Mathematical Principles of Theoretical Physics

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.