2.6. PRINCIPLE OF HAMILTONIAN DYNAMICS (PHD) 93
whereH=H(q,p,t)is the total energy,Lis the Lagrangian, andq,psatisfy the Hamilton
equations
(2.5.62)
dq
dt
=
∂H
∂p
,
dp
dt
=−
∂H
∂q
.
IfL=L(q,q ̇,t)is invariant under the translation (2.5.60), thenLdoes not explicitly con-
tain timet, i.e.
∂L
∂t
= 0.
Then it follows from (2.5.61) and (2.5.62) that
dH
dt
=−
∂L
∂t
= 0.
Hence the energy conservation is deduced using the time translation invariance.
We end this section with some relations of symmetries and conservation laws:
energy ⇔ time translation,
momentum ⇔ space translation,
angular Momentum ⇔ space rotation,
particle number ⇔ phase rotation,
parity ⇔ space reflection.
2.6 Principle of Hamiltonian Dynamics (PHD)
2.6.1 Hamiltonian systems in classical mechanics
In classical mechanics, the principle of Hamiltonian dynamics (PHD) consists of the follow-
ing three main ingredients:
1) for an isolated (conserved) mechanical system, its states are described by a set of state
variables given by
(2.6.1) q 1 ,···,qN, p 1 ,···,pN;
2) its total energyHis a function of (2.6.1), i.e.
(2.6.2) H=H(q,p),
3) the state variablesqandpsatisfy
(2.6.3)
dqk
dt
=
∂H
∂pk
,
dpk
dt
=−
∂H
∂qk
for 1≤k≤N.