26
1 General Problem Formulation
then equation (1.54) can be rewritten in the form of the full time derivative:
dN _ oN oN dr oN dk oN dw _ G
dt - 7it + or dt + ok dt + aw dt - 0
(1.56)
The equations (1.54) and (1.56) are called kinetic equations. They are
well-known in theoretical physics. It is a generalization of Liouville's theorem
of the conservation of the distribution function of the gas as a particle system
moving in phase space (Landau & Lifshits, 1973). The value in the right-hand
side of (1.56), is called the collision integral. The integral-differential equation
(1.56) with the collision integral describing a molecule collision in phase space
is called the Boltzmann equation, which was proposed in 1872.
As noted in Sect. 1.2, the ratios (1.55) present equations of motion for
wave packets in terms of the variables r and k (it follows that F = 1l,
Sect. 1.3). These equations coincide in form with the Hamiltonian equations, being the central point in classic mechanics (Lantsosh, 1965; Landau & Lifshits, 1973). They are solved in terms of the particle momentum p
and its coordinate q. The canonical Hamiltonian equations represent a system of 2s (in our case s = 3) first-order differential equations for 2s unknown
functions p(t) and q(t). They substitute s of the second-order equations for
the Lagrange motion simulation method.
The total time derivative of the Hamilton function 1l is written as:
(1.57)
Substituting Qi and Pi into (1.57) from (1.55), the last two terms are cancelled
reciprocally:
d1l
81l
dt
7it 0
(1.58)
In particular, if the Hamiltonian function is not dependent on time explicitly, then d1l/ dt = 0, i.e. 1l is conserved.
If the Hamiltonian function is not dependent on the coordinates, the corresponding component of the generalized momentum is preserved while the
system is moving. This can be written as:
Pi=-: =0.
(1.59)
Such a coordinate is called cyclic.
Let f be some function of the coordinates q, momentum p and timet. Its
full time derivative is as follows:
df of
of.
of .
dt = ot + L EJ.% + L ~Pj 0
j
qJ
j
'PJ
(1.60)
1 General Problem Formulation
then equation (1.54) can be rewritten in the form of the full time derivative:
dN _ oN oN dr oN dk oN dw _ G
dt - 7it + or dt + ok dt + aw dt - 0
(1.56)
The equations (1.54) and (1.56) are called kinetic equations. They are
well-known in theoretical physics. It is a generalization of Liouville's theorem
of the conservation of the distribution function of the gas as a particle system
moving in phase space (Landau & Lifshits, 1973). The value in the right-hand
side of (1.56), is called the collision integral. The integral-differential equation
(1.56) with the collision integral describing a molecule collision in phase space
is called the Boltzmann equation, which was proposed in 1872.
As noted in Sect. 1.2, the ratios (1.55) present equations of motion for
wave packets in terms of the variables r and k (it follows that F = 1l,
Sect. 1.3). These equations coincide in form with the Hamiltonian equations, being the central point in classic mechanics (Lantsosh, 1965; Landau & Lifshits, 1973). They are solved in terms of the particle momentum p
and its coordinate q. The canonical Hamiltonian equations represent a system of 2s (in our case s = 3) first-order differential equations for 2s unknown
functions p(t) and q(t). They substitute s of the second-order equations for
the Lagrange motion simulation method.
The total time derivative of the Hamilton function 1l is written as:
(1.57)
Substituting Qi and Pi into (1.57) from (1.55), the last two terms are cancelled
reciprocally:
d1l
81l
dt
7it 0
(1.58)
In particular, if the Hamiltonian function is not dependent on time explicitly, then d1l/ dt = 0, i.e. 1l is conserved.
If the Hamiltonian function is not dependent on the coordinates, the corresponding component of the generalized momentum is preserved while the
system is moving. This can be written as:
Pi=-: =0.
(1.59)
Such a coordinate is called cyclic.
Let f be some function of the coordinates q, momentum p and timet. Its
full time derivative is as follows:
df of
of.
of .
dt = ot + L EJ.% + L ~Pj 0
j
qJ
j
'PJ
(1.60)