1549380323-Statistical Mechanics Theory and Molecular Simulation

Lagrangian formulation 11

It can be easily verified that substitution of eqn. (1.4.5) into eqn. (1.4.6) gives eqn.
(1.2.10):

∂L

∂r ̇i

=mir ̇i

d

dt

(

∂L

∂r ̇i

)

=mi ̈ri

∂L

∂ri

=−

∂U

∂ri

=Fi

d

dt

(

∂L

∂r ̇i

)

−

∂L

∂ri

=mi ̈ri−Fi= 0, (1.4.7)

which is just Newton’s second law of motion.
As an example of the application of the Euler–Lagrange equation, consider the
one-dimensional harmonic oscillator discussed in the previous section. The Hooke’s
law forceF(x) =−kxcan be derived from a potential

U(x) =

1

2

kx^2 , (1.4.8)

so that the Lagrangian takes the form

L(x,x ̇) =

1

2

mx ̇^2 −

1

2

kx^2. (1.4.9)

Thus, the equation of motion is derived as follows:

∂L

∂x ̇

=mx ̇

d

dt

(

∂L

∂x ̇

)

=mx ̈

∂L

∂x

=−kx

d

dt

(

∂L

∂x ̇

)

−

∂L

∂x

=mx ̈+kx= 0, (1.4.10)

which is the same as eqn. (1.3.5).
It is important to note that when the forces in a particular system are conserva-
tive, then the equations of motion satisfy an important conservation law, namely the
conservation of energy. The total energy is given by the sum of kinetic and potential
energies:

E=

∑N

i=1

1

2

mir ̇^2 i+U(r 1 ,...,rN). (1.4.11)