96 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS
Conversely, if we know the PHD system (2.6.14) and (2.6.15), then it follows from
(2.6.16) and (2.6.17) that the Lagrange densityLsatisfies
(2.6.20) H
(
x,∂L
∂y)
−yk∂L
∂yk+L(x,y) = 0.Theoretically we can solve the differential equation (2.6.20), and obtain the solutionL=
L(x,y). Letx=qandy=q ̇, then we deduce the expression for the Lagrange density from
the Hamilton functionH(q,p).
Remark 2.44.We remark that both PLD and PHD are independent to each other.However,
the PLD dynamical system (2.1.5)-(2.1.8) and the PHD dynamical system (2.6.1)-(2.6.3) are
equivalent in mechanics by (2.6.16) and (2.6.17). But we shall see later that the relations
(2.6.16) and (2.6.17) are not valid in electromagnetism where PLD and PHD still hold true.
In addition, the differentialdHis given by
(2.6.21) dH=
∂H
∂qkdqk+∂H
∂pkdpk+∂H
∂tdt.By (2.6.17),
(2.6.22) dH=q ̇kdpk−
∂L
∂qk
dqk−∂L
∂t
dt.It follows from (2.6.21) and (2.6.22) that
q ̇k=∂H
∂pk(2.6.23) ,
∂H
∂qk=−
∂L
∂qk(2.6.24) ,
∂H
∂t=−
∂L
∂t(2.6.25).
From (2.6.21) and (2.6.25) we deduce (2.5.61). Then by (2.6.16) and the Lagrange equation
(2.1.8), we have
dpk
dt
=
d
dt(
δL
δq ̇k)
=
∂L
∂qk.
Hence (2.6.24) becomes
(2.6.26) p ̇k=−
∂H
∂qk.
Thus, by the relations (2.6.16) and (2.6.17) we deduce the equivalence of PLD dynamics and
PHD dynamics in another fashion.
2.6.2 Dynamics of conservative systems
Energy conservation is a universal law in physics. It implies that the PHD is a universal
principle to describe conservative physical systems. In other words, the Hamilton principle