014 -- Lecture 12 20
