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General canonical equations of the mechanics of a [[Holonomic system|holonomic system]], presented in terms of a certain Lie algebra of infinitesimal transformations, and equivalent to the [[Poincaré equations|Poincaré equations]].
 
General canonical equations of the mechanics of a [[Holonomic system|holonomic system]], presented in terms of a certain Lie algebra of infinitesimal transformations, and equivalent to the [[Poincaré equations|Poincaré equations]].
  
If instead of independent variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c0220501.png" /> defining real displacements one introduces the quantities
+
If instead of independent variables $  \eta _ {j} $
 +
defining real displacements one introduces the quantities
 +
 
 +
$$
 +
y _ {j}  = 
 +
\frac{\partial  L }{\partial  \eta _ {j} }
 +
,\ \
 +
j = 1 \dots k ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c0220502.png" /></td> </tr></table>
+
where  $  L ( t ; x _ {1} \dots x _ {n} ; \eta _ {1} \dots \eta _ {k} ) $
 +
is the Lagrangian, then the Poincaré equations take the simpler form of the Chetaev equations
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c0220503.png" /> is the Lagrangian, then the Poincaré equations take the simpler form of the Chetaev equations
+
$$ \tag{1 }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c0220504.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
\frac{d y _ {j} }{dt}
 +
  = \sum _ {\alpha , \beta = 1 } ^ { k }
 +
c _ {\alpha j \beta } 
 +
\frac{\partial  H }{\partial  y _  \alpha  }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c0220505.png" /></td> </tr></table>
+
y _  \beta  - X _ {j} H ,\  \eta _ {j}  = \
 +
 
 +
\frac{\partial  H }{\partial  y _ {j} }
 +
,\ \
 +
$$
 +
 
 +
$$
 +
= 1 \dots k ,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c0220506.png" /></td> </tr></table>
+
$$
 +
H ( t ;  x _ {1} \dots x _ {n} ; y _ {1} \dots y _ {k} )
 +
= \sum _ { j= } 1 ^ { k }  \eta _ {j} y _ {j} - L
 +
$$
  
 
is the Hamiltonian. The second group of equations (1) may be replaced by the equations
 
is the Hamiltonian. The second group of equations (1) may be replaced by the equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c0220507.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
  
(Cf. [[Poincaré equations|Poincaré equations]] for the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c0220508.png" />.)
+
\frac{dx _ i}{dt}
 +
  =  \left ( X _ {0} + \sum _ {\alpha = 1 } ^ { k } 
 +
\frac{\partial  H }{\partial  y _  \alpha  }
 +
X _  \alpha  \right ) x _ {i} ,\  i = 1 \dots n .
 +
$$
 +
 
 +
(Cf. [[Poincaré equations|Poincaré equations]] for the operators $  X _ {0} , X _  \alpha  $.)
  
 
Introducing the action function by the formula
 
Introducing the action function by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c0220509.png" /></td> </tr></table>
+
$$
 +
V ( t ; x _ {1} \dots x _ {n} ; x _ {1}  ^ {0} \dots x _ {n}  ^ {0} )
 +
= \int\limits _ {t _ {0} } ^ { t }
 +
\left ( \sum _ {\alpha = 1 } ^ { k }  y _  \alpha  \eta _  \alpha  - H \
 +
\right )  dt ,
 +
$$
  
 
where the integration takes place over the real trajectories of the system, one obtains the relations
 
where the integration takes place over the real trajectories of the system, one obtains the relations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
y _  \alpha  = X _  \alpha  V ,\  y _  \alpha  ^ {0}  = \
 +
- X _  \alpha  ^ { 0 } V ,\  \alpha = 1 \dots k .
 +
$$
  
Here the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205011.png" /> denote the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205012.png" /> applied to the initial momentum at time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205013.png" /> and the initial position of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205014.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205015.png" /> are the initial values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205016.png" />. If the action function is known, then the equations (3) solve the mechanical problem in which the second group of equations (3) defines, in an implicit way, the law of motion of the system.
+
Here the $  X _  \alpha  ^ { 0 } $
 +
denote the operators $  X _  \alpha  $
 +
applied to the initial momentum at time $  t _ {0} $
 +
and the initial position of the system $  x _ {i}  ^ {0} $;  
 +
and $  y _  \alpha  ^ {0} $
 +
are the initial values of $  y _  \alpha  $.  
 +
If the action function is known, then the equations (3) solve the mechanical problem in which the second group of equations (3) defines, in an implicit way, the law of motion of the system.
  
 
The action function satisfies the first-order partial differential equation
 
The action function satisfies the first-order partial differential equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
X _ {0} V + H ( t , x _ {1} \dots x _ {n} , X _ {1} V \dots
 +
X _ {k} V  )  = 0 .
 +
$$
  
If a complete integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205018.png" /> of (4) is known, then the solutions to the Chetaev equations are determined by the relations
+
If a complete integral $  V ( t , x _ {1} \dots x _ {n} ;  a _ {1} \dots a _ {n} ) $
 +
of (4) is known, then the solutions to the Chetaev equations are determined by the relations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205019.png" /></td> </tr></table>
+
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205021.png" /> are arbitrary constants, restricted by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205022.png" /> integrability equations of the constraint.
+
\frac{\partial  V }{\partial  a _ {i} }
 +
  = b _ {i} ,\ \
 +
y _ {j}  = X _ {j} V ,\ \
 +
i = 1 \dots n ,\  j = 1 \dots k ,
 +
$$
  
Instead of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205023.png" /> one can consider new variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205024.png" /> defining the position of the system. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205027.png" />, represent the Lie algebra of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205028.png" />-parameter Lie group of continuous transformations in the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205029.png" /> with structure constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205030.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205031.png" />; and suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205033.png" /> are variables defining possible and actual displacements, so that for some function
+
where  $  a _ {i} $
 +
and  $  b _ {i} $
 +
are arbitrary constants, restricted by the $  n - k $
 +
integrability equations of the constraint.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205034.png" /></td> </tr></table>
+
Instead of the variables  $  x _ {i} $
 +
one can consider new variables  $  \alpha _ {i} $
 +
defining the position of the system. Suppose that  $  A _ {0} = \partial  / \partial  t $,
 +
$  A _ {s} $,
 +
$  s = 1 \dots k $,
 +
represent the Lie algebra of a  $  ( k + 1 ) $-
 +
parameter Lie group of continuous transformations in the variables  $  \alpha _ {i} $
 +
with structure constants  $  \gamma _ {srj} $,
 +
where  $  \gamma _ {0rj} = 0 $;
 +
and suppose that  $  \pi _ {s} $
 +
and  $  \theta _ {s} $
 +
are variables defining possible and actual displacements, so that for some function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205035.png" /></td> </tr></table>
+
$$
 +
f ( t , \alpha _ {1} \dots \alpha _ {n} ),
 +
$$
 +
 
 +
$$
 +
\delta f  = \sum _ { s= } 1 ^ { k }  \pi _ {s} A _ {s} f ,\  d
 +
= \left (
 +
\frac{\partial  f }{\partial  t }
 +
+ \sum _ { s= } 1 ^ { k }  \theta _ {s} A _ {s} f  \right )  d t .
 +
$$
  
 
The transformation of variables is determined by the characteristic function
 
The transformation of variables is determined by the characteristic function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205036.png" /></td> </tr></table>
+
$$
 +
V ( t , x _ {1} \dots x _ {n} , \alpha _ {1} \dots \alpha _ {n} ),\ \
 +
\left \|
 +
\frac{\partial  ^ {2} V }{\partial  x _ {i} \partial  \alpha _ {j} }
 +
 
 +
\right \|  \neq  0 ,
 +
$$
  
 
and the formulas
 
and the formulas
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205037.png" /></td> </tr></table>
+
$$
 +
y _ {s}  = X _ {s} V ,\ \
 +
\beta _ {s}  = - A _ {s} V ,\ \
 +
s = 1 \dots k ,
 +
$$
  
 
together with the integrability equations of the constraints. Such transformations are called canonical transformations, they preserve the canonical form of the equations of motion, in which the Hamilton function in the new variables takes the form
 
together with the integrability equations of the constraints. Such transformations are called canonical transformations, they preserve the canonical form of the equations of motion, in which the Hamilton function in the new variables takes the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205038.png" /></td> </tr></table>
+
$$
 +
H  ^ {*} ( t , \alpha , \beta )  = \
 +
 
 +
\frac{\partial  V }{\partial  t }
 +
+ H .
 +
$$
 +
 
 +
(Cf. also [[Hamiltonian system|Hamiltonian system]].) If the characteristic function of the transformation is a complete integral of equation (4) (for  $  X _ {0} = \partial  / \partial  t $),
 +
then  $  H  ^ {*} = 0 $
 +
and the Chetaev equations (1) and (2) in the new variables take the form
 +
 
 +
$$
  
(Cf. also [[Hamiltonian system|Hamiltonian system]].) If the characteristic function of the transformation is a complete integral of equation (4) (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205039.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205040.png" /> and the Chetaev equations (1) and (2) in the new variables take the form
+
\frac{d \beta _ {s} }{dt}
 +
  = \sum _ { r,j= } 1 ^ { k }
 +
\gamma _ {rsj} \theta _ {r} \beta _ {j} ,\  \theta _ {s}  = 0 ,\ \
 +
s = 1 \dots k ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205041.png" /></td> </tr></table>
+
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205042.png" /></td> </tr></table>
+
\frac{d \alpha _ {i} }{dt}
 +
  = \sum _ { s } \theta _ {s} A _ {s} \alpha _ {i} ,
 +
$$
  
that is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205046.png" />.
+
that is $  \alpha _ {i} = \textrm{ const } $,  
 +
$  \beta _ {s} = \textrm{ const } $,  
 +
$  i = 1 \dots n $,  
 +
$  s = 1 \dots k $.
  
The linear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205047.png" /> defines the fundamental relative integral invariant of dynamics.
+
The linear form $  \Omega = \sum _ {s=} 1  ^ {k} y _ {s} \omega _ {s} $
 +
defines the fundamental relative integral invariant of dynamics.
  
The condition for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205048.png" /> to be constant is that the first integral of Chetaev's equations has the form
+
The condition for $  f ( t , x _ {1} \dots x _ {n} , y _ {1} \dots y _ {k)} $
 +
to be constant is that the first integral of Chetaev's equations has the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205049.png" /></td> </tr></table>
+
$$
 +
X _ {0} f + ( H , f  )  = 0 ,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205050.png" /></td> </tr></table>
+
$$
 +
( f , g )  = \sum _ {\alpha = 1 } ^ { k }
 +
\left (
 +
\frac{\partial  f }{\partial  y _  \alpha  }
 +
X _  \alpha  g -
 +
 
 +
\frac{\partial  g }{\partial  y _  \alpha  }
 +
X _  \alpha  f  \right ) +
 +
$$
 +
 
 +
$$
 +
+
 +
\sum _ {\alpha , \beta , s = 1 } ^ { k }  c _ {\alpha \beta s
 +
}  y _ {s}
 +
\frac{\partial  f }{\partial  y _  \alpha  }
 +
 +
\frac{\partial  g }{\partial  y _  \beta  }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205051.png" /></td> </tr></table>
+
$$
  
 
defines the Poisson bracket.
 
defines the Poisson bracket.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205053.png" /> are first integrals, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022050/c02205054.png" /> is also an integral (a generalization of the [[Poisson theorem|Poisson theorem]]).
+
If $  f = a $
 +
and $  g = b $
 +
are first integrals, then $  ( f , g ) = c $
 +
is also an integral (a generalization of the [[Poisson theorem|Poisson theorem]]).
  
 
The Chetaev equations were introduced by N.G. Chetaev [[#References|[1]]]–[[#References|[3]]], who also developed their theory.
 
The Chetaev equations were introduced by N.G. Chetaev [[#References|[1]]]–[[#References|[3]]], who also developed their theory.

Revision as of 16:43, 4 June 2020


General canonical equations of the mechanics of a holonomic system, presented in terms of a certain Lie algebra of infinitesimal transformations, and equivalent to the Poincaré equations.

If instead of independent variables $ \eta _ {j} $ defining real displacements one introduces the quantities

$$ y _ {j} = \frac{\partial L }{\partial \eta _ {j} } ,\ \ j = 1 \dots k , $$

where $ L ( t ; x _ {1} \dots x _ {n} ; \eta _ {1} \dots \eta _ {k} ) $ is the Lagrangian, then the Poincaré equations take the simpler form of the Chetaev equations

$$ \tag{1 } \frac{d y _ {j} }{dt} = \sum _ {\alpha , \beta = 1 } ^ { k } c _ {\alpha j \beta } \frac{\partial H }{\partial y _ \alpha } y _ \beta - X _ {j} H ,\ \eta _ {j} = \ \frac{\partial H }{\partial y _ {j} } ,\ \ $$

$$ j = 1 \dots k , $$

where

$$ H ( t ; x _ {1} \dots x _ {n} ; y _ {1} \dots y _ {k} ) = \sum _ { j= } 1 ^ { k } \eta _ {j} y _ {j} - L $$

is the Hamiltonian. The second group of equations (1) may be replaced by the equations

$$ \tag{2 } \frac{dx _ i}{dt} = \left ( X _ {0} + \sum _ {\alpha = 1 } ^ { k } \frac{\partial H }{\partial y _ \alpha } X _ \alpha \right ) x _ {i} ,\ i = 1 \dots n . $$

(Cf. Poincaré equations for the operators $ X _ {0} , X _ \alpha $.)

Introducing the action function by the formula

$$ V ( t ; x _ {1} \dots x _ {n} ; x _ {1} ^ {0} \dots x _ {n} ^ {0} ) = \int\limits _ {t _ {0} } ^ { t } \left ( \sum _ {\alpha = 1 } ^ { k } y _ \alpha \eta _ \alpha - H \ \right ) dt , $$

where the integration takes place over the real trajectories of the system, one obtains the relations

$$ \tag{3 } y _ \alpha = X _ \alpha V ,\ y _ \alpha ^ {0} = \ - X _ \alpha ^ { 0 } V ,\ \alpha = 1 \dots k . $$

Here the $ X _ \alpha ^ { 0 } $ denote the operators $ X _ \alpha $ applied to the initial momentum at time $ t _ {0} $ and the initial position of the system $ x _ {i} ^ {0} $; and $ y _ \alpha ^ {0} $ are the initial values of $ y _ \alpha $. If the action function is known, then the equations (3) solve the mechanical problem in which the second group of equations (3) defines, in an implicit way, the law of motion of the system.

The action function satisfies the first-order partial differential equation

$$ \tag{4 } X _ {0} V + H ( t , x _ {1} \dots x _ {n} , X _ {1} V \dots X _ {k} V ) = 0 . $$

If a complete integral $ V ( t , x _ {1} \dots x _ {n} ; a _ {1} \dots a _ {n} ) $ of (4) is known, then the solutions to the Chetaev equations are determined by the relations

$$ \frac{\partial V }{\partial a _ {i} } = b _ {i} ,\ \ y _ {j} = X _ {j} V ,\ \ i = 1 \dots n ,\ j = 1 \dots k , $$

where $ a _ {i} $ and $ b _ {i} $ are arbitrary constants, restricted by the $ n - k $ integrability equations of the constraint.

Instead of the variables $ x _ {i} $ one can consider new variables $ \alpha _ {i} $ defining the position of the system. Suppose that $ A _ {0} = \partial / \partial t $, $ A _ {s} $, $ s = 1 \dots k $, represent the Lie algebra of a $ ( k + 1 ) $- parameter Lie group of continuous transformations in the variables $ \alpha _ {i} $ with structure constants $ \gamma _ {srj} $, where $ \gamma _ {0rj} = 0 $; and suppose that $ \pi _ {s} $ and $ \theta _ {s} $ are variables defining possible and actual displacements, so that for some function

$$ f ( t , \alpha _ {1} \dots \alpha _ {n} ), $$

$$ \delta f = \sum _ { s= } 1 ^ { k } \pi _ {s} A _ {s} f ,\ d f = \left ( \frac{\partial f }{\partial t } + \sum _ { s= } 1 ^ { k } \theta _ {s} A _ {s} f \right ) d t . $$

The transformation of variables is determined by the characteristic function

$$ V ( t , x _ {1} \dots x _ {n} , \alpha _ {1} \dots \alpha _ {n} ),\ \ \left \| \frac{\partial ^ {2} V }{\partial x _ {i} \partial \alpha _ {j} } \right \| \neq 0 , $$

and the formulas

$$ y _ {s} = X _ {s} V ,\ \ \beta _ {s} = - A _ {s} V ,\ \ s = 1 \dots k , $$

together with the integrability equations of the constraints. Such transformations are called canonical transformations, they preserve the canonical form of the equations of motion, in which the Hamilton function in the new variables takes the form

$$ H ^ {*} ( t , \alpha , \beta ) = \ \frac{\partial V }{\partial t } + H . $$

(Cf. also Hamiltonian system.) If the characteristic function of the transformation is a complete integral of equation (4) (for $ X _ {0} = \partial / \partial t $), then $ H ^ {*} = 0 $ and the Chetaev equations (1) and (2) in the new variables take the form

$$ \frac{d \beta _ {s} }{dt} = \sum _ { r,j= } 1 ^ { k } \gamma _ {rsj} \theta _ {r} \beta _ {j} ,\ \theta _ {s} = 0 ,\ \ s = 1 \dots k , $$

$$ \frac{d \alpha _ {i} }{dt} = \sum _ { s } \theta _ {s} A _ {s} \alpha _ {i} , $$

that is $ \alpha _ {i} = \textrm{ const } $, $ \beta _ {s} = \textrm{ const } $, $ i = 1 \dots n $, $ s = 1 \dots k $.

The linear form $ \Omega = \sum _ {s=} 1 ^ {k} y _ {s} \omega _ {s} $ defines the fundamental relative integral invariant of dynamics.

The condition for $ f ( t , x _ {1} \dots x _ {n} , y _ {1} \dots y _ {k)} $ to be constant is that the first integral of Chetaev's equations has the form

$$ X _ {0} f + ( H , f ) = 0 , $$

where

$$ ( f , g ) = \sum _ {\alpha = 1 } ^ { k } \left ( \frac{\partial f }{\partial y _ \alpha } X _ \alpha g - \frac{\partial g }{\partial y _ \alpha } X _ \alpha f \right ) + $$

$$ + \sum _ {\alpha , \beta , s = 1 } ^ { k } c _ {\alpha \beta s } y _ {s} \frac{\partial f }{\partial y _ \alpha } \frac{\partial g }{\partial y _ \beta } $$

defines the Poisson bracket.

If $ f = a $ and $ g = b $ are first integrals, then $ ( f , g ) = c $ is also an integral (a generalization of the Poisson theorem).

The Chetaev equations were introduced by N.G. Chetaev [1][3], who also developed their theory.

References

[1] N.G. Chetaev, "Sur les équations de Poincaré" C.R. Acad. Sci. Paris , 185 (1927) pp. 1577–1578
[2] N.G. Chetaev, "Sur les équations de Poincaré" Dokl. Akad. Nauk SSSR Ser A. : 7 (1928) pp. 103–104
[3] N.G. Chetaev, "On the equations of Poincaré" Prikl. Mat. i Mekh. , 5 : 2 (1941) pp. 253–262 (In Russian)
How to Cite This Entry:
Chetaev equations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Chetaev_equations&oldid=46335
This article was adapted from an original article by V.V. Rumyantsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article