Mathematical Principles of Theoretical Physics

(Rick Simeone) #1

2.6. PRINCIPLE OF HAMILTONIAN DYNAMICS (PHD) 99


which means that the rate of energy change inΩequals to the difference of the input and
output of energy flow crossing the boundary ofΩper unit time. Consequently (2.6.34) is
equivalent to energy conservation.


In the Lagrangian dynamics, Noether Theorem2.38gives a correspondence between
symmetries and conservations, and provides a way to seek theconservation laws. Likewise,
the Hamiltonian dynamics provides another criterion to findconservation laws.
LetS(u,v)be a functional given by


(2.6.38) S(u,v) =




S(u,v,Du,Dv)dx.

The following theorem provides a condition forS(u,v)to be a conserved quantity.


Theorem 2.48(Conservation Laws of Hamiltonian System).Let S(u,v)be a functional as
given by (2.6.38). If S and the Hamilton energy H satisfy the following relation




[


∂S


∂uk

−∂i

(


∂S


∂ ξik

)][


∂H


∂vk

−∂i

(


∂H


∂ ζik

)]


(2.6.39) dx


=




[


∂S


∂vk

−∂i

(


∂S


∂ ζik

)][


∂H


∂uk

−∂i

(


∂H


∂ ξik

)]


dx,

for solutions(u,v)of (2.6.33), then S is a conserved quantity of the Hamiltonian system.
Namely, S satisfies that


(2.6.40)


dS
dt

=−



∂Ω

Ps·nds,

where Ps= (Ps^1 ,Ps^2 ,Ps^3 )is the flux given by


Psk=−

[


∂S


∂ ξk j

∂uj
∂t

+


∂S


∂ ζk j

∂vj
∂t

]


.


Proof.The proof is similar to that of Theorem2.46. By


dS
dt

=




[


∂S


∂uj

∂uj
∂t

+


∂S


∂ ξij

∂i

(


∂uj
∂t

)


+


∂S


∂vj

∂vj
∂t

+


∂S


∂ ζij

∂i

(


∂vj
∂t

)]


dx

=




[(


∂S


∂uj

−∂i

(


∂S


∂ ξij

))


∂uj
∂t

+


(


∂S


∂vj

−∂i

(


∂S


∂ ζij

))


∂vj
∂t

]


dx

+



∂Ω

Psk·nkdS.

By (2.6.33) and (2.6.39) we deduce (2.6.40). The proof is complete.


In fact, Theorem2.46is a special case of Theorem2.48forS=H. Theorem2.48is useful

2.6.4 Quantum Hamiltonian systems.

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