Hamiltonian Formalism and Phase Space in Classical Mechanics

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 9/21/2015 PHY 711 Fall 2015 -- Lecture 12 1

9/21/2015 PHY 711 Fall 2015 -- Lecture 12 2

Hamiltonian formalism ( , ) t ) = ( ( , ) t H H q p t Canonical dq = equations H of motion dt p dp H = dt q 9/21/2015 PHY 711 Fall 2015 -- 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 9/21/2015 PHY 711 Fall 2015 -- 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 9/21/2015 PHY 711 Fall 2015 -- 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 9/21/2015 PHY 711 Fall 2015 -- 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. 9/21/2015 PHY 711 Fall 2015 -- 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 9/21/2015 PHY 711 Fall 2015 -- 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 9/21/2015 PHY 711 Fall 2015 -- 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 9/21/2015 PHY 711 Fall 2015 -- Lecture 12 10

Liouvilles theorem (x+ x,p+ p) (x,p+ p) p D x t (x+ x,p) (x,p) p x 9/21/2015 PHY 711 Fall 2015 -- 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 t time rate of change of particles within volume = time rate of particle entering minus particles leaving D D x p x p = 9/21/2015 PHY 711 Fall 2015 -- 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 9/21/2015 PHY 711 Fall 2015 -- 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 9/21/2015 PHY 711 Fall 2015 -- Lecture 12 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 9/21/2015 PHY 711 Fall 2015 -- Lecture 12 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 9/21/2015 PHY 711 Fall 2015 -- Lecture 12 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 9/21/2015 PHY 711 Fall 2015 -- Lecture 12 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. 9/21/2015 PHY 711 Fall 2015 -- Lecture 12 18

Modern usage of Lagrangian and Hamiltonian formalisms J. Chem. Physics 72 2384-2393 (1980) 9/24/2014 PHY 711 Fall 2014 -- Lecture 13 19

Molecular dynamics is a subfield of computational physics focused on analyzing the motions of atoms in fluids and solids with the goal of relating the atomistic and macroscopic properties of materials. Ideally molecular dynamics calculations can numerically realize the statistical mechanics viewpoint. Imagine N that the generalize coordinate d s ( represent ) q t atoms, L = each with spacial 3 t = coordinate U : s ( , ) t ) ( ( , ) t L q q T simplicity For assumed is it , that the potential interactio n a is sum of pairwise interactio ns : 9/24/2014 PHY 711 Fall 2014 -- Lecture 13 20

rij ( ) ( , ) ) i i 2 = = r r r r r ( ( ) L L t t m u 1 i i i i i j 2 j From this Lagrangian, can find the 3N coupled 2nd order differential equations of motion and/or find the corresponding Hamiltonian, representing the system at constant energy, volume, and particle number N (N,V,E ensemble). 9/24/2014 PHY 711 Fall 2014 -- Lecture 13 21

Lagrangian and Hamiltonian forms ( ) ( , ) ) 2 = = r r r r r ( ( ) L L t t m u 1 i i i i i j 2 i i j = p r m i i i 2 ( ) p 2 = + i r r H u i j m i i j i Canonical equations : ( ) r r r p p d d i j = = r r ' i i i u i j dt m dt r r i j i i j 9/24/2014 PHY 711 Fall 2014 -- Lecture 13 22

H. C. Andersen wanted to adapt the formalism for modeling an (N,V,E) ensemble to one which could model a system at constant pressure (P). P V constant P constant, V variable 9/24/2014 PHY 711 Fall 2014 -- Lecture 13 23

PV contribution to potential energy Andersen' clever tra s nsformatio n : = / 1 3 r Let / Q i i ( ) ( , ) ) 2 = = r r r r r ( ( ) L L t t m u 1 i i i i i j 2 i i j ( ) ( ) , ) 2 Q Q = = + / 2 3 / 1 3 2 ( ( , ) , L L t t Q Q m u Q M Q 1 1 i i i i i j 2 2 i i j kinetic energy of balloon 9/24/2014 PHY 711 Fall 2014 -- Lecture 13 24

( ) ( ) , ) i i 2 Q Q = = + / 2 3 / 1 3 2 ( ( , ) , L L t t Q Q m u Q M Q 1 1 i i i i i j 2 2 j L = = / 2 3 mQ i i i L Q = = M Q 2 ( ) 2 2 i i = + + + i / 1 3 H u Q Q i j / 2 3 2 m Q M j i M d dQ = = i i / 2 3 2 dt m Q dt i ( ) d i i j = / 1 3 / 1 3 ' i Q u Q i j dt j i j 2 ( ) dt 2 1 d i i = i / 1 3 ' u Q i j i j / 2 3 / 2 3 3 2 3 Q m Q Q j i 9/24/2014 PHY 711 Fall 2014 -- Lecture 13 25

Relationship between system representations Scaled Original ( ) ( ) i Q / Equations of motion in original coordinates: ( ) ( ) t = Q t V r t = 3 / 1 Q t i = 3 / 1 p i i r p 1 ln d d V = + r i i i 3 dt p m dt i ( ) r r 1 ln d d V j i j = r r p ' i u i j i 3 dt dt r r i i j ( ) 2 p p 1 2 1 d V 2 i j = + r r r r ' i m i M u i j i j 3 3 dt V i i 9/24/2014 PHY 711 Fall 2014 -- Lecture 13 26

Physical interpretation: Imposed (target) pressure ( ) p p 1 2 1 i j r r r r ' Internal pressure of system i m i u i j i j 3 3 V i i dependence Time ( ) 2 p p 1 2 1 d V 2 i j = + r r r r ' i m i M u i j i j 3 3 dt V i i 9/24/2014 PHY 711 Fall 2014 -- Lecture 13 27

Digression on numerical evaluation of differential equations Example differenti equation al dimension) (one ; 2 d x 2 ( ) = = = ( ) Let 3 , 2 , 1 f t t nh n dt ( ) ( ) Euler' method s ; x x nh f f nh n n : 1 = + + 2 x x hv h f + 1 n n n n 2 = + v v hf + 1 n n n Velocity Verlet algorithm : 1 = + + 2 x x hv h f + 1 n n n n 2 1 ( ) = + + v v h f f + + 1 1 n n n n 2 9/24/2014 PHY 711 Fall 2014 -- Lecture 13 28