1549380323-Statistical Mechanics Theory and Molecular Simulation

24 Classical mechanics structure. Consider a solution xtto eqn. (1.6.26) starting from an initial condition x 0. Because the sol ...

Polymer model 25 =   0 1 −1 0   = M, (1.6.34) showing that the symplectic condition is satisfied. 1.7 A simple classical pol ...

26 Classical mechanics ηi=xi−xi 0 , (1.7.3) wherexi 0 −x(i+1)0=bi. The Hamiltonian in terms of the new variables and their conju ...

Polymer model 27 ω^2 kak=Aak. (1.7.9) Here,Ais a matrix given by A=ω^2      1 −1 0 0 0 ··· 0 0 −1 2 −1 0 0 ··· 0 0 0 −1 2 − ...

28 Classical mechanics ζk(t) =ζk(0) cosωkt+ pζk(0) mωk sinωkt k= 2,...,N, (1.7.15) whereζ 1 (0),...,ζN(0),pζ 1 (0),...,pζN(0) ar ...

The action integral 29 w 1 : w 2 : w 3 : Fig. 1.8 Normal modes of the harmonic polymer model forN= 3 particles. 1.8 The action i ...

30 Classical mechanics Q Q (Q(t 1 ), Q(t 1 )) (Q(t 2 ), Q(t 2 )) Fig. 1.9 Two proposed paths joining the fixed endpoints. The ac ...

The action integral 31 satisfy δQ(t 1 ) =δQ(t 2 ) = 0, δQ ̇(t 1 ) =δQ ̇(t 2 ) = 0. (1.8.3) The variation in the action is define ...

32 Classical mechanics particular initial condition, and in fact, initial conditions for Hamilton’sequations are generally chose ...

Constraints 33 1 2 ∑N i=1 mir ̇^2 i−C= 0, (1.9.3) whereCis a constant. Since constraints reduce the number of degrees of freedom ...

34 Classical mechanics ∑N i=1 1 2 mir ̇i· ( dri dt ) −C= 0 ∑N i=1 1 2 mir ̇i·dri−Cdt= 0 (1.9.7) so that a 1 i= 1 2 mir ̇i, a 1 t ...

Gauss’s principle 35 Note that, even if a system is subject to a set oftime-independentholonomic constraints (akt= 0), the Hamil ...

36 Classical mechanics the trajectory, the conditionσ(r) = 0 will be satisfied. The latter condition defines a surface on which ...

Rigid body motion 37 where the double dot-product notation in the expression∇∇σ··r ̇r ̇indicates a full contraction of the two t ...

38 Classical mechanics between the two hydrogens for a total of three holonomic constraint conditions. An am- monia molecule (NH ...

Rigid body motion 39 C Q S P O θ n r r’ Fig. 1.12 Rotation of the vectorrtor′about an axisn. a massm. We shall assume that the m ...

40 Classical mechanics there is a constraint in the form ofx^2 +y^2 =d^2. Rather than treating the constraint via a Lagrange mul ...

Rigid body motion 41 ω(lz=Iω), the angular velocity must also be a vector whose direction is along the z-axis. Thus, we write th ...

42 Classical mechanics inthe body-fixed frame. A similar relation can be derived for the time derivative of the positionr 2 of a ...

Rigid body motion 43 =− ∂U ∂θ . (1.11.23) Therefore, we see that the torque is simply the force on an angularcoordinate. The equ ...