1549380323-Statistical Mechanics Theory and Molecular Simulation

18 Classical mechanics

The generalization of the Legendre transform to a functionfofnvariablesx 1 ,...,xn
is straightforward. In this case, there will be a variable transformation of the form

s 1 =

∂f

∂x 1

=g 1 (x 1 ,...,xn)

..

.

sn=

∂f

∂xn

=gn(x 1 ,...,xn). (1.5.6)

Again, it is assumed that this transformation is invertible so that it is possible to
express eachxias a functionxi(s 1 ,...,sn) of the new variables. The Legendre transform
offwill then be

f ̃(s 1 ,...,sn) =f(x 1 (s 1 ,...,sn),...,xn(s 1 ,...,sn))−

∑n

i=1

sixi(s 1 ,...,sn). (1.5.7)

Note that it is also possible to perform the Legendre transform of afunction with
respect to any subset of the variables on which the function depends.

1.6 Generalized momenta and the Hamiltonian formulation of

classical mechanics

For a first application of the Legendre transform technique, we willderive a new
formulation of classical mechanics in terms of positions and momentarather than
positions and velocities. The Legendre transform will appear again numerous times
in subsequent chapters. Recall that the Cartesian momentum of aparticlepiis just
pi=mir ̇i. Interestingly, the momentum can also be obtained as a derivative of the
Lagrangian with respect tor ̇i:

pi=

∂L

∂r ̇i

=

∂

∂r ̇i





∑N

j=1

1

2

mjr ̇^2 j−U(r 1 ,...,rN)



=mir ̇i. (1.6.1)

For this reason, it is clear how the Legendre transform method canbe applied. We
seek to derive a new function of positions and momenta as a Legendre transform of
the Lagrangian with respect to the velocities. Note that, by way ofeqn. (1.6.1), the
velocities can be easily expressed as functions of momenta,r ̇i=r ̇i(pi) =pi/mi. There-
fore, substituting the transformation into eqn. (1.5.7), the new Lagrangian, denoted
L ̃(r 1 ,...,rN,p 1 ,...,pN), is given by

L ̃(r 1 ,...,rN,p 1 ,...,pN) =L(r 1 ,...,rN,r ̇ 1 (p 1 ),...,r ̇N(pN))−

∑N

i=1

pi·r ̇i(pi)

=

1

2

∑N

i=1

mi

[

pi

mi

] 2

−U(r 1 ,...,rN)−

∑N

i=1

pi·

pi

mi