Mathematical Principles of Theoretical Physics

94 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS

The state variablesqk( 1 ≤k≤N)represent positions, andpk( 1 ≤k≤N)represent momen-
tums. The system (2.6.1)-(2.6.3) is called Hamiltonian system.
In physics, PLD and PHD are two independent fundamental principles. However, the two
sets of equations derived from PLD and PHD are usually equivalent.
In classical mechanics, Hamiltonian systems and Lagrange systems can be transformed
to each other by the Legendre transformation. We start with asimple example. Consider a
particle with massmin a force fieldF. The Lagrange action for this system is given by

(2.6.4)

L=

∫T

0

L(q,q ̇)dt,

L=T−V=

1

2

mq ̇^2 −Fq,

whereqstands for position, and the Euler-Lagrange equation of (2.6.4) is the Newtonian
Second Law, written as

(2.6.5)

d

dt

δL

δq ̇

=

δL

δq

⇒ mq ̈=F.

Based on (2.6.1)-(2.6.3), the state variablesq,pof PHD are

(2.6.6) qas in (2.6.4) and p=mq ̇.

The total energy is

(2.6.7) H=

1

2 m

p^2 +Fq,

and the Hamilton equations of (2.6.7) are given by

(2.6.8)

dp

dt

=

∂H

∂q

=F,

dq

dt

=−

∂H

∂p

=

1

m

p.

By (2.6.6), it is clear that

(2.6.9) Hamilton Eqs( 2. 6. 8 ) =Lagrange Eq( 2. 6. 5 ) =Newton 2nd Law.

The equivalences in (2.6.9) only manifest that the three principles have an intrinsic relation.
In particular, by (2.6.4)-(2.6.7), the variablesq,q ̇of PLD and the variablesq,pof PHD have
the relation:

(2.6.10) p=

δL

δq ̇

(by

δL

δq ̇

=

∂L

∂q ̇

=mq ̇),

and the Lagrange densityL(q,q ̇)and the Hamilton energyH(q,p)are related by

(2.6.11) pq ̇−L(q,q ̇) =H(q,p).