Physical Chemistry Third Edition

1270 E Classical Mechanics

E.3 Lagrangian Mechanics

Lagrangian mechanics is a way of writing the classical mechanics of Newton in a

way that has the same form in any coordinate system. It is convenient for problems in

which Cartesian coordinates cannot conveniently be used. We specify the positions of

the particles in a system by the coordinatesq 1 ,q 2 ,q 3 ,...,qn, wherenis the number of

coordinates, equal to three times the number of particles if they are point-mass particles

that move in three dimensions. These coordinates can be any kind of coordinates, such

as spherical polar coordinates, cylindrical polar coordinates, and so on. To specify the

state of the system, some measures of the particles’ velocities are needed in addition

to coordinates. The Lagrangian method uses the time derivatives of the coordinates:

.

qi

dqi

dt

(i1, 2,...,n) (E-11)

We use a symbol with a dot over it to represent a time derivative.

The Lagrangian function is defined by

LK −V (definition of the Lagrangian) (E-12)

whereK andVare the kinetic and potential energy expressed in terms of theqsand

the

.

qs. TheLagrangian equations of motionare:

d

dt

(

∂L

∂qi

)

−

∂L

∂qi

0(i1, 2,...,n) (E-13)

These equations are equivalent to Newton’s second law and we present them without

derivation. These equations have exactly the same form for all coordinate systems, so

that the work of transforming Newton’s second law into a particular coordinate system

can be avoided, once we have expressed the kinetic and potential energy in terms of

the appropriate coordinates.^6

Particle of

massm

Plane of

orbit

Fixed

center

Orbit

Vector r

Velocity

vector

Angular

momentum

vector

Figure E.2 Diagram to Illustrate the
Definition of the Angular Momentum
Vector.The angular momentum vector,
L, is perpendicular to the vectorrand
the vectorv, and has magnitude equal
tom|r||v|.

One application of Lagrange’s equations of motion is to a particle orbiting about

a fixed point in a plane. We use plane polar coordinatesρandφ(withφmeasured in

radians), and place the origin of coordinates at the fixed point. The component of the

velocity parallel to the position vector is

.

ρ, and the component perpendicular to this

direction isρ

.

φ, so that the Lagrangian is

L

1

2

m

.

ρ^2 +

1

2

mρ^2

.

φ^2 −V(ρ,φ) (E-14)

The Lagrangian equations of motion are

d(2m

.

ρ)

dt

−mρ

.

φ

2

+

∂V

∂ρ

 0 (E-15)

d(mρ^2

.

φ)

dt

+

∂V

∂φ

 0 (E-16)

(^6) This fact is proved in J. C. Slater and N. Frank,Mechanics, McGraw-Hill, New York, 1947, p. 69ff, or
in any other book on the same subject.