Mathematical Principles of Theoretical Physics

(Rick Simeone) #1

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.
Free download pdf