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\begin{document}
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\begin{English}
\title{On the difference between admissible and
``differentiable'' Hamiltonians}
\author{Vladimir O. Soloviev\\
\small Institute for High Energy Physics\\
\small  Protvino, 142284, Moscow Region, Russia\\
\small e-mail: vosoloviev@mx.ihep.su}
\date{October 30, 1996}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
It is shown that the Regge-Teitelboim criterion for fixing
the unique boundary contribution to the Hamiltonian  
compatible with free boundary conditions
should be modified if the Poisson structure is noncanonical.
The new criterion requires cancellation  of 
boundary contributions
to the Hamiltonian equations of motion. In the same time, 
boundary contributions
to the variation of Hamiltonian are allowed. The Ashtekar formalism
for gravity and hydrodynamics of the ideal fluid with a free
surface in the Clebsch variables are treated as examples.

\medskip
PACS: 11.10.Ef, 04.20.Fy, 03.40.Gc

\medskip
Keywords: Hamiltonian formalism, canonical gravity, 
Ashtekar variables, hydrodynamics, Clebsch variables,
Poisson brackets, free boundary
\end{abstract}
%\vskip 2mm
\end{English}
\begin{Russian}
\title{   
 ``'' }
\author{..~\\
\small    \\
\small  , 142284,  , \\
\small e-mail: vosoloviev@mx.ihep.su}
\date{30  1996}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
,   -,  
   ,     
   ,     
  .   
      .
   ,      .
        
        
. 
\end{abstract}
\end{Russian}
\newpage
%\section {Introduction}
\begin{English}
In this publication 
we consider the case of free boundary conditions in field theory, i.e., the
situation when variations of the field variables and their spatial
derivatives are not necessarily equal to zero on the boundary of the relevant
space domain. Therefore 
it would be incorrect to treat Hamiltonians (or Lagrangians)
differing in spatial divergences as representatives of the same equivalence
class. We are to fix a unique
admissible Hamiltonian (or Lagrangian) 
from that class according to some rule.
\end{English}
\begin{Russian}
        ,  ,
,        
         .
      ( ),
   ,   
 .    
 ( )       
.
\end{Russian}

\begin{English}
The importance of such problems for physics can be easily seen from
the long history of discussion on the role of surface integrals in
the Hamiltonian of General Relativity
\end{English}
\begin{Russian}
      , ,  
       
   
\end{Russian}
\cite{ADM} --
%\cite{Dirac}, \cite{Higgs},
%\cite{Schwinger},
\cite{DeWitt}. 
\begin{English}
This discussion had been resolved in a publication
by Regge and Teitelboim
\end{English}
\begin{Russian}
         ,
\end{Russian}
\cite{RT}
\begin{English}
where a criterion for the adequate choice of divergences
in the Hamiltonian  was proposed. It was called the requirement of
``differentiability'' of the Hamiltonian (see also
\end{English}
\begin{Russian}
        .
    ``'' 
(. 
\end{Russian}
\cite{FT}
).

\begin{English}
Here we intend to show that this criterion, which is applicable for
the canonical Poisson structure,  should be revised in a more general
situation, especially if boundary contributions appear in the symplectic
form and (or) in the Poisson brackets.
In the same time the general mathematical background
staying behind the concrete formulation given to it by Regge and Teitelboim
is untouched by our improvement.
\end{English}
\begin{Russian}
   ,   , 
 
 ,       ,
         ()
  .       ,  
 ,     ,  
    .
\end{Russian}

\begin{English}
This general background is the ``natural boundary conditions'' criterion
of the variational calculus
\end{English}
\begin{Russian}
     ``  ''
  
\end{Russian}
\cite{HC}, \cite{Lancz}.
%It is well known that variational principles are applied not only
%to dynamical problems but to static as well. 
\begin{English}
The general idea of the
natural boundary conditions is
that we can get from a variational principle not only the Euler-Lagrange
equations
%inside the relevant domain
but also some equations on the boundary.
Both of them follow from the requirement
that the functional under study should have a stationary point.
It is possible to derive the natural boundary conditions 
when arbitrary field variations on the boundary contribute to the
functional variation,
and so, their coefficients must be equal to
zero to make the functional stationary.
\end{English}
\begin{Russian}
       ,  
       -,
    .       ,
     .  
  ,     
      ,   , 
     ,
   . 
\end{Russian}

\begin{English}
The difference between admissible and ``differentiable'' Hamiltonians,
which is discussed in the paper, arises when noncanonical variables
are used in the Hamiltonian formalism. This situation is not unfamiliar.
For example, noncanonical Poisson brackets may originate as Dirac
brackets after some reduction procedure. Below we will consider two
examples. The first one is the Ashtekar formulation of General Relativity
where the noncanonicity of Poisson brackets 
arises as a result of the change of variables
\end{English}
\begin{Russian}
    ``'' 
, 
  ,  ,    
  .     .
,   \-     
   .     .
        , 
      
\end{Russian}
\cite{Sol92}.
\begin{English}
The second example is the Hamiltonian formalism for surface waves in
ideal fluid. Here just a position of the boundary becomes a dynamical
variable. In both cases we will see that the relation between
the Lagrangian and the Hamiltonian is not so simple as in the 
canonical situation.
As a result we can see that the correspondence between the action functional
and the boundary conditions 
survives in the general case whereas the requirement
of ``differentiability'' for the Hamiltonian should be replaced by another one.
The new criterion requires a cancellation of 
boundary terms in the
Hamiltonian vector field, or saying it in other words, in the Hamiltonian
equations of motion. The Hamiltonian vector field is to be constructed
according to the so-called formal variational calculus
\end{English}
\begin{Russian}
       
  ,      
.     ,     
   ,      
.    ,    
       ,  
 ``''    
.        
 , ,  ,    .
        
   
\end{Russian}
\cite{GD}
\begin{English}
and its extension on divergences proposed in previous publications
of the author 
\end{English}
\begin{Russian}
    ,    

\end{Russian}
\cite{Sol93}, \cite{Sol95}, \cite{Sol96}.


\begin{English}
Throughout this paper we will use the characteristic function of a
relevant compact space domain 
\end{English}
\begin{Russian}
       
   
\end{Russian}
$\Omega$
\begin{English}
constructed by means of the Heaviside 
\end{English}
\begin{Russian}
   
\end{Russian}
$\theta$
\begin{English}
-function
and smooth function 
\end{English}
\begin{Russian}
-    
\end{Russian}
$P(x)$ 
\begin{English}
with the following properties
\be
P(x)= \cases {
>0 &  if $x\in\Omega$; \cr
=0  & if $x\in\partial\Omega$; \cr
<0 & otherwise. \cr }
\ee
\end{English}
\begin{Russian}
  
\be
P(x)= \cases {
>0 &   $x\in\Omega$; \cr
=0  &  $x\in\partial\Omega$; \cr
<0 & . \cr }
\ee
\end{Russian}
\begin{English}
It allows us to write any integral over 
\end{English}
\begin{Russian}
      
\end{Russian}
$\Omega$ 
\begin{English}
formally as an integral over infinite space 
\end{English}
\begin{Russian}
       
\end{Russian}
${\rm R}^3$ 
\begin{English}
and freely integrate by parts. We will omit
\end{English}
\begin{Russian}
    .   
\end{Russian}
$d^3x$ 
\begin{English}
in these integrals.
\end{English}
\begin{Russian}
  .
\end{Russian}

\begin{English}
As a first example let us consider the Ashtekar formalism for canonical
gravity
\end{English}
\begin{Russian}
        

\end{Russian}
\cite{Ash}. 
\begin{English}
It is well known that this formulation can be constructed as a
result of field transformations starting from the tetrad variables in time 
gauge where the unit normal to spacelike hypersurface is taken as one leg
of the tetrad
\end{English}
\begin{Russian}
 ,        
  ,     \' ,
    - 
      
\end{Russian}
\cite{Henn}.
\begin{English}
The initial pair of canonical variables 
\end{English}
\begin{Russian}
   
\end{Russian}
$(E_{ia},\pi^{ia})$
\begin{English}
can first be changed for another one 
\end{English}
\begin{Russian}
     
\end{Russian}
$(\tilde E^{ia},K_{ia})$
\be
\{\tilde E^{ia}(x),K_j^b(y)\}=\frac{1}{2}\delta^i_j\delta^{ab}\delta(x,y),
\ee
\begin{English}
where
\end{English}
\begin{Russian}

\end{Russian}
\be
\tilde E^{ia}=EE^{ia},\quad K^a_i=K_{ij}E^{ja}+E^{-1}E_{ib}J^{ab},
\ee
\begin{English}
and 
\end{English}
\begin{Russian}

\end{Russian}
$E^a_iE^j_a=\delta^j_i$,
$E^a_iE^i_b=\delta^a_b$, $E=\det|E_{ia}|$, $K_{ij}$ 
\begin{English}
is the second fundamental
form of the spacelike hypersurface,
and the generator of triad (the three remaining vectors of the tetrad)
rotations is
\end{English}
\begin{Russian}
    -
,     ( ,  
 ) 
\end{Russian}
\be
J^{ab}=\frac{1}{2}(K^a_i\tilde E^{bi}-K^b_i\tilde E^{ai}).\label{eq:Jab}
\ee

\begin{English}
Then the second transformation is used which introduce the Ashtekar
connection variable 
\end{English}
\begin{Russian}
   ,   
 
\end{Russian}
$A^a_i$ 
\begin{English}
instead of 
\end{English}
\begin{Russian}

\end{Russian}
$K_{ia}$
\be
A^a_i=iK^a_i+\Gamma^a_i,\qquad
\Gamma^a_i=\frac{1}{2}\epsilon^{abc}\tilde E_{jc}\tilde E^{jb}_{\ |i},
\ee
\begin{English}
where the vertical line denotes the standard Riemannian covariant
derivative and
\end{English}
\begin{Russian}
       
 
\end{Russian}
\be
\{\tilde
E^{ia}(x),A^b_j(y)\}=\frac{i}{2}\delta^i_j\delta^{ab}\delta(x,y).
\ee

\begin{English}
This transformation leads to the new form of the internal rotation
generator which we take as a primer of the admissible Hamiltonian
\end{English}
\begin{Russian}
       
,        
\end{Russian}
\be
H(\hat\lambda^a)=
2\int \theta(P)\hat\lambda^a{\cal D}_i\tilde E^{ia}\equiv 2i
\int \theta(P)\hat\lambda^a\epsilon^{abc}J^{bc}.
\ee
\begin{English}
Here the covariant derivative 
\end{English}
\begin{Russian}
 
\end{Russian}
${\cal D}_i$ 
\begin{English}
is defined by the following relation
\end{English}
\begin{Russian}
  
\end{Russian}
\be
{\cal D}_i\xi^{ka}=\xi^{ka}_{\ |i}+\epsilon^{abc}A^b_i\xi^{kc}.
\ee

\begin{English}
In contrast to its previous form given according to the formula 
\end{English}
\begin{Russian}
     ,  
\end{Russian}
(\ref{eq:Jab}), 
\begin{English}
in the Ashtekar variables the generator density depends on
the spatial derivatives of 
\end{English}
\begin{Russian}
       
 
\end{Russian}
$\tilde E^{ia}$, 
\begin{English}
and so its variation 
\end{English}
\begin{Russian}
   
\end{Russian}
\be
\delta H=
\int \left(\frac{\delta H}{\delta\tilde E^{ia}}\delta\tilde E^{ia}
+\frac{\delta H}{\delta A^a_i}\delta A^a_i
\right),
\ee
\begin{English}
contains a boundary contribution
\end{English}
\begin{Russian}
  
\end{Russian}
\begin{eqnarray}
\frac{\delta H}{\delta\tilde E^{ia}}&=&-\theta_{,i}2\hat\lambda^a-
\theta 2{\cal D}_i\hat\lambda^a,\label{eq:eder}\\
\frac{\delta H}{\delta A^a_i}&=&-\theta 2\epsilon^{abc}\hat\lambda^b
\tilde E^{ic}.
\end{eqnarray}
\begin{English}
If we deal with a free boundary condition case then this functional is
not  ``differentiable'' in the Regge-Teitelboim terminology.
Now we will show that it is nevertheless admissible as it
gives regular Hamiltonian equations. 

The explanation follows from the fact that the Ashtekar variables are
canonical only up to the boundary term
\end{English}
\begin{Russian}
       
``''   -.  
,      ,     
 .

  ,    
      
\end{Russian}
\cite{Sol92}
\be
\left\{A^a_i(x),A^b_j(y)\right\}=\theta_{,k}C^k_{[(ia)(jb)]}\delta(x,y),
\ee
\begin{English}
where the square brackets denote antisymmetrization 
\end{English}
\begin{Russian}
    
\end{Russian}
$i\leftrightarrow j$,
$a\leftrightarrow b$ 
\begin{English}
and
\end{English}
\begin{Russian}

\end{Russian}
\be
C^k_{(ia)(jb)}=\frac{i}{2E}(\epsilon^{acb}\delta^k_jE_{ic}-
\epsilon^{acd}E_{ib}E_{jc}E^{kd}).
\ee

\begin{English}
Due to this noncanonicity the Hamiltonian equations aquire a form
\end{English}
\begin{Russian}
     
 
\end{Russian}
\begin{eqnarray}
\dot {\tilde E}^{ia}(x)&=&\int \{\tilde E^{ia}(x),A^b_j(y)\}
\frac{\delta H}{\delta
A^b_j(y)},\\
\dot A^a_i(x)&=&\int \{A^a_i(x),\tilde E^{jb}(y)\}
\frac{\delta H}{\delta\tilde E^{jb}(y)}+\int\{A^a_i(x),A^b_j(y)\}
\frac{\delta H}{\delta A^b_j(y)},
\end{eqnarray}
\begin{English}
or, in the explicit form obtained by using the prescription from
papers
\end{English}
\begin{Russian}
   ,      
\end{Russian}
\cite{Sol93}-\cite{Sol96}, 
\begin{English}
(here it is simply 
\end{English}
\begin{Russian}
(    
\end{Russian}
$\theta(P)\cdot\theta'(P)=\theta'(P)$), 
\begin{English}
they are
\end{English}
\begin{Russian}
  
\end{Russian}
\begin{eqnarray}
\dot{\tilde E}^{ia}&=& i\epsilon^{abc}\hat\lambda^c\tilde E^{ib},\\
\dot A^a_i&=& i{\cal D}_i\hat\lambda^a,
\end{eqnarray}
\begin{English}
where 
\end{English}
\begin{Russian}

\end{Russian}
$\theta (P)$
\begin{English}
-factors are omitted.

We can see that the singular on the boundary
terms in the second equation have been mutually
canceled despite their presence in the full variational derivative 
\end{English}
\begin{Russian}
-   .

 ,        
 ,       

\end{Russian}
(\ref{eq:eder}). 
\begin{English}
It means that our Hamiltonian is admissible under
arbitrary boundary conditions.

It is not surprising then that the Poisson algebra of these generators is
closed irrespectable to boundary conditions
\end{English}
\begin{Russian}
 ,        
.

 ,       
    
\end{Russian}
\begin{equation}
\{H(\hat\xi^a),H(\hat\eta^b)\}=
H\left(i\epsilon^{cab}\hat\xi^a\hat\eta^b\right).
\end{equation}

\begin{English}
In the Lagrangian treatment it is evident that due to the absence
of spatial derivatives in the action
\end{English}
\begin{Russian}
   ,   
\end{Russian}
\begin{equation}
S=2\int\limits_{t_1}^{t_2}dt\int\theta(P)\left(K^a_i\dot{\tilde E^{ia}}-
i\hat\lambda^a\varepsilon^{abc}J^{bc}\right),
\end{equation}
\begin{English}
any boundary conditions are natural. The Ashtekar transformation
gives it a new appearance
\end{English}
\begin{Russian}
    ,  
  .  , 
,     
\end{Russian}
\begin{equation}
S=-2i\int\limits_{t_1}^{t_2}dt\int\theta(P)\left((A^a_i-\Gamma^a_i)
\dot{\tilde E^{ia}}-
i\hat\lambda^a{\cal D}_i\tilde E^{ia}\right),
\end{equation}
\begin{English}
where this property is no more evident. But let us derive a
variation of the action
\end{English}
\begin{Russian}
      . , , 
\end{Russian}
\begin{eqnarray}
\delta S&=&-2i\int\limits_{t_1}^{t_2}dt\int\theta(P)
\Biggl(\delta{\tilde E^{ia}}
\left[-\dot A^a_i+i{\cal D}_i\tilde E^{ia}\right]+\delta A^a_i\left[
\dot{\tilde E^{ia}}+i\varepsilon^{abc}\lambda^b\tilde E^{ic}\right]+\nonumber\\
&+&\frac{\partial}{\partial t}\left((A^a_i-\Gamma^a_i)
\delta\tilde E^{ia}\right)+\left((\dot\Gamma^a_i\delta\tilde E^{ia}-
\dot{\tilde E^{ia}}\delta\Gamma^a_i)-\partial_i(i\hat\lambda^a
\delta\tilde E^{ia})\right)\Biggr).
\end{eqnarray}
\begin{English}
Then the total time derivative term gives zero contribution as field
variations on the time interval boundary are zero, the terms in square brackets
give the equations of motion and the rest terms give no contribution
due to relation
\end{English}
\begin{Russian}
          
        ,
      ,   
  , 
\end{Russian}
\begin{equation}
(\dot\Gamma^a_i\delta\tilde E^{ia}-
\dot{\tilde E^{ia}}\delta\Gamma^a_i)-\partial_i(i\hat\lambda^a
\delta\tilde E^{ia})=0\ {\rm mod}\left(\dot{\tilde E^{ia}}
+i\varepsilon^{abc}\lambda^b\tilde E^{ic}=0\right),
\end{equation}
\begin{English}
valid for arbitrary functions
\end{English}
\begin{Russian}
  
\end{Russian}
$\delta\tilde  E^{ia}$,
\begin{English}
that can be verified by the straightforward calculation.
\end{English}
\begin{Russian}
   .
\end{Russian}


\begin{English}
As the second example we consider the Hamiltonian description of ideal
fluid with a free surface. In Eulerian variables the action can be
written
\end{English}
\begin{Russian}
       
   .    
   
\end{Russian}
\cite{SW}
\begin{English}
by means of Clebsch potentials
\end{English}
\begin{Russian}
   
\end{Russian}
\be
{\bf v}=\nabla\phi+\frac{\eta}{\rho}\nabla s+\frac{\beta}{\rho}\nabla\alpha,
\ee
\begin{English}
in the following way
\end{English}
\begin{Russian}
 
\end{Russian}
\begin{equation}
S=\int\limits_{t_1}^{t_2}dt\int\theta(P)\left[
\rho\left(\frac{{\bf v}^2}{2}-\Phi({\bf x})-
\varepsilon(\rho,s)\right)-\rho\frac{D\phi}{Dt}
-\eta\frac{Ds}{Dt}-
\beta\frac{D\alpha}{Dt}-\tau K\right].\label{eq:action_Efbtrue}
\end{equation}
\begin{English}
where 
\end{English}
\begin{Russian}

\end{Russian}
$\rho$ 
\begin{English}
is the fluid mass density, 
\end{English}
\begin{Russian}
-   ,
\end{Russian}
$s$ 
\begin{English}
is the specific entropy,
\end{English}
\begin{Russian}
--    ,
\end{Russian}
$\varepsilon=\varepsilon(\rho,s)$ 
\begin{English}
is the specific internal energy density,
\end{English}
\begin{Russian}
--   ,
\end{Russian}
$\tau$ 
\begin{English}
is the surface tension coefficient,
\end{English}
\begin{Russian}
--   ,
\end{Russian}
\be
\frac{D}{Dt}=\frac{\partial}{\partial t}+{\bf v}\cdot\nabla,\qquad
K=-\nabla\cdot\left(\frac{\nabla P}{|\nabla P|}\right).
\ee
\begin{English}
On the boundary the last formula gives the external curvature of it
whereas 
\end{English}
\begin{Russian}
       ,  
\end{Russian}
$\nabla P$ 
\begin{English}
is proportional to the normal, the proposal to use it
is due to Abarbanel et al
\end{English}
\begin{Russian}
  ,    
    .
\end{Russian}
\cite{Abarb}.

\begin{English}
The corresponding symplectic form 
\end{English}
\begin{Russian}
  
\end{Russian}
\begin{equation}
-\int \left[ \theta(P)(\delta\rho\wedge\delta\phi+
\delta\eta\wedge\delta s+\delta\beta\wedge\delta\alpha)+
\theta'(P)\delta P\wedge(\rho\delta\phi+\eta\delta s+\beta
\delta\alpha)\right].\label{eq:symplectic}
\end{equation}
\begin{English}
is degenerate. To go to the Hamiltonian formalism we can use the Dirac
procedure
\end{English}
\begin{Russian}
 .      
  
\end{Russian}
\cite{Dirac64} 
\begin{English}
or the Faddeev-Jackiw approach
\end{English}
\begin{Russian}
  -
\end{Russian}
\cite{FJ},
\begin{English}
but really both of them lead to the same result as a simple trick.
If we introduce  a canonical variable 
\end{English}
\begin{Russian}
         ,  
 .    
\end{Russian}
$\pi$, 
\begin{English}
conjugate to 
\end{English}
\begin{Russian}
 
\end{Russian}
$P$
\be
S\rightarrow S+\int\limits^{t_2}_{t_1}dt\int (\pi\dot P-
\lambda\pi),
\ee
\begin{English}
and so add to the symplectic form the standard term
\end{English}
\begin{Russian}
       
\end{Russian}
\be
\int \delta\pi\wedge \delta P,
\ee
\begin{English}
then the modified symplectic form becomes nondegenerate and can be
inverted for getting the Poisson bivector
\end{English}
\begin{Russian}
       
   ,      

\end{Russian}
\begin{eqnarray}
\Psi&=&\int \Biggl[\theta(P)\left(
\frac{\delta}{\delta\rho}\wedge\frac{\delta}{\delta\phi}+
\frac{\delta}{\delta\eta}\wedge\frac{\delta}{\delta s}+\frac{\delta}
{\delta\beta}\wedge\frac{\delta}{\delta\alpha}
\right)+\nonumber\\
&+&\frac{\delta}{\delta P}\wedge\frac{\delta}{\delta\pi}+
\theta'(P)\frac{\delta}{\delta\pi}\wedge\left(
\rho\frac{\delta}{\delta\rho}+\eta\frac{\delta}{\delta\eta}+
\beta\frac{\delta}{\delta\beta}\right)\Biggr],
\end{eqnarray}
\begin{English}
which explicitely contains the boundary 
\end{English}
\begin{Russian}
    
\end{Russian}
$\delta$
\begin{English}
-function.
\end{English}
\begin{Russian}
-.
\end{Russian}

\begin{English}
The Hamiltonian has the following form
\end{English}
\begin{Russian}
   
\end{Russian}
\begin{equation}
{\rm H}=\int \left[\theta(P)\left(\frac{\rho{\bf v}^2}{2}
+\rho\Phi+\rho\varepsilon
(\rho,s)+\tau K\right)+\lambda \pi\right],
\end{equation}
\begin{English}
and its variation also have singular boundary contributions,
for example,
\end{English}
\begin{Russian}
       , ,
\end{Russian}
\begin{eqnarray}
\frac{\delta{\rm H}}{\delta P}&=&\theta'(P)\left(
\frac{\rho {\bf v}^2}{2}+\rho\Phi+\rho\varepsilon+\tau K
\right),\\
%\frac{\delta{\rm H}}{\delta\rho}&=&\theta(P)\left(
%{\bf v}\nabla\phi-\frac{v^2}{2}+\Phi+\varepsilon+\frac{p}{\rho}\right),\\
\frac{\delta{\rm H}}{\delta\phi}&=&-\theta'(P)\rho{\bf v}\cdot\nabla P-
\theta(P)\nabla(\rho{\bf v}),
%\\
%\frac{\delta{\rm H}}{\delta\eta}&=&\theta(P){\bf v}\cdot\nabla s,\\
%\frac{\delta{\rm H}}{\delta s}&=&-\theta'(P)\eta{\bf v}\cdot\nabla P-
%\theta(P)\left(\nabla(\eta{\bf v})-\rho T\right),\\
%\frac{\delta{\rm H}}{\delta\beta}&=&\theta(P){\bf v}\cdot\nabla\alpha,\\
%\frac{\delta{\rm H}}{\delta\alpha}&=&-\theta'(P)\beta{\bf v}\cdot\nabla P-
%\theta(P)\nabla(\beta{\bf v}),\\
%\qquad \frac{\delta{\rm H}}{\delta \pi_P}=\lambda_P.
\end{eqnarray}
\begin{English}
and so on.
\end{English}
\begin{Russian}
  .
\end{Russian}

\begin{English}
Let us estimate the Hamiltonian vector field according to
the standard formula where the interior product should be
understood according to the definition
given in Refs.
\end{English}
\begin{Russian}
       ,
       , 
 
\end{Russian}
\cite{Sol95}, \cite{Sol96}
\begin{eqnarray}
-\d{\rm H}\inprod\Psi
&=&
\left[-\theta'(P)\rho(\lambda+{\bf v}\cdot\nabla P)-\theta(P)
\nabla(\rho{\bf v})\right]\frac{\delta}{\delta\rho}+
\nonumber\\
&+&\left[-\theta'(P)\eta(\lambda+{\bf v}\cdot\nabla P)-
\theta(P)(\nabla(\eta{\bf v})-\rho T)\right]\frac{\delta}{\delta\eta}
+\nonumber\\
&+&\left[-\theta'(P)\beta(\lambda+{\bf v}\cdot\nabla P)-\theta(P)
\nabla(\beta{\bf v})\right]\frac{\delta}{\delta\beta}
+\nonumber\\
&+&\theta'(P)(p-\tau K)\frac{\delta}
{\delta\pi}
+\nonumber\\
&+&\theta(P)\left[\frac{v^2}{2}-{\bf v}\cdot\nabla\phi-\Phi-
\varepsilon-\frac{p}{\rho}\right]\frac{\delta}{\delta\phi}
-\theta(P)\left[{\bf v}\cdot\nabla s\right]\frac{\delta}{\delta s}
-\nonumber\\
&-&\theta(P){\bf v}\cdot\nabla\alpha\frac{\delta}{\delta\alpha}
+\theta(P)\lambda\frac{\delta}{\delta P},
\end{eqnarray}
\begin{English}
here 
\end{English}
\begin{Russian}

\end{Russian}
$p=\rho^2\frac{\partial\varepsilon}{\partial\rho}$ 
\begin{English}
is the pressure and 
\end{English}
\begin{Russian}
--  
\end{Russian}
$T=\frac{\partial\varepsilon}{\partial s}$ 
\begin{English}
is the temperature.



Then the requirement that this Hamiltonian vector field must  not contain
boundary terms is equivalent to the standard boundary conditions
for the problem
\end{English}
\begin{Russian}
-- .

  ,       
 ,      
 
\end{Russian}
\cite{Lamb}
\begin{eqnarray}
\theta'(P)(\dot P+{\bf v}\cdot\nabla P)&=&0,\label{eq:1}\\
\theta'(P)(p-\tau K)&=&0,\label{eq:2}
\end{eqnarray}
\begin{English}
if we take into account equation of motion
\end{English}
\begin{Russian}
      
\end{Russian}
\begin{equation}
\dot P=
%\theta(P)
\lambda.
\end{equation}
\begin{English}
It is useful to compare this approach with the analisys made in other
variables
\end{English}
\begin{Russian}
     ,    

\end{Russian}
\cite{Abarb}.

\begin{English}
If we try to use the criterion by Regge and Teitelboim here we will get the
wrong boundary conditions, for example, one of them will be
\end{English}
\begin{Russian}
         ,
      , ,   
  
\end{Russian}
\be
{\bf v}\cdot\nabla P=0,
\ee
\begin{English}
that is, the requirement for the boundary to be fixed.
\end{English}
\begin{Russian}
 ,      .
\end{Russian}

\begin{English}
Lagrangian approach to the problem consists in studying the variation
of the action (\ref{eq:action_Efbtrue}). Apart from terms giving equations
of motion this variation also contains boundary terms
\end{English}
\begin{Russian}
        
(\ref{eq:action_Efbtrue}).  ,   
,      
\end{Russian}
\begin{eqnarray}
\delta'S=\int\limits^{t_2}_{t_1}dt\int\Biggl(-\frac{\partial}{\partial t}\Bigl(
\theta\rho\delta\phi+\theta\eta\delta s+\theta\beta\delta\alpha\Bigr)
+\nonumber\\
+\theta'\left(\dot P+{\bf v}\cdot\nabla P\right)\left(\rho\delta\phi+
\eta\delta s+\beta\delta\alpha\right)+\theta'(p-\tau K)\delta P\Biggr).
\end{eqnarray}
\begin{English}
The total time derivative gives the symplectic form (\ref{eq:symplectic})
and does not contribute to the variational principle because the field
variations are zero on the time boundary. The other terms contribute
at the spatial boundary where the field variations are arbitrary
and just give us the natural boundary conditions which are the same as 
(\ref{eq:1}), (\ref{eq:2}).
\end{English}
\begin{Russian}
       
(\ref{eq:symplectic}),       ,  
     \' .  
   ,    
,        
  ,  
(\ref{eq:1}), (\ref{eq:2}).
\end{Russian}
%\section{Conclusion}

\begin{English}
We have shown that in general situation when the Hamiltonian variables
are not necessarily canonical and their Poisson brackets may contain
boundary terms the Regge-Teitelboim criterion of
``differentiability'' of the Hamiltonian must be replaced by a new one.
A Hamiltonian may be considered as admissible if the Hamiltonian vector
field constructed 
according to the extended definitions given in 
\end{English}
\begin{Russian}
 ,    ,     
         ,
 ``'' ,    
,     .  
 ,    , 
  ,   
\end{Russian}
\cite{Sol95}, \cite{Sol96}, 
\begin{English}
does not
contain any boundary contribution. This completes the search for a new
criterion started in 
\end{English}
\begin{Russian}
    .     
,   
\end{Russian}
\cite{Sol92}. 
\begin{English}
In general form the idea can be also
traced to a publication by Mason 
\end{English}
\begin{Russian}
           

\end{Russian}
\cite{Mason}.
\begin{English}
More detailed treatment will be given elsewhere.

We hope that the Hamiltonian approach to field theory with free
boundary conditions will be useful in dealing with different physical
problems, especially those where the Lagrangian approach meets with
difficulties.
\end{English}
\begin{Russian}
       .

 ,         
       
,   ,      .
\end{Russian}

\vspace{12pt}
\begin{English}
{\large\bf Acknowledgements}

This work has been completed during a visit of the author to the International
Centre for Theoretical Physics in Trieste. The author is most grateful
to Professor S.~Randjbar-Daemi for the invitation and kind hospitality, 
partial support from ICTP is gratefully acknowledged. It is a pleasure
to thank A.~Zheltukhin for discussion.
\end{English}
\begin{Russian}
{\large\bf }

          
   .   . .~-
   .     
 .   . .~  
 .
\end{Russian}

\newpage
\hfill
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\begin{English}
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 ,  1, ., 1951)
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\begin{English}
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\bibitem{Sol95}
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\bibitem{Ash}
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%\bibitem{Ash2}
%A.~Ashtekar,
{ Phys. Rev.} { D36}   (1987) 1587;
%\bibitem{Ash3}
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\bibitem{Henn}
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\bibitem{Abarb}
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\bibitem{FJ}
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\end{thebibliography}
\end{document}
\bye


