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General equations of the mechanics of holonomic systems (cf. [[Holonomic system|Holonomic system]]) presented with the help of a certain Lie algebra of infinitesimal transformations.
 
General equations of the mechanics of holonomic systems (cf. [[Holonomic system|Holonomic system]]) presented with the help of a certain Lie algebra of infinitesimal transformations.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p0730401.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p0730402.png" />, be variables determining the position of a holonomic mechanical system constrained by ideal constraints depending explicitly on time. If the system has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p0730403.png" /> degrees of freedom, then there is an intransitive algebra of infinitesimal transformations
+
Let $  x _ {i} $,  
 +
$  i = 1 \dots n $,  
 +
be variables determining the position of a holonomic mechanical system constrained by ideal constraints depending explicitly on time. If the system has $  k $
 +
degrees of freedom, then there is an intransitive algebra of infinitesimal transformations
 +
 
 +
$$
 +
X _ {0}  = 
 +
\frac \partial {\partial  t }
 +
 
 +
+ \sum _ { j= } 1 ^ { n }
 +
\xi _ {0}  ^ {j}
 +
\frac \partial {\partial  x _ {j} }
 +
,\ \
 +
X _ {a}  =  \sum _ { j= } 1 ^ { n }  \xi _  \alpha  ^ {j}
  
<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/p/p073/p073040/p0730404.png" /></td> </tr></table>
+
\frac \partial {\partial  x _ {j} }
 +
,
 +
$$
  
<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/p/p073/p073040/p0730405.png" /></td> </tr></table>
+
$$
 +
\alpha  = 1 \dots k ,
 +
$$
  
enabling the system to be carried at the moment of time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p0730406.png" /> from the position <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p0730407.png" /> to the infinitesimally-near actual position <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p0730408.png" /> and the virtual position <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p0730409.png" /> by infinitesimal transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304010.png" /> of the algebra and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304011.png" /> of the subalgebra, respectively. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304013.png" />, the independent variables determining the virtual and actual displacements respectively, are connected by the equations
+
enabling the system to be carried at the moment of time $  t $
 +
from the position $  x _ {i} $
 +
to the infinitesimally-near actual position $  x _ {i} + d x _ {i} $
 +
and the virtual position $  x _ {i} + \delta x _ {i} $
 +
by infinitesimal transformations $  ( X _ {0} + \sum _ {\alpha = 1 }  ^ {k} \eta _  \alpha  X _  \alpha  )  d t $
 +
of the algebra and $  \sum _ {\alpha = 1 }  ^ {k} \omega _ {i} X _ {i} $
 +
of the subalgebra, respectively. Here $  \omega _  \alpha  $
 +
and $  \eta _  \alpha  $,  
 +
the independent variables determining the virtual and actual displacements respectively, are connected 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/p/p073/p073040/p07304014.png" /></td> </tr></table>
+
$$
 +
\delta {\eta _ {i} }  = \
  
if the algebra of virtual displacements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304015.png" /> is determined by its structure constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304016.png" />:
+
\frac{d \omega _ {i} }{dt}
 +
-
 +
\sum _ {\alpha , \beta = 1 } ^ { k }  c _ {\alpha \beta i }  \omega _  \alpha  \eta _  \beta  ,\ \
 +
i = 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/p/p073/p073040/p07304017.png" /></td> </tr></table>
+
if the algebra of virtual displacements  $  X _  \alpha  $
 +
is determined by its structure constants  $  c _ {\alpha \beta i }  $:
  
<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/p/p073/p073040/p07304018.png" /></td> </tr></table>
+
$$
 +
( X _  \alpha  X _  \beta  )  = \
 +
X _  \alpha  X _  \beta  - X _  \beta  X _  \alpha  = \
 +
\sum _ { i= } 1 ^ { k }  c _ {\alpha \beta i }  X _ {i} ,
 +
$$
  
and the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304019.png" /> commutes with the algebra of virtual displacements
+
$$
 +
\alpha , \beta  = 1 \dots k ,
 +
$$
  
<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/p/p073/p073040/p07304020.png" /></td> </tr></table>
+
and the operator  $  X _ {0} $
 +
commutes with the algebra of virtual displacements
 +
 
 +
$$
 +
( X _ {0} X _  \alpha  )  = 0 ,\  \alpha = 1 \dots k .
 +
$$
  
 
Below these conditions are assumed to hold.
 
Below these conditions are assumed to hold.
  
The Poincaré equations are ordinary first-order (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304021.png" />) differential equations
+
The Poincaré equations are ordinary first-order (with respect to $  \eta _ {j} $)  
 +
differential 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/p/p073/p073040/p07304022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304023.png" />,
+
\frac{d}{dt}
  
<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/p/p073/p073040/p07304024.png" /></td> </tr></table>
+
\frac{\partial  L }{\partial  \eta _ {j} }
 +
\
 +
= \sum _ {\alpha , \beta = 1 } ^ { k }  c _ {\alpha j \beta }
 +
\eta _  \alpha 
 +
\frac{\partial  L }{\partial  \eta _  \beta  }
  
is the Lagrange function, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304025.png" /> is the kinetic energy and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304026.png" /> is the force function.
+
+ X _ {j} L ,
 +
$$
 +
 
 +
where  $  j = 1 \dots k $,
 +
 
 +
$$
 +
L ( t , x _ {1} \dots x _ {n} , \eta _ {1} \dots \eta _ {k} )  = \
 +
T + U
 +
$$
 +
 
 +
is the Lagrange function, $  T ( t , x , \eta ) $
 +
is the kinetic energy and $  U ( t , x ) $
 +
is the force function.
  
 
The equations (1) were first obtained by H. Poincaré (see [[#References|[1]]]) in the case when the algebra of virtual displacements is transitive and the constraints do not depend explicitly on time, and he applied them (see [[#References|[2]]]) to investigate the motion of a solid body with an ellipsoidal cavity, entirely filled by a uniformly vortex moving ideal fluid. N.G. Chetaev (see [[#References|[3]]]) generalized and developed the theory of the Poincaré equations to the case where the algebra of displacements is intransitive and the constraints depend explicitly on time (see [[#References|[3]]]–[[#References|[5]]]) and also converted them to a simpler canonical form (see [[Chetaev equations|Chetaev equations]]). In particular, he gave (see [[#References|[5]]]) a method for constructing the algebra of virtual and actual displacements when the holonomic constraints are given by a differential form and he introduced the important concept of cyclic displacements.
 
The equations (1) were first obtained by H. Poincaré (see [[#References|[1]]]) in the case when the algebra of virtual displacements is transitive and the constraints do not depend explicitly on time, and he applied them (see [[#References|[2]]]) to investigate the motion of a solid body with an ellipsoidal cavity, entirely filled by a uniformly vortex moving ideal fluid. N.G. Chetaev (see [[#References|[3]]]) generalized and developed the theory of the Poincaré equations to the case where the algebra of displacements is intransitive and the constraints depend explicitly on time (see [[#References|[3]]]–[[#References|[5]]]) and also converted them to a simpler canonical form (see [[Chetaev equations|Chetaev equations]]). In particular, he gave (see [[#References|[5]]]) a method for constructing the algebra of virtual and actual displacements when the holonomic constraints are given by a differential form and he introduced the important concept of cyclic displacements.
  
Displacements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304028.png" />, are said to be cyclic if they satisfy the conditions: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304029.png" />; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304032.png" />.
+
Displacements $  X _ {r} $,  
 +
$  r = s + 1 \dots k $,  
 +
are said to be cyclic if they satisfy the conditions: 1) $  X _ {r} L = 0 $;  
 +
2) $  ( X _ {r} X _  \beta  ) = 0 $,  
 +
$  r = s + 1 \dots k $,  
 +
$  \beta = 1 \dots k $.
  
According to 2) the cyclic displacements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304033.png" /> form an Abelian subalgebra of the algebra of virtual displacements which commutes with all the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304034.png" />. For cyclic displacements the first integrals of the Poincaré equations,
+
According to 2) the cyclic displacements $  X _ {r} $
 +
form an Abelian subalgebra of the algebra of virtual displacements which commutes with all the operators $  X _  \beta  $.  
 +
For cyclic displacements the first integrals of the Poincaré 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/p/p073/p073040/p07304035.png" /></td> </tr></table>
+
$$
  
exist. From these relations the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304036.png" /> can be expressed in terms of the constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304037.png" /> and the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304040.png" />, and one can introduce the Routh function
+
\frac{\partial  L }{\partial  \eta _ {r} }
 +
\
 +
= a _ {r}  = \textrm{ const } ,\ \
 +
r = 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/p/p073/p073040/p07304041.png" /></td> </tr></table>
+
exist. From these relations the variables  $  \eta _ {r} $
 +
can be expressed in terms of the constants  $  a _ {r} $
 +
and the variables  $  t $,
 +
$  x _ {i} $,
 +
$  \eta _ {1} \dots \eta _ {s} $,
 +
and one can introduce the Routh 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/p/p073/p073040/p07304042.png" /></td> </tr></table>
+
$$
 +
R ( t , x _ {1} \dots x _ {n} ;  \eta _ {1} \dots \eta _ {s} ; \
 +
a _ {s+} 1 \dots a _ {k} ) =
 +
$$
 +
 
 +
$$
 +
= \
 +
L - \sum _ {r = s + 1 } ^ { k } 
 +
\frac{\partial  L }{\partial  \eta _ {r} }
 +
\eta _ {r} .
 +
$$
  
 
Then for the non-cyclic displacements the Poincaré equations take the form
 
Then for the non-cyclic displacements the Poincaré equations take 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/p/p073/p073040/p07304043.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
 
 +
\frac{d}{dt}
 +
 +
\frac{\partial  R }{\partial  \eta _ {j} }
 +
\
 +
= \sum c _ {\alpha j \beta }  \eta _  \alpha 
 +
\frac{\partial  R }{\partial  \eta _  \beta  }
 +
+
 +
\sum c _ {\alpha j \gamma }  \eta _  \alpha  a _  \gamma  + X _ {j} R ,
 +
$$
 +
 
 +
$$
 +
\alpha , j , \beta  = 1 \dots s ; \  \gamma  = s + 1 \dots k .
 +
$$
  
<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/p/p073/p073040/p07304044.png" /></td> </tr></table>
+
After integrating the equations (2) the values  $  \eta _ {r} $
 +
are determined by the equations
  
After integrating the equations (2) the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304045.png" /> are determined by the equations
+
$$
 +
\eta _ {r}  = -
  
<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/p/p073/p073040/p07304046.png" /></td> </tr></table>
+
\frac{\partial  R }{\partial  a _ {r} }
 +
,\ \
 +
r = s + 1 \dots k .
 +
$$
  
If, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304049.png" />, that is, if the non-cyclic displacements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304051.png" />, form a subalgebra of the algebra of virtual displacements, then the mechanical system corresponding to this subalgebra forms some sort of independent holonomic system with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304052.png" /> degrees of freedom, described by the equations (1) when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304053.png" />, where the role of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304054.png" /> is played by the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304055.png" />.
+
If, in addition, $  c _ {\alpha j \gamma }  = 0 $,
 +
$  \alpha , j = 1 \dots s $,  
 +
$  \gamma = s + 1 \dots k $,  
 +
that is, if the non-cyclic displacements $  X _  \beta  $,  
 +
$  \beta = 1 \dots k $,  
 +
form a subalgebra of the algebra of virtual displacements, then the mechanical system corresponding to this subalgebra forms some sort of independent holonomic system with $  s $
 +
degrees of freedom, described by the equations (1) when $  \alpha , j , \beta = 1 \dots s $,  
 +
where the role of the function $  L $
 +
is played by the function $  R $.
  
The Poincaré equations contain as particular cases: the [[Lagrange equations (in mechanics)|Lagrange equations (in mechanics)]] when the transformation algebra of infinitesimal increments in one of the variables reduces to a commutative algebra of transformations; and the [[Euler equation|Euler equation]] for the rotation of a solid body, when the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304056.png" /> are taken to be the projections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073040/p07304059.png" /> of the instantaneous angular velocity.
+
The Poincaré equations contain as particular cases: the [[Lagrange equations (in mechanics)|Lagrange equations (in mechanics)]] when the transformation algebra of infinitesimal increments in one of the variables reduces to a commutative algebra of transformations; and the [[Euler equation|Euler equation]] for the rotation of a solid body, when the $  \eta _ {i} $
 +
are taken to be the projections p $,  
 +
$  q $
 +
and $  r $
 +
of the instantaneous angular velocity.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Poincaré,  ''C.R. Acad. Sci. Paris'' , '''132'''  (1901)  pp. 369–371</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Poincaré,  ''Bull. Astron.'' , '''27'''  (1910)  pp. 321–356</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.G. Chetaev,  ''Dokl. Akad. Nauk SSSR'' , '''7'''  (1928)  pp. 103–104</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.G. Chetaev,  "Sur les équations de Poincaré"  ''C.R. Acad. Sci. Paris'' , '''185'''  (1927)  pp. 1577–1578</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.G. Chetaev,  ''Prikl. Mat. i. Mekh.'' , '''5''' :  2  (1941)  pp. 253–262</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Poincaré,  ''C.R. Acad. Sci. Paris'' , '''132'''  (1901)  pp. 369–371</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Poincaré,  ''Bull. Astron.'' , '''27'''  (1910)  pp. 321–356</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.G. Chetaev,  ''Dokl. Akad. Nauk SSSR'' , '''7'''  (1928)  pp. 103–104</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.G. Chetaev,  "Sur les équations de Poincaré"  ''C.R. Acad. Sci. Paris'' , '''185'''  (1927)  pp. 1577–1578</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.G. Chetaev,  ''Prikl. Mat. i. Mekh.'' , '''5''' :  2  (1941)  pp. 253–262</TD></TR></table>

Revision as of 08:06, 6 June 2020


General equations of the mechanics of holonomic systems (cf. Holonomic system) presented with the help of a certain Lie algebra of infinitesimal transformations.

Let $ x _ {i} $, $ i = 1 \dots n $, be variables determining the position of a holonomic mechanical system constrained by ideal constraints depending explicitly on time. If the system has $ k $ degrees of freedom, then there is an intransitive algebra of infinitesimal transformations

$$ X _ {0} = \frac \partial {\partial t } + \sum _ { j= } 1 ^ { n } \xi _ {0} ^ {j} \frac \partial {\partial x _ {j} } ,\ \ X _ {a} = \sum _ { j= } 1 ^ { n } \xi _ \alpha ^ {j} \frac \partial {\partial x _ {j} } , $$

$$ \alpha = 1 \dots k , $$

enabling the system to be carried at the moment of time $ t $ from the position $ x _ {i} $ to the infinitesimally-near actual position $ x _ {i} + d x _ {i} $ and the virtual position $ x _ {i} + \delta x _ {i} $ by infinitesimal transformations $ ( X _ {0} + \sum _ {\alpha = 1 } ^ {k} \eta _ \alpha X _ \alpha ) d t $ of the algebra and $ \sum _ {\alpha = 1 } ^ {k} \omega _ {i} X _ {i} $ of the subalgebra, respectively. Here $ \omega _ \alpha $ and $ \eta _ \alpha $, the independent variables determining the virtual and actual displacements respectively, are connected by the equations

$$ \delta {\eta _ {i} } = \ \frac{d \omega _ {i} }{dt} - \sum _ {\alpha , \beta = 1 } ^ { k } c _ {\alpha \beta i } \omega _ \alpha \eta _ \beta ,\ \ i = 1 \dots k , $$

if the algebra of virtual displacements $ X _ \alpha $ is determined by its structure constants $ c _ {\alpha \beta i } $:

$$ ( X _ \alpha X _ \beta ) = \ X _ \alpha X _ \beta - X _ \beta X _ \alpha = \ \sum _ { i= } 1 ^ { k } c _ {\alpha \beta i } X _ {i} , $$

$$ \alpha , \beta = 1 \dots k , $$

and the operator $ X _ {0} $ commutes with the algebra of virtual displacements

$$ ( X _ {0} X _ \alpha ) = 0 ,\ \alpha = 1 \dots k . $$

Below these conditions are assumed to hold.

The Poincaré equations are ordinary first-order (with respect to $ \eta _ {j} $) differential equations

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

where $ j = 1 \dots k $,

$$ L ( t , x _ {1} \dots x _ {n} , \eta _ {1} \dots \eta _ {k} ) = \ T + U $$

is the Lagrange function, $ T ( t , x , \eta ) $ is the kinetic energy and $ U ( t , x ) $ is the force function.

The equations (1) were first obtained by H. Poincaré (see [1]) in the case when the algebra of virtual displacements is transitive and the constraints do not depend explicitly on time, and he applied them (see [2]) to investigate the motion of a solid body with an ellipsoidal cavity, entirely filled by a uniformly vortex moving ideal fluid. N.G. Chetaev (see [3]) generalized and developed the theory of the Poincaré equations to the case where the algebra of displacements is intransitive and the constraints depend explicitly on time (see [3][5]) and also converted them to a simpler canonical form (see Chetaev equations). In particular, he gave (see [5]) a method for constructing the algebra of virtual and actual displacements when the holonomic constraints are given by a differential form and he introduced the important concept of cyclic displacements.

Displacements $ X _ {r} $, $ r = s + 1 \dots k $, are said to be cyclic if they satisfy the conditions: 1) $ X _ {r} L = 0 $; 2) $ ( X _ {r} X _ \beta ) = 0 $, $ r = s + 1 \dots k $, $ \beta = 1 \dots k $.

According to 2) the cyclic displacements $ X _ {r} $ form an Abelian subalgebra of the algebra of virtual displacements which commutes with all the operators $ X _ \beta $. For cyclic displacements the first integrals of the Poincaré equations,

$$ \frac{\partial L }{\partial \eta _ {r} } \ = a _ {r} = \textrm{ const } ,\ \ r = s + 1 \dots k , $$

exist. From these relations the variables $ \eta _ {r} $ can be expressed in terms of the constants $ a _ {r} $ and the variables $ t $, $ x _ {i} $, $ \eta _ {1} \dots \eta _ {s} $, and one can introduce the Routh function

$$ R ( t , x _ {1} \dots x _ {n} ; \eta _ {1} \dots \eta _ {s} ; \ a _ {s+} 1 \dots a _ {k} ) = $$

$$ = \ L - \sum _ {r = s + 1 } ^ { k } \frac{\partial L }{\partial \eta _ {r} } \eta _ {r} . $$

Then for the non-cyclic displacements the Poincaré equations take the form

$$ \tag{2 } \frac{d}{dt} \frac{\partial R }{\partial \eta _ {j} } \ = \sum c _ {\alpha j \beta } \eta _ \alpha \frac{\partial R }{\partial \eta _ \beta } + \sum c _ {\alpha j \gamma } \eta _ \alpha a _ \gamma + X _ {j} R , $$

$$ \alpha , j , \beta = 1 \dots s ; \ \gamma = s + 1 \dots k . $$

After integrating the equations (2) the values $ \eta _ {r} $ are determined by the equations

$$ \eta _ {r} = - \frac{\partial R }{\partial a _ {r} } ,\ \ r = s + 1 \dots k . $$

If, in addition, $ c _ {\alpha j \gamma } = 0 $, $ \alpha , j = 1 \dots s $, $ \gamma = s + 1 \dots k $, that is, if the non-cyclic displacements $ X _ \beta $, $ \beta = 1 \dots k $, form a subalgebra of the algebra of virtual displacements, then the mechanical system corresponding to this subalgebra forms some sort of independent holonomic system with $ s $ degrees of freedom, described by the equations (1) when $ \alpha , j , \beta = 1 \dots s $, where the role of the function $ L $ is played by the function $ R $.

The Poincaré equations contain as particular cases: the Lagrange equations (in mechanics) when the transformation algebra of infinitesimal increments in one of the variables reduces to a commutative algebra of transformations; and the Euler equation for the rotation of a solid body, when the $ \eta _ {i} $ are taken to be the projections $ p $, $ q $ and $ r $ of the instantaneous angular velocity.

References

[1] H. Poincaré, C.R. Acad. Sci. Paris , 132 (1901) pp. 369–371
[2] H. Poincaré, Bull. Astron. , 27 (1910) pp. 321–356
[3] N.G. Chetaev, Dokl. Akad. Nauk SSSR , 7 (1928) pp. 103–104
[4] N.G. Chetaev, "Sur les équations de Poincaré" C.R. Acad. Sci. Paris , 185 (1927) pp. 1577–1578
[5] N.G. Chetaev, Prikl. Mat. i. Mekh. , 5 : 2 (1941) pp. 253–262
How to Cite This Entry:
Poincaré equations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poincar%C3%A9_equations&oldid=48204
This article was adapted from an original article by V.V. Rumyantsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article