1.1 Mechanics The study of motion 2 Notes...mechanics apply to the motion on atomic and molecular scales is the working assumption in developing methods for molecular simulations.

Introduction to Molecular Simulation Chapter 1. Classical mechanics

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Chapter 1. Classical Mechanics and Numerical Methods

1.1 Mechanics – The study of motion Humans long ago observed motions of earthbound and celestial objects and intuitively

discovered that the motions of inanimate objects (“mechanics”) follow certain predictable

patterns. Without this realization pre-modern architects, astronomers, navigators, etc.

would not have been able to achieve many of their accomplishments which in many cases

are as impressive as modern ones. Indeed animals must observe and intuitively

understand the operation of laws of nature. Without this understanding, a hawk would not

know how steep and fast to dive to have a chance at catching a rabbit, or a gibbon would

not know how fast and what angle to pounce to reach the next branch of a tree high above

the forest floor.

A great discovery of modern science is that mathematical laws governing

mechanics quantitatively determine how positions and velocities of objects change with

time and how they are affected by forces. The great insight of Sir Isaac Newton in

discovering the laws of mechanics was that the same mathematic principles which apply

to motion of objects on earth, which move within distance scales of 1 to 100s of meters

and times scales of seconds to hours, also apply to the motion of celestial objects like the

moon, earth, and sun, which move on distance scales of 108 to 10

11 meters and time

scales in the range of hours to years. Limiting ourselves to motions encountered on earth

and objects within the solar system, the applicability of the Newton’s laws of mechanics

spans a 1011

range of distances and a 109 range of times.

Over time scientists became familiar with the structure of matter and discovered

that the atomic and molecular building blocks of materials have sizes in the range of 10–9

to 10–7

meters. The motions of these molecules would occur on time scales much shorter

than seconds. The question naturally arose whether on the opposing end of small length

and time scales, it is possible to assume that the same mechanical laws which govern

human scale motions, also govern the motion of molecules in solid, liquid, and gas which

occur on length scales of 10–9

to 10–7

m. That indeed (with caveats) the laws of classical

mechanics apply to the motion on atomic and molecular scales is the working assumption

in developing methods for molecular simulations.1

In classical molecular simulations the laws of mechanics are applied to predict the

motions and energies of molecules under different external thermodynamic conditions. In

molecular systems, the positions and velocities of atoms and the nature and magnitude of

forces acting on atoms depend on the chemical structure, temperature, and pressure of the

simulated system. The mechanical approach can be used to study diverse phenomena,

such as a solvated protein interacting with a drug substrate, a DNA molecule in a saline

solution, an organic material adsorbing on the surface of a solid, or a solid material

undergoing a melting transition.

The mechanical laws governing the positions, velocities, and forces between

molecules at different times are expressed as differential equations. The particular form

of the differential equations, and the meaning of the mechanical variables themselves,

depend on whether classical or quantum mechanics are used to describe the system. Most

of our focus is on the classical mechanical description, but parallel quantum mechanical

1 These statements, of course, neglect effects related to relativity and quantum mechanics.


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descriptions for the motions of molecules can be formulated and will be occasionally

discussed.

We begin this first chapter by reviewing analytical solutions of some simple

systems using classical Newtonian mechanics in Section 1.1 and 1.2. These systems serve

to introduce some of the concepts and notation used later in the chapter and throughout

this book. While the systems described are macroscopic, they serve as models for

describing atomic and molecular motions in later chapters. An introduction of numerical

computation techniques, namely the finite-difference (Euler) method and more

sophisticated Verlet and leapfrog methods to solve Newton’s equations of motion follow

in Sections 1.3 and 1.4. These methods form the core of any molecular dynamics

simulation and all further developments are constructed on the foundation of these

numerical methods. The numerical solution of the harmonic oscillator is discussed in

detail in Section 1.5. The generalization of the mechanical ideas to many-atom systems is

briefly discussed in Section 1.6. Finally, in anticipation of their use in developing

molecular dynamics simulation methods, the Lagrangian and Hamiltonian formulations

of mechanics are introduced in the last section of this chapter. These formulations are

alternatives to Newton’s laws of motion which are much more appropriate for linking

mechanical motions of molecules in the system to the external environment in a way that

satisfies the laws of thermodynamics and statistical mechanics.

1.2 Classical Newtonian Mechanics The three laws that govern the motion of macroscopic objects moving at low speeds

compared to the speed of light where first stated together by Isaac Newton in 1687. These

laws are: 1) Any object moves in a straight line with constant speed (i.e., with constant

velocity) unless acted on by a force; 2) The acceleration (change of velocity with time) of

the object is proportional to the force acting on it, the proportionality constant is the mass

of the object. This law is summarized in the vector formula F = ma. If more than one

force acts on the object, the vector sum of the forces determine the acceleration; 3) For

each force on an object, the object exerts a force of equal magnitude pointing in the

opposite direction.[Young, Halliday, Fowles] Newton’s laws do not specify if and how

the forces depend on the position of the object or time and this is the subject of additional

empirical observation and analysis of the motion. In actuality, the force laws for any

specific type of interaction (gravity, mass connected to a spring, electro-magnetic

interactions, etc.) are devised so that the laws of motion are satisfied.

In systems with many interacting molecules, Newton’s three laws of motion give a

set of equations which describe the time dependence of position, ri(t), velocity, vi(t) [or

momentum pi(t) = mivi(t)], and force Fi(t) [or equivalently, the acceleration ai(t)] for all

atoms i. Other mechanical quantities for each atom and molecule, such as energy and

angular momentum, can be calculated from these fundamental mechanical variables at

any time as needed.

In most mechanical systems, the force on an object varies with its position and

proximity to other objects. In these cases, velocities and forces vary dynamically and the

simple algebraic formula F = ma does not suffice to determine the motion of the

constituting particles in the system over all times. Newton invented the calculus of

infinitesimals to predict motions in cases of position-dependent forces, but as we will see,


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he was also the first to suggest to what amounts to a numerical algebraic method to deal

with this problem of position dependent forces.

In modern notation, Newton’s second law of motion is written as a set of

differential equations, second order in time, the solutions of which give the time variation

of Cartesian coordinates. For the xi, yi, and zi components of the position vector ri of atom

i in an N-atom system, Newton’s second law is written as,

i

iiz

ii

i

iiy

ii

i

iix

ii

z

UF

dt

zdm

y

UF

dt

ydm

x

UF

dt

xdm

}{

}{

}{

,2

2

,2

2

,2

2

rr

rr

rr

(1.1)

Fi({r}) is the force vector on atom i, which can depend on the set {r} of positions of all

other atoms in the system. The positions and velocities of different atoms are coupled

through the forces acting between them.

In Eq. (1.1) forces are written in terms of partial derivatives of the scalar potential

energy, Ui({r}) of atom i with respect to its three position components. This is

convenient since in many cases, the mathematical form of the potential energy function is

more readily determined than the force.

For a system with N-atoms, Newton’s equations of motion give a set of 3N coupled

second-order differential equations. These equations can be solved by analytical methods

(very rarely) or using numerical methods (most of the time), the latter of which are the

main focus of molecular dynamics methodology. Solutions of these coupled equations

give the time dependence of the set of coordinates {r(t)} and velocities {v(t)} [or

momenta {p(t)}] for all atoms in the system. To determine a unique solution for the

positions and velocities, 6N initial conditions for the coordinates and velocities of all

atoms at time t = 0 are required. The positions of the atoms at different times, {r(t)}

constitutes the spatial trajectory (orbit) of the system. The sets of {r(t)} and {p(t)} at

different times constitute the phase space trajectory of the system.

For a limited number of low-dimensional systems where the coupled equations of

motion are separable, Newton’s equations can be solved analytically to give a closed-

form solution of the spatial and phase space trajectory. In Section 1.2, solutions to

Newton’s equations of motion for some simple mechanical systems are reviewed and the

concept of phase space is introduced. The phase space trajectory of a system is important

in describing the mechanics of many-atom systems and plays a central role in statistical

mechanics and its application to molecular dynamics simulation methodology.

1.3 Analytical solutions of Newton’s equations and phase space 1.2.1 Motion of an object under constant gravitational force

A mechanical system studied by Newton (and Galileo Galilei among others before him)

was the motion of an object near the Earth’s surface where there is a constant

gravitational acceleration of a = g = –9.8 m/s2 pointing towards the center of the Earth,

see Figure 1.1(a). Newton’s equation of motion for a mass thrown perpendicularly

upwards (in the positive y-direction) in the Earth’s gravitational field is,


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mgFdt

ydm y

2

2

. (1.2)

Starting at an initial position y(0) and initial velocity of vy(0) at a time t = 0, integrating

this equation once with respect to time and using the initial conditions, gives the time

variation of the velocity of the particle,

)0()( yy vgttvdt

dy . (1.3)

Integrating Eq. (1.3) with respect to time, gives the time variation of the position,

)0()0(2

1)( 2 ytvgtty y (1.4)

The spatial trajectories for a mass at two sets of initial conditions y(0) and vy(0), and the

time dependence of the momentum are shown in Fig. 1.1(b), and the corresponding phase

space trajectories are shown in Figure 1.1(c).

The potential energy of a mass in the Earth’s gravitational field at any time is,

( ) [ ( ) (0)]U t mg y t y . (1.5)

The gravitational potential energy near the surface of the Earth increases linearly with

position above the reference point, y(0), usually taken to be the surface of the Earth. The

total mechanical energy which is the sum of kinetic and potential energies of the mass

during any time t of its motion is,

)]0()([)(2

1)( 2 ytymgtmvtE . (1.6)

Substituting the velocity and position from Eqs. (1.3) and (1.4), respectively, into Eq.

(1.6), shows that for a particular trajectory, the total energy is constant at all times and

depends on the initial conditions through the value of the parameters y(0) and vy(0),

21( ) (0) (0) .

2yE t mv mgy const (1.7)


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Figure 1.1. (a) The coordinate system for a mass moving under the influence of constant

gravitational acceleration. (b) The time dependence of the position and momentum for a particle

of mass 1 kg starting at y(0) = 0, thrown upwards with initial speed of v(0) = 10 m/s (full lines)

and 5 m/s (dashed line). (c) Two y-py phase space trajectories for the motions in part (b). All

points in the y-py phase plane are covered by trajectories which are determined by the initial

conditions of the motion. Two “states” corresponding to volume elements dxdp in the phase space

are shown in (c).

The phase space trajectory or streamline of the projectile is determined by explicitly

eliminating time from Eqs. (1.3) and (1.4),

22 2 0

2

( )1( ) (0) ( ) (0)

2 2

yy y

p t Ey t y v t v

g mgm g

. (1.8)

For the motion of a mass in a constant gravitational field, the coordinate y, and the

“conjugate” momentum py form the phase space of the mechanical system. Trajectories

for two sets of initial conditions, or more fundamentally for two specific energy values,

are shown in Figure 1.1(c). Time does not enter the phase space description but implicitly

determines the direction of motion along the trajectory.

The concept of phase space is used extensively when discussing statistical

mechanics. For a one-dimensional motion of a single mass in the y-direction, phase space

is two-dimensional and consists of coordinate and its conjugate momentum, {y, py}. For a

three-dimensional motion of a single mass, the phase space is six dimensional and

consists of coordinate-momentum pairs, {x, y, z, px , py , pz}.


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The “state” in phase space is determined by the volume element about each phase

space point. For example, in the one dimension motion mentioned above, a state is a

volume element dydpy about the point {y, py} shown in Fig. 1.1(c). States on the same

phase space trajectory all have the same energy. All points in phase space correspond to

states which belong to a unique trajectory. As we shall see, the smallest phase space

volume element (state) is determined by the Heisenberg uncertainty principle, dx∙dpx = h.

1.3.2 One-dimensional harmonic oscillator

One of the most important mechanical systems in physics and chemistry is the harmonic

oscillator which describes the motion of a mass m connected to a spring governed by

Hooke’s law (1660, after the English scientist Robert Hooke), F = –k(x – x0). In the

harmonic oscillator, the force is linearly proportional to the displacement ξ = x – x0 of the

mass from a relaxed position x0 and points in the direction opposite the displacement and

towards the relaxed position x0. The force constant of the spring, k, determines the

“stiffness”, i.e., how much force must be exerted to extend or compress the spring by a

unit length. Note that a harmonic spring behaves symmetrically with respect to extension

or compression. The potential energy of the spring is a quadratic function of the

displacement, U = ½k(x – x0)2. These relations are shown in Figure 1.2(a).

The one-dimensional single-mass harmonic oscillator can also represent the relative

motion of two masses connected by a spring with force constant k, see Appendix A1.1.

For the one-dimensional harmonic oscillator, Newton’s second law is written as,

)]0([2

2

xxkFdt

xdm x . (1.9)

This equation is simplified by using the displacement, ξ, as the variable, and defining the

angular frequency ω = (k/m)1/2

to give,

022

2

dt

d. (1.10)

Equation (1.10) is a homogenous second-order differential equation with constant

coefficients.[Boyce] The general solution of Eq. (1.10) gives the time dependence of the

displacement (t) as a sum of complex exponential functions, or equivalently as a sum of

sine and cosine functions,

1 2 1 2cos( ) sin( ) sin( )i t i tc e c e C t C t A t . (1.11)

In the final form, the parameters A and represent the amplitude and phase of the

motion, respectively. These solutions can be verified by substituting Eq.(1.11) into

Eq.(1.10). Pairs of constants (c1,c2), (C1,C2), or (A, ) characterize the specific trajectory

of the mass. The sinusoidal motion in the last expression in Eq. (1.11) gives the

harmonic oscillator its name. The frequency and period of the harmonic oscillator are =

/2π and τ = 1/, respectively. The time-dependence of the velocity of the mass is

calculated from the time-derivative of Eq. (1.11),

)cos()( tAdtdtv . (1.12)

The constants (A,) are determined by the two initial conditions, namely the values of the

initial displacement (0) and velocity v(0) at t = 0.


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Figure 1.2 (a) The quadratic potential energy function and linear force function for a one-

dimensional harmonic oscillator with an angular frequency ω = 5 s-1

. In the rest state, the mass is

at ξ = x – x0 = 0. (b) The time dependence of the displacement ξ and momentum for a particle of

mass 1 kg with ξ(0) = 0 and pξ(0) = 1.0 kgm/s (full line) and 0.5 m/s (dashed line). (c) The ξ-pξ

ellipses characterizing the phase space trajectory of the harmonic oscillator. The major and minor

axes of the ellipse depend on the initial conditions of the spring. All points in the ξ-pξ phase space

are covered by trajectories. For a specific spring and mass, initial conditions determine which

elliptical trajectory passes through a point in phase space.

As an example of a specific trajectory, consider a mass, m = 1 kg connected to a

harmonic spring that gives it an angular frequency of = 5 s-1

. If initially the mass is at

x(0) = x0 (i.e., (0) = 0) and has an initial velocity v(0) = 1.0 m/s, the specific solutions of

the harmonic oscillator, Eqs. (1.11) and (1.12) are )sin(2.0)( tt and )cos()( ttv

, respectively, shown in Figure 1.2(b). A second trajectory with the initial conditions, (0)

= 0 and v(0) = 0.5 m/s is also shown in this figure.

The total mechanical energy of the harmonic oscillator system is a sum of the

kinetic and potential energies determined using Eqs. (1.11) and (1.12),

E(t) = K(t) + U(t) = mv(t)2/2 + k(t)

2/2 = (m2

A2 + kA

2)/2 = kA

2. (1.13)

The total energy is constant and depends on the initial conditions through the amplitude

parameter A.

Eliminating the time variable between Eqs. (1.11) and (1.12) gives the phase

space trajectory of the harmonic oscillator,

1)(

)(

)()()()()(2

2

2

222

2

2

2

22

2

2

2

mE

tp

mE

t

Am

tp

A

t

A

tv

A

t

. (1.14)

which is an ellipse in {, p} phase space. The trajectories for two different initial

conditions corresponding to different energy values are shown in Figure 1.2(c). Each

state in phase space is represented by a volume element ddp around the point {, p}


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and is associated with a unique trajectory. Note that the phase space trajectory of each

mechanical system is a reflection of the specific nature of the forces, or more exactly, the

“Hamiltonian” of the system, see below.

1.3.3 Radial force functions in three-dimensions

Determining the spatial and phase space trajectories of a mass subject to a radially

directed force proportional to 1/r2 requires considerably greater of mathematical effort.

This force describes the motion of particles interacting with gravitational and electrostatic

forces. For radial 1/r2 forces, Newton’s second law is,

2/3222

2

2/3222

2

32

2

)(

)(

yx

yK

dt

ydm

yx

xK

dt

xdm

r

KF

dt

dm

rr

r (1.15)

Details of the analytical solution of Newton’s equations of motion for these cases are

given in Appendix A1.2 where we prove that the motion of the mass subject to a radial

force in Eq. (1.15) remains confined to the xy-plane.[Synge, Fowles] The two

equations in Eq. (1.15) cannot be solved directly in the Cartesian coordinate system,

however, they can be solved after transformation to polar coordinates {r(t), θ(t)}. The

spatial trajectory or orbit of the motion of the mass in polar coordinates is,

)(cos

211

)(

1

2

2

2t

mK

EmK

tr

(1.16)

where ℓ is the angular momentum of the mass with respect to the origin r = 0, see Eq.

(A1.12), and E is the energy determined by initial conditions of the motion. See

Appendix A1.2 for full details. For non-zero angular momenta (i.e., where the mass is not

moving head-on towards the center of force), the orbit is elliptic if E < 0, parabolic for E

= 0, and hyperbolic for E > 0. These three cases are shown in Figure 1.3. States with E <

0 orbit around the origin (one of the foci of the ellipse) and for obvious reasons are called

bound states.


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Figure 1.3. The spatial orbit of a mass moving in a 1/r2 force field (1/r potential energy

function) in three energy regimes, E < 0 (elliptic motion), E = 0 (parabolic motion), and E > 0

(hyperbolic motion). Coordinates are plotted in reduced units as x* = xℓ2/mK and y* = yℓ

2/mK.

The radial force has a four-dimensional Cartesian {x, y, px, py} or polar {r, θ, pr, pθ}

phase space. For each trajectory, the angular momentum pθ = ℓ is constant and so with

known constant E and ℓ values (determined from initial conditions), the motion is limited

to a two-dimensional hypersurface in phase space. The phase space state of this system is

determined by a volume element dxdydpxdpy (or the corresponding volume element in

polar coordinates) about each point which is on a unique phase space trajectory. Similar

to spatial trajectories, phase space trajectories indicate whether the system is bound or

free.

Similar to gravitation, electrostatic forces have 1/r2 distance dependence and

although the study of planetary motion is more familiar, the trajectories mentioned above

are also observed in molecular systems as a result of electrostatic attractions. For

electrostatic repulsions, only positive energy trajectories are possible.

The harmonic oscillator is model for interactions between bound atoms within a

molecule. As discussed in detail in Chapter 2, van der Waals intermolecular forces and

corresponding potential energy functions between non-bonded atoms (in different

molecules or distant atoms on the same molecule), require a different mathematical form.

The Lennard-Jones potential is a commonly used function form for van der Waals

interactions between non-bonded atoms,

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12

12

1212 4

rrrU

. (1.17)

In Eq. (1.17), r12 = (r1 – r2)1/2

is the radial distance between two atoms, ε is the depth of

the potential well and σ is the distance at which the potential becomes zero, see Figure

1.4(a). The Lennard-Jones potential includes contributions from a repulsive term,

(σ/r12)12

, which dominates at short distances and an attractive term, –(σ/r12)6, which

dominates at large distances, see Chapter 2. In Appendix A1.1 it is shown that the relative

motion of two atoms interacting with a Lennard-Jones potential can represent the motion

of single object with a reduced mass as a function of the relative distance r = r12. The

Lennard-Jones potential is often written in terms of the reduced distance r* = r/σ and

potential energy ]*)1(*)1[(4*)(* 612 rrUrU . The Lennard-Jones force is,

])()(2[)24()()( 713 rrrrUrF . The force is negative (attractive)

when r > and positive (repulsive) when r < .

An object moving in regions of the Lennard-Jones potential phase space where the

total system energy is negative, is in a bound state and undergoes anharmonic oscillatory

motion. Unlike the harmonic oscillator, the Lennard-Jones potential decays to zero as the

mass moves away from the origin of force. As the mass moves inwards to distances less

than the potential minimum, the repulsion increases sharply with a slope significantly

greater than a harmonic spring. The Lennard-Jones potential and the corresponding force

are plotted in Figure 1.4(a).


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Figure 1.4. (a) The reduced Lennard-Jones potential, U*=U/ε and the corresponding reduced

force F*=F/(24ε/σ). (b) The reduced position (r* = r/σ) and momentum, p* = p/(εm)1/2

as a

function of reduced time, t* = t/[σ(m/ε)1/2

], for trajectories with head-on collisions with initial

positions r0* = 0.95, 1.005, 1.05, 1.1, 1.2 and 0 initial momentum. Note the anharmonic nature of

the oscillations. (c) The phase space trajectories of the Lennard-Jones potential for the initial

conditions given in part (b). The closed phase space trajectories of the bound states are not

elliptical.

The one-dimensional motion of a mass subject to the Lennard-Jones potential

placed at five initial placements of the mass at positions near r*(0) = r(0)/σ = 1, with

initial momentum of p*(0) = 0 are shown in Figure 1.4(b). Unlike the harmonic

oscillator, the motion is not symmetric with respect to stretching and compression about

the minimum of the potential. For the trajectory with initial position at r*(0) = 1.10 which

is near the minimum of the Lennard-Jones potential ( *

minr = 1.122), the change of the

position and velocity with time are approximately harmonic and position-momentum (r-

pr) phase space trajectories are close to an ellipse in shape, see Fig. 1.4(c). For initial

displacements r*(0) away from the minimum of the potential, the motion becomes non-

sinusoidal and the trajectory deviates from the elliptic shape; see for example, the r*(0) =

1.005 trajectory in Figure 1.4(c).

The states with r*(0) > 1 have a net negative energy and give rise to “bound states”

with closed phase space trajectories. In these case, the mass remains in the vicinity of the

force center at all times. “Free state” trajectories with r0* < 1 have a net positive energy,

with non-periodic motion and trajectories in phase space which are not closed curves, see

corresponding trajectories for r0* = 0.95 in Fig. 1.4(b) and 1.4(c).

For systems with masses interacting with Lennard-Jones or other realistic potential

energy functions, analytical solutions of Newton’s equations are difficult to determine or

unavailable. The fact that the positions, velocities, and forces on the molecules obey


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Newton’s equations of motion is sufficient to allow the use numerical methods to

calculate the trajectory. Indeed, the trajectories shown in Figures 1.4(b) and 1.4(c) are

calculated for a one particle head-on (one-dimensional) collision in a Lennard-Jones

potential using numerical methods discussed in the next section.

The Lennard-Jones potential is a radial potential and motion of a mass subject to

this potential has constant angular momentum. The phase space trajectories shown in

Figure 1.4(c) are for head-on, one dimensional collisions with zero angular momentum,

pθ = 0.

1.3.4 Motion under the influence of a drag force

A mass moving in air is subject to air resistance or drag which imparts a frictional

decelerating force to its motion.[Bradbury,Fowles] If the mass is not moving at too great

a speed, the drag force is considered to be proportional to its speed. The drag force

defined in this manner is non-conservative, meaning that it cannot be expressed as a

derivative of a corresponding potential. The drag force is actually the result of the

collision of the object with molecules in the atmosphere. This leads to energy exchange

between the object and the atmosphere and the speed dependent drag force is an

approximate way of capturing this effect. The energy of the moving object in isolation is

not conserved when it moves under a drag force.

For the object moving perpendicular to the surface of the earth under the operation

of a velocity-dependent atmospheric drag force, Newton’s second law is,

2

2 y yd y dy

m F mg bv mg bdtdt

. (1.18)

The positive constant b is called the drag coefficient. For an object like a parachute with a

large cross section, the drag coefficient is large, but for a compact object like a ball

bearing, the drag coefficient may be relatively small. Equation (1.18) is a second order

ordinary differential equation with constant coefficients which can be solved by

rearranging and multiplying both sides by exp(bt/m), [Boyce]

2

2

bt m bt m bt md y b dy d dye e ge

m dt dt dtdt

. (1.19)

Integrating both sides of the Eq. (1.19) from the initial time t = 0 where the speed is vy(0)

gives,

( ) (0) 1bt m bt my y

mgv t v e e

b

. (1.20)

At long times, the velocity approaches the so-called the terminal velocity mg/b. In this

long time limit, gravitation and drag forces cancel each other and the mass moves with

constant velocity in air. The quantity τ = m/b determines the time scale it takes for the

mass to reach the terminal velocity. For a mass originally at y(0), integrating Eq. (1.20)

gives the time dependence of the position of the falling mass,

( ) (0) (0) 1bt my

mg m mgy t y v e t

b b b

(1.21)


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The time dependence of the position and speed of the object are shown in Fig. 1.5(a).

Figure 1.5. (a) The time variation of the position and velocity of a mass dropped from a reduced

height yb2/gm

2 = 100, with an initial velocity vyb/mg = 0. After an initial period of increasing

velocity, the mass approaches a terminal velocity. (b) The phase space trajectory of the mass as it

is released from a height above the surface of the earth and falls downwards.

In this case, since the force is non-conservative, an explicit expression for the

potential energy for motion under the influence of the drag force cannot be written.

However, we can gain insight into the behavior of the system by first multiplying both

sides of Eq. (1.18) by vy = dvy/dt,

2

2

2 y y yd y

m v mgv bvdt

, (1.22)

which can be written as,

2

2

2

1ybvmgy

dt

dym

dt

d

. (1.23)

The term in brackets is the energy of the object falling in the absence of the drag force. If

we identify the brackets with the energy of the object, E, the change of energy of the

object as it falls is,

2y

dEbv

dt . (1.24)

With a positive constant b, the mechanical energy of the isolate moving object decreases

as it falls. The change in mechanical energy is converted to heat in the object and the

atmosphere medium in which the object is falling. [Bradbury] Non-conservative forces

such as the drag force often summarize complex interactions of the system with the

environment in relatively simple force terms.

The phase space trajectory of an object moving under a drag force is shown in Fig.

15(b). Unlike the previous cases discussed in Sec. 1.2, the energy of the object along the

phase space trajectory is not constant.

An application of motion subject to drag forces in molecular simulations is given in

Chapter 6 where the coupling of a molecular system to its environment for reaching

temperature and/or pressure equilibrium is effectively modeled as a type of drag force.


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1.8 The Lagrangian and Hamiltonian formulations of classical mechanics

Newton’s laws in Cartesian coordinates are intuitive and but as seen in the case of

radially symmetric forces, these forms of the equations are not always straightforward to

work with when seeking analytical solutions for the trajectories. In these cases, the

Cartesian coordinates in the equations of motion are transformed to use the knowledge of

the symmetry in the forces to simplify the analysis.

In contrast, the Lagrangian and Hamiltonian formulations of classical mechanics

can be written for any coordinate system. These approaches furthermore incorporate

energy concepts into the formulation of the equations of motion which makes their

analysis straightforward to interpret and physically insightful.[Fowles,Berendsen 2007]

Additionally, the Lagrangian and Hamiltonian formulations allow the introduction of

“non-conservative” forces which couple the mechanical system to external bodies. These

applications are important when implementing thermostats and barostats in molecular

simulations.

The Lagrangian function, L, is defined as the difference between the kinetic, K, and

potential energy U of a system. The Lagrangian is written in terms of coordinates, q, and

their conjugate velocities, q of the particles in the system. The “dot” notation on the

coordinate to represent velocity was first introduced by Newton and emphasizes the

relation between a coordinate and the velocity corresponding to the time derivative

(“fluxion”) of that coordinate. In this notation, the Lagrangian is given as,

})({2

1ii iii UmUKL qqq (1.42)

The coordinates and velocities in Eq. (1.42) can be expressed in Cartesian coordinates or

any other system such as spherical or cylindrical coordinates. The Euler–Lagrange

equations for each particle i with mass mi are a generalization of Newton’s equations of

motion using the Lagrangian function,

0

ii

LL

dt

d

qq. (1.43)

For simple conservative mechanical systems where the potential only depends on the

coordinates, the Euler-Lagrange equations reduce Newton’s equations of motion. For

example, the Lagrangian for the one-dimensional harmonic oscillator is,

22

2

1

2

1kxxmL . (1.44)

Substituting this Lagrangian in Eq. (1.42) gives Newton’s second law for this system.

An important advantage of the Euler-Lagrange equations is that they are valid for

mechanical variables q and q in any coordinate system. As illustrated in Appendix A1.2

for the case of radial forces, the Newtonian equations of motion in the form of a = F/m

are only strictly valid for Cartesian coordinates. In other coordinate systems, additional

“force” terms related to the change of coordinate system may be introduced in Newton’s

equations.

In the Hamiltonian approach, first introduced by the Irish physicist Sir Rowan

Hamilton in 1835, the dynamics are described in terms of the coordinates and their

conjugate momenta in any coordinate system. The momentum pi, conjugate to a

coordinate qi is defined as,


14

i

iL

qp

(1.45)

The Hamiltonian function for a system is the sum of kinetic and potential energy, written

in terms of coordinates qi, and their conjugate momenta, pi,

)}{()}{(),( iiii UKH qpqp . (1.46)

Mathematically, the Hamiltonian function is a Legendre transformation of the Lagrangian

function as defined by,

),(),(),( iii iiiiii

iii LLL

H qqpqqqq

qqp

, (1.47)

but this connection is not further explored here.[McQuarrie] The Hamiltonian function

represents the total energy of a system and is widely used in mechanical analyses.

Hamiltonian equations of motion are a set of two first-order differential equations in time,

i

ii

i

ii

H

dt

d

H

dt

d

q

pp

p

qq

(1.48)

Similar to the Euler-Lagrange equation, for cases where the potential is conservative (and

the force is the negative derivative of the potential), Hamilton’s equations of motion

reduce to Newton’s equations of motion. The Hamiltonian formulation is the basis of

describing the phase space trajectory in terms of the positions and momenta.

As an illustration of the use of Hamilton’s equations of motion, the momentum for

the one-dimensional harmonic oscillator is,

xmL

pi

i

, (1.49)

which gives the Hamiltonian for this system as,

2

2

12

2

k

m

pH . (1.50)

Hamilton’s equations of motion for the harmonic oscillator thus are,

kdt

dpp

m

p

dt

d

(1.51)

These two equations may be solved separately to give the time variation of the coordinate

and momentum, as given in Eqs.(1.11) and (1.12). The structure of the Hamiltonian

equations gives direct insight into the dynamical nature of the motion. Eliminating time

between the two equations in Eq. (1.51) directly gives the phase space trajectory of the

harmonic oscillator,

.22 constmkpdkmdpp

dtkdp

dtm

pd

(1.52)

The last equation is the elliptical trajectory in the two-dimensional phase space of the

dynamic variables ξ and its conjugate momentum pξ shown in Fig. 1.2. This example


15

illustrates the utility of the Hamiltonian approach for extracting information about the

phase space trajectory of a system without having to solve for the time dependence of the

individual coordinates and momenta first.

* * *


16

Appendices

Appendix A1.1 Separation of motion in two-particle systems with radial forces

If two objects interact with a force that depends only on the distance between the two and

not their relative orientations, the equations of motion describing the system can be

separated into two one-particle equations. The two masses connected by a harmonic

spring or interacting with gravitational, electrostatic, and Lennard-Jones potentials are

examples of these types of motion. The proof is elementary and follows writing the

Newton’s second law for the two particles in Cartesian coordinates. Consider two

particles at positions r1 and r2,

211221212

22

2

21122

12

1

rrrrr

rrr

FFdt

dm

Fdt

dm

(A1.1)

The center of mass position, Rcm, and relative coordinates, rrel, for the two particles are

defined as,

21

21

2211

rrr

rrR

rel

cmmm

mm

(A1.2)

The relation between the two set of coordinates is shown in Figure A1.1.

Figure A1.1. The coordinate system transformation from r1 and r2 to Rcm and r12 = rrel. The

position of the center of mass depends on the relative mass of the two particles.

The Cartesian position vectors of the two particles in terms of the center of mass

and relative positions are,

relcm

relcm

mm

m

mm

m

rRr

rRr

21

12

21

21

(A1.3)

Substituting the second derivatives of Rcm and rrel from Eq. (A1.3) into Eq.(A.1.1) gives

two new equations of motion for the center of mass and relative motion,


17

relrelrel

cm

FFmm

mm

dt

ddt

d

rrr

R

12

12

12

21

21

2

2

2

2

1

0

(A1.4)

In the new coordinate system, the motion of the center of mass and relative coordinates

are no longer coupled. The center of mass moves with no acceleration (constant velocity)

and the equation of motion for the rrel has the same functional form as the Cartesian

equations of motions in Eq. (A.1.1), but in terms of the relative coordinate and reduced

mass μ = m1m2/(m1+m2).

The one-particle systems described in the Secs. 1.2.2 and 1.2.3 apply to the relative

motion of two-particle systems interacting with radial force functions. For example, the

bound or free nature of the relative motion in the gravitational or Lennard-Jones

potentials determines whether the two particles in the system stay in a closed trajectory or

have sufficient kinetic energy to overcome the potential energy well acting between them

and move apart.

Appendix A1.2 Motion under spherically symmetric forces

To illustrate the difficulty of solving Newton’s equations of motion to a system with a

spherically symmetric (radial) force function, consider a mass m subject to a radially

directed 1/r2 gravitational or Coulombic force shown in Figure A1.2(a).

Figure A1.2. (a) The motion of a mass subject to a radial force F(r) pointing towards an origin.

The motion remains confined to the xy-plane. (b) Points in a plane can be described by Cartesian

coordinates x and y or polar coordinates r and θ.

Newton’s second law for the motion of this mass in vector form of the three-

dimensional Cartesian coordinates is,

32

2

r

KF

dt

dm

rr

r . (A1.5)

The force is proportional to 1/r2 and points towards the center of force in the direction

opposite the vector r. For gravitational motion of a comet of mass m around the sun, for

example, the constant K = GMsm where G is the gravitational constant and Ms is the mass

of the sun. In the case of Coulombic force, for the motion of a charge q subject to a force

from the charge Q located at the origin, K = –Qq/4πε0, where ε0 is the permittivity of

vacuum. The force is always attractive for gravitational interactions, but may be attractive

or repulsive for Coulombic interactions based on the relative signs of the charges.


18

The motion of the mass subject to a radial force may at first seem three-

dimensional. However, calculating the time variation of the angular momentum (L =

r×mv) shows that the motion of the mass is confined to a plane. The time variation of the

angular momentum L of a mass subject to a radial force is,

00 Frarvvv

rvr

vrL mmdt

dm

dt

dmm

dt

d

dt

d. (A.1.6)

We used the fact that for radial forces, r and F vectors are parallel and oppositely

directed and so their cross product is zero. The angular momentum vector can only be

constant if both r and mv vectors always lie in the same plane. This observation, which

holds for any radially directed force, reduces the dimensionality of the motion. Assuming

the mass moves in the xy-plane, the radial distance r is (x2+ y

2)1/2

. From Eq. (A1.5), the

two Newtonian equations of motion for the Cartesian coordinates are,

2/3222

2

2/3222

2

)(

)(

yx

yKym

dt

ydm

yx

xKxm

dt

xdm

(A1.7)

We used Newton’s notation with each dot representing one order of the time derivative.

Analytical solutions to these two coupled second-order differential equations cannot be

directly found. However, since the force only involves the radial distance, a change of

coordinates from Cartesian to polar coordinates, r and θ, shown in Figure A1.2(b)

simplifies the description.

The transformation of Cartesian to polar coordinates is,

sin

cos

ry

rx (A1.8)

The Cartesian velocities can be written in terms of the time derivatives of the polar

coordinates,

cossin

sincos

rry

rrx (A1.9)

Substituting the polar equivalents to the Cartesian accelerations in Eq. (A1.8) gives,

0cos)2(sin

0sin)2(cos

2

2

2

2

rrmr

Krr

rrmr

Krr

(A1.10)

Multiplying the first line of Eq. (A1.10) by cosθ and the second line by sinθ and adding

the two resulting equations, gives,

0)(2

2 r

Krrm . (A1.11)

Substituting Eq. (A1.11) into either of the two equations in

(A1.10), we get a second equation,

.0)(1

)2( 22 constmrmrdt

d

rrrm , (A1.12)


19

where the constant, , is identified as the magnitude of the angular momentum of the

mass. That 2mr is the magnitude of the angular momentum can be verified by writing

out the Cartesian coordinates and velocities in the cross product L = r×mv in terms of

polar coordinates and velocities given in Eqs. (A1.8) and (A1.9).

Substituting from Eq. (A1.12) in Eq. (A1.11) gives the radial equation which

determines the time variation of the radial coordinate,

023

2

r

K

mrrm

. (A1.13)

Note the analogy between Eq. (A1.13) and Newton’s second law in Cartesian

coordinates, ma = F. In Eq. (A1.13), in addition to the force related to the gravitation (or

Coulombic interaction) K/r2, there is a contribution to the radial acceleration from

32 mr which is called the centripetal force which originates from the coordinate

transformation. The form ma = F does not hold for the polar coordinates since 2rKrm .

The change from Cartesian to polar coordinates which better reflect the symmetry

of the forces, introduces relevant mechanical quantities, such as angular momentum, into

the analysis in a natural manner as we solve the Newtonian equations of motion. These

secondary quantities clarify the content of the equations and simplify the interpretation of

the results.

The radial equation can be solved to get mathematical form of the trajectory (orbit)

in polar coordinates by defining a new variable u = 1/r. Applying the chain rule,

2

22

2

2

222

111

d

ud

m

u

d

ud

md

du

dt

d

mr

d

du

md

du

udt

d

d

du

udt

du

uudt

dr

(A1.14)

Substituting the time derivatives of Eq. (A1.14) into Eq. (A1.13) gives a parametric

equation for the coordinates,

22

2

mKu

d

ud

. (A1.15)

The general solution of this inhomogeneous second-order linear differential equation

(verify by substitution) is,

cos2

CmK

u

. (A1.16)

To determine the constant C, we write the kinetic energy as the difference between the

total energy and the potential energy U = K/r = Ku,

2 2 2 2 2

2 2

m mx y r r E U E Ku (A1.17)

Substituting Eqs. (A1.16), (A1.12), and (A1.14) into (A1.17)

gives the value for C as,

4

22

2

2

KmmEC . (A1.18)


20

Finally, substituting this value of C into Eq. (A1.16) gives a description of the (u,θ) orbit

in terms of the energy of the system,

cos2

112

2

2 mK

EmKu

. (A1.19)

Equation (A1.19) is the equation for the conical intersections in polar coordinates. As

shown in Fig. 1.3, the orbit of the particle is an ellipse if E < 0, a parabola if E = 0, and a

hyperbola if E > 0. The ellipse, of course, is a closed curve where the mass remains

bound to the center of force, whereas in parabolic and hyperbolic orbits where the motion

is unbound, the mass approaches the center of force from large distances, but maintains

enough kinetic energy to ultimately escape from the potential well. The initial conditions,

which determine the energy and angular momentum of the motion, also determine the

nature of the orbit.

A similar analysis can be carried out for the Lennard-Jones force corresponding to

the potential of Eq. (1.17). The analog to Eq. (A1.13) is,

0212

713

3

2

rrmrrm

. (A1.20)

Solving this equation gives the trajectory of two interacting Lennard-Jones molecules.

Similar to the case of a 1/r potential, the molecules interacting with a Lennard-Jones

potential are constrained to move in two dimensions and have bound and unbound

trajectories. Some bound trajectories for the Lennard-Jones potential are shown in Figure

1.5. Further details of the Lennard-Jones potential trajectories are given in Stogryn and

Hirschfelder.[Stogryn]

In the above, the dynamics of systems with radial forces have been analyzed by

transforming the Cartesian coordinate system to polar coordinates. This transformation

can be performed, but as illustrated, is somewhat cumbersome. To demonstrate the utility

of the Lagrangian and Hamiltonian approaches for this problem, using Eqs. (A1.9) we

write the Lagrangian for this system,

r

Krr

mrUyxUKL 22222

2)( . (A1.21)

The Lagrangian equations of motion Eq. (1.43) for this system become,

0)2(0

00

2

2

2

rrrmLL

dt

d

r

Krrm

r

L

r

L

dt

d

(A1.22)

These equations are identical to Eqs. (A1.13) and (A1.12) but have been derived in a

more direct manner. The momenta conjugate to the radial and angular coordinates are,

2mrL

p

rmr

Lpr

(A1.23)

and the Hamiltonian for this system is,


21

r

Kp

mrp

mr

Krr

mLpprH rr 2

2

2222

2

1

2

1)(

2 (A1.24)

Hamilton’s equations of motion for this system are,

0

2

23

2

Hp

mr

p

p

H

r

K

mr

p

r

Hp

mpp

Hr

r

r

r

(A1.25)

The second equation is the radial equation and the fourth equation gives the conservation

of angular momentum. The phase space for the radial motion has four dimensions,

namely r, pr, θ, and pθ, but the conservation of angular momentum shows that pθ is

constant.

1.1 Mechanics The study of motion 2 Notes...mechanics apply to the motion on atomic and molecular scales is the working assumption in developing methods for molecular simulations.

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