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A variational principle for the stationary single-velocity homogeneous transport equation (cf. [[Transport equations, numerical methods|Transport equations, numerical methods]])

$$ \tag{1 }
( \overline \Omega \; , \nabla \psi ) +
\Sigma ( x) \psi  = \ 
\lambda \int\limits _ {| {\overline \Omega \; }  ^  \prime  | = 1 }
\theta ( x, \mu _ {0} )
\psi  ( \overline \Omega \; {}  ^  \prime  , x)
 d \overline \Omega \; {}  ^  \prime  ,
$$

$$ 
\mu _ {0}  =  ( \overline \Omega \; {}  ^  \prime  , \overline \Omega \; ),
$$

with boundary condition

$$ \tag{2 }
\psi \mid  _ {x \in \Gamma }   =  0,\ \ 
( \overline \Omega \; , \overline{n}\; )  <  0,
$$

where  $  \Gamma $
is the boundary of a convex bounded domain  $  G $. 
If the scatter indicatrix  $  \theta ( x, \mu _ {0} ) $
is an even function of  $  \mu _ {0} $, 
transition to the new unknown function

$$ 
u  =  
\frac{[ \psi ( \overline \Omega \; , x) +
\psi (- \overline \Omega \; , x)] }{2}

$$

reduces the problem (1), (2) to self-adjoint form. In the problem thus obtained, Vladimirov's variational principle for the smallest eigenvalue  $  \lambda _ {1} $
states that  $  \lambda _ {1} $
is the minimum of the functional

$$ 
\int\limits _ {| \overline \Omega \; | = 1 }
\int\limits _  \Gamma 
| ( \overline \Omega \; , \overline{n}\; ) |
u  ^ {2} ( \overline \Omega \; , x)
 d \overline \Omega \;  dS _ {x} +
$$

$$ 
+ 
\int\limits _ {| \overline \Omega \; | = 1 } \int\limits _ { G } 
\frac{1}{\Sigma ( x) }
 ( \overline \Omega \; , \nabla u)  ^ {2}  dx  d \overline \Omega \; +
$$

$$ 
+ 
\int\limits _ {| \overline \Omega \; | = 1 } \int\limits _ { G } \Sigma ( x)
u  ^ {2} ( \overline \Omega \; , x)  dx  d \overline \Omega \;
$$

on the set of functions  $  u ( \overline \Omega \; , x) $
which satisfy the condition

$$ 
\int\limits _ {| \overline \Omega \; | = 1 }
\int\limits _ {| \overline \Omega \; {}  ^  \prime  | = 1 }
\int\limits \theta ( x, \mu _ {0} )
u ( \overline \Omega \; {}  ^  \prime  , x)
 dx  d {\overline \Omega \; {}  ^  \prime  }  d \overline \Omega \;  =  1.
$$

The corresponding (non-negative) eigenfunction realizes the minimum of the functional [[#References|[3]]]. In this variational principle the respective boundary conditions are natural. Variational principles for higher eigenvalues and for the non-homogeneous problem are formulated in a similar manner.

The principle was first obtained by V.S. Vladimirov [[#References|[1]]], and it yielded the optimum boundary conditions in the method of spherical harmonics (cf. [[Spherical harmonics, method of|Spherical harmonics, method of]]). Vladimirov's variational principle, in conjunction with finite difference methods, is extensively employed in numerical computations of neutron physics.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.S. Vladimirov,   "Mathematical problems in the theory of single-velocity particle transfer" , Moscow  (1961)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.I. Marchuk,   "Methods of calculation of nuclear reactors" , Moscow  (1961)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B. Davison,   "Neutron transport theory" , Oxford Univ. Press  (1957)</TD></TR></table>

====Comments====
Cf. also [[Vladimirov method|Vladimirov method]].