Mathematical Principles of Theoretical Physics

2.6. PRINCIPLE OF HAMILTONIAN DYNAMICS (PHD) 95

In this example, the two relations (2.6.10) and (2.6.11) are obtained directly by their
physical meanings, and by which we can deduce one by another.
In fact, the relations (2.6.10) and (2.6.11) are also valid in general.We discuss this problem
as follows.
Consider a mechanical system. For PLD, the state variables are positionsqkand velocities
q ̇k:

(2.6.12) q 1 ,···,qN and q ̇ 1 ,···,q ̇N,

The Lagrange action is given by

(2.6.13) L=

∫T

0

L(q,q ̇)dt.

For PHD, the state variables are:

(2.6.14) q 1 ,···,qN and p 1 ,···,pN,

The Hamilton energy is:

(2.6.15) H=H(q,p).

The two systems (2.6.12)-(2.6.13) and (2.6.14)-(2.6.15) of PLD and PHD satisfy the fol-
lowing relations, which implies the equivalence of PLD and PHD in classical mechanics.

Dynamical Relation 2.43(PLD and PHD).For the two systems of PLD and PHD in classical
mechanics, the following conclusions hold true:

1) The two sets (2.6.12) and (2.6.14) of variables satisfy the following relation:

(2.6.16) pk=

∂L(q,q ̇)

∂q ̇k

for 1 ≤k≤N.

2) The two functions (2.6.13) and (2.6.15) satisfy the following relation, which is usually

called the Legendre transformation:

(2.6.17) pkq ̇k−L(q,q ̇) =H(q,p).

By (2.6.16)-(2.6.17), if we have obtained the PLD system (2.6.12)-(2.6.13), then by the
implicit function theorem we can solve from (2.6.16) the functions

(2.6.18) q ̇k=fk(q,p).

Then, inserting (2.6.18) in the left-hand side of (2.6.17) we deduce the expression of the
Hamilton energy:

(2.6.19) H(q,p) =pkfk(q,p)−L(q,f(q,p)),

which gives rise to the Hamiltonian system (2.6.3). In other words, we can derive the Hamil-
tonian dynamics from the Lagrangian dynamics by the relations (2.6.16) and (2.6.17).