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Ordinary second-order differential equations which describe the motions of mechanical systems under the action of forces applied to them. The equations were established by J.L. Lagrange [[#References|[1]]] in two forms: Lagrange's equations of the first kind, or equations in Cartesian coordinates with undetermined Lagrange multipliers, and of the second kind, or equations in generalized Lagrange coordinates.
 
Ordinary second-order differential equations which describe the motions of mechanical systems under the action of forces applied to them. The equations were established by J.L. Lagrange [[#References|[1]]] in two forms: Lagrange's equations of the first kind, or equations in Cartesian coordinates with undetermined Lagrange multipliers, and of the second kind, or equations in generalized Lagrange coordinates.
  
 
Lagrange's equations of the first kind describe motions of both holonomic systems, constrained only by geometrical relations of the form
 
Lagrange's equations of the first kind describe motions of both holonomic systems, constrained only by geometrical relations of 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/l/l057/l057150/l0571501.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
f _ {s} ( x _ {1} \dots x _ {3N} , t )  = 0 ,\ \
 +
s = 1 \dots k ,\  f _ {s} ( x , t ) \in  C  ^ {2} ,
 +
$$
  
 
and non-holonomic systems, on which one imposes, as well as relations (1), kinematic relations of the form
 
and non-holonomic systems, on which one imposes, as well as relations (1), kinematic relations of 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/l/l057/l057150/l0571502.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\phi _ {r} ( x _ {1} \dots x _ {3N} , \dot{x}  _ {1} \dots
 +
\dot{x}  _ {3N} , t )  = 0 ,
 +
$$
  
<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/l/l057/l057150/l0571503.png" /></td> </tr></table>
+
$$
 +
= 1 \dots m ,\  \phi _ {r} ( x , \dot{x} , t )  \in  C  ^ {1} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l0571504.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l0571505.png" /> are the Cartesian coordinates and the velocities of the points, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l0571506.png" /> is the number of points of the system, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l0571507.png" /> is the time, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l0571508.png" /> is the mass of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l0571509.png" />-th point, which has coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715012.png" />.
+
where $  x _  \nu  $
 +
and $  \dot{x}  _  \nu  = d x _  \nu  / d t $
 +
are the Cartesian coordinates and the velocities of the points, $  N $
 +
is the number of points of the system, $  t $
 +
is the time, and $  m _ {3p-} 2 = m _ {3p-} 1 = m _ {3p} $
 +
is the mass of the $  p $-
 +
th point, which has coordinates $  x _ {3p-} 2 $,  
 +
$  x _ {3p-} 1 $,  
 +
$  x _ {3p} $.
  
The relations (1) and (2) are assumed to be independent, that is, the ranks of the matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715014.png" /> are equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715016.png" />, respectively. Lagrange's equations of the first kind have the form
+
The relations (1) and (2) are assumed to be independent, that is, the ranks of the matrices $  \| \partial  f _ {s} / \partial  x _  \nu  \| $
 +
and $  \| \partial  \phi _ {r} / \partial  \dot{x}  _  \nu  \| $
 +
are equal to $  k $
 +
and $  m $,  
 +
respectively. Lagrange's equations of the first kind have 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/l/l057/l057150/l05715017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
m _  \nu  \dot{x} dot {} _  \nu  = X _  \nu  +
 +
\sum _ { s= } 1 ^ { k }  \lambda _ {s}
 +
\frac{\partial  f _ {s} }{\partial  x _  \nu  }
 +
+
 +
\sum _ { r= } 1 ^ { m }  \mu _ {r}
 +
\frac{\partial  \phi _ {r} }{\partial  \dot{x}  _  \nu  }
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715019.png" /> are undetermined Lagrange multipliers, proportional to the reactions of the constraints, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715020.png" /> are the projections on the coordinate axes of the given active forces, and the force <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715021.png" />, acting on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715022.png" />-th point, has the projections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715025.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715026.png" />.
+
where $  \lambda _ {s} $
 +
and $  \mu _ {r} $
 +
are undetermined Lagrange multipliers, proportional to the reactions of the constraints, $  X _  \nu  $
 +
are the projections on the coordinate axes of the given active forces, and the force $  F _ {p} $,  
 +
acting on the $  p $-
 +
th point, has the projections $  X _ {3p-} 2 $,  
 +
$  X _ {3p-} 1 $,  
 +
$  X _ {3p} $;  
 +
$  \dot{x} dot {} _  \nu  = d \dot{x}  _  \nu  / d t $.
  
To the differential equations (3) one must adjoin the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715027.png" /> equations (1) and (2), as a result of which one obtains a system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715028.png" /> equations in the same number of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715031.png" />. In practice, Lagrange's equations of the first kind are usually applied to systems with a small number of unknowns.
+
To the differential equations (3) one must adjoin the $  k + m $
 +
equations (1) and (2), as a result of which one obtains a system of $  3 N + k + m $
 +
equations in the same number of variables $  x _  \nu  $,  
 +
$  \lambda _ {s} $,  
 +
$  \mu _ {r} $.  
 +
In practice, Lagrange's equations of the first kind are usually applied to systems with a small number of unknowns.
  
Lagrange's equations of the second kind describe only motions of holonomic systems restricted by constraints of the form (1). By introducing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715032.png" /> independent generalized Lagrange coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715033.png" />, by means of which any possible position of the system can be obtained for certain values of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715034.png" /> from the equalities
+
Lagrange's equations of the second kind describe only motions of holonomic systems restricted by constraints of the form (1). By introducing $  n = 3 N - k $
 +
independent generalized Lagrange coordinates $  q _ {i} $,  
 +
by means of which any possible position of the system can be obtained for certain values of the $  q _ {i} $
 +
from the equalities
  
<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/l/l057/l057150/l05715035.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
x _  \nu  = x _  \nu  ( q _ {1} \dots q _ {n} , t ) ,\ \
 +
x _  \nu  ( q _ {i} , t ) \in  C  ^ {2} ,
 +
$$
  
that convert equations (1) into identities, one can establish for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715036.png" /> a one-to-one correspondence between the possible positions of the system and the points of some region of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715037.png" />-dimensional configuration space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715038.png" />. In the case of stationary constraints (1) it is always possible to choose the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715039.png" /> so that the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715040.png" /> does not occur in (4). Also, by means of (4) one can write down expressions for the sum of the elementary works of all the active forces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715041.png" /> corresponding to all possible displacements of the system:
+
that convert equations (1) into identities, one can establish for every $  t $
 +
a one-to-one correspondence between the possible positions of the system and the points of some region of the $  n $-
 +
dimensional configuration space $  ( q _ {1} \dots q _ {n} ) $.  
 +
In the case of stationary constraints (1) it is always possible to choose the variables $  q _ {i} $
 +
so that the time $  t $
 +
does not occur in (4). Also, by means of (4) one can write down expressions for the sum of the elementary works of all the active forces $  F _ {p} $
 +
corresponding to all possible displacements of the system:
  
<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/l/l057/l057150/l05715042.png" /></td> </tr></table>
+
$$
 +
\sum _ { p= } 1 ^ { N }  F _ {p} \cdot \delta r _ {p}  = \
 +
\sum _ {\nu = 1 } ^ { 3N }  X _  \nu  \delta x _  \nu  = \
 +
\sum _ { i= } 1 ^ { n }  Q _ {i} \delta {q _ {i} } ,
 +
$$
  
 
and the kinetic energy of the system:
 
and the kinetic energy of the system:
  
<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/l/l057/l057150/l05715043.png" /></td> </tr></table>
+
$$
 +
T ( q _ {i} , \dot{q}  _ {i} , t )  =
 +
\frac{1}{2}
 +
 
 +
\sum _ {\nu = 1 } ^ { 3N }  m _  \nu  \dot{x}  _  \nu  ^ {2}  = \
 +
T _ {2} + T _ {1} + T _ {0} .
 +
$$
  
 
Here
 
Here
  
<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/l/l057/l057150/l05715044.png" /></td> </tr></table>
+
$$
 +
Q _ {i}  = \sum _ {\nu = 1 } ^ { 3N }  X _  \nu 
 +
\frac{\partial  x _  \nu  }{\partial  q _ {i} }
 +
 
 +
$$
  
is the generalized force corresponding to the coordinate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715045.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715046.png" /> are homogeneous forms of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715047.png" /> in the generalized velocities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715048.png" />, and
+
is the generalized force corresponding to the coordinate $  q _ {i} $,  
 +
the $  T _ {s} ( q _ {i} , \dot{q} _ {i} , t ) $
 +
are homogeneous forms of degree $  s $
 +
in the generalized velocities $  \dot{q} _ {i} $,  
 +
and
  
<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/l/l057/l057150/l05715049.png" /></td> </tr></table>
+
$$
 +
T _ {2}  =
 +
\frac{1}{2}
 +
\sum _ { i,j= } 1 ^ { n }
 +
a _ {ij} \dot{q} _ {i} \dot{q} _ {j} ,\ \
 +
T _ {1}  = \sum _ { i= } 1 ^ { n }  a _ {i} \dot{q} _ {i} ,
 +
$$
  
<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/l/l057/l057150/l05715050.png" /></td> </tr></table>
+
$$
 +
T _ {0}  =
 +
\frac{1}{2}
 +
\sum _ {\nu = 1 } ^ { 3N }  m _  \nu  \left (
 +
\frac{\partial  x _  \nu  }{\partial  t }
 +
\right )  ^ {2} ,
 +
$$
  
<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/l/l057/l057150/l05715051.png" /></td> </tr></table>
+
$$
 +
a _ {ij}  = \sum _ {\nu = 1 } ^ { 3N }  m _  \nu 
 +
\frac{\partial  x _  \nu  }{\partial  q _ {i} }
 +
 +
\frac{\partial  x _  \nu  }{\partial  q _ {j} }
 +
,
 +
$$
  
<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/l/l057/l057150/l05715052.png" /></td> </tr></table>
+
$$
 +
a _ {i}  = \sum _ {\nu = 1 } ^ { 3N }  m _  \nu 
 +
\frac{\partial  x _  \nu  }{\partial  q _ {i} }
 +
 +
\frac{\partial  x _  \nu  }{\partial  t }
 +
.
 +
$$
  
In the case of stationary constraints <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715053.png" />.
+
In the case of stationary constraints $  T = T _ {2} $.
  
 
Lagrange's equations of the second kind have the form
 
Lagrange's equations of the second kind have 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/l/l057/l057150/l05715054.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
$$ \tag{5 }
 +
 
 +
\frac{d}{dt}
 +
 
 +
\frac{\partial  T }{\partial  \dot{q} _ {i} }
 +
-
 +
 
 +
\frac{\partial  T }{\partial  q _ {i} }
 +
  = \
 +
Q _ {i} ,\  i = 1 \dots n .
 +
$$
 +
 
 +
Equations (5) form a system of  $  n $
 +
ordinary second-order differential equations with unknowns  $  q _ {i} $.
 +
Their form is invariant with respect to the choice of Lagrange coordinates. This system of equations of motion has least possible order  $  2n $.
 +
In this respect, and also in the absence of reactions of the constraints in (5), equations (5) have a great advantage compared to Lagrange's equations of the first kind (3). After integrating (5) one can determine the reactions of the constraints from the equations that express Newton's second law for the points of the system (cf. also [[Newton laws of mechanics|Newton laws of mechanics]]).
 +
 
 +
In the case of potential generalized forces, when there is a force function  $  U ( q _ {1} \dots q _ {n} , t ) $
 +
such that  $  Q _ {i} = \partial  U / \partial  q _ {i} $,
 +
$  i = 1 \dots n $,
 +
equations (5) take the form
 +
 
 +
$$ \tag{6 }
 +
 
 +
\frac{d}{dt}
 +
 
 +
\frac{\partial  L }{\partial  \dot{q} _ {i} }
 +
-
  
Equations (5) form a system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715055.png" /> ordinary second-order differential equations with unknowns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715056.png" />. Their form is invariant with respect to the choice of Lagrange coordinates. This system of equations of motion has least possible order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715057.png" />. In this respect, and also in the absence of reactions of the constraints in (5), equations (5) have a great advantage compared to Lagrange's equations of the first kind (3). After integrating (5) one can determine the reactions of the constraints from the equations that express Newton's second law for the points of the system (cf. also [[Newton laws of mechanics|Newton laws of mechanics]]).
+
\frac{\partial  L }{\partial  q _ {i} }
 +
  = \
 +
0 ,\  i = 1 \dots n ,
 +
$$
  
In the case of potential generalized forces, when there is a force function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715058.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715060.png" />, equations (5) take the form
+
where  $  L ( q _ {i} , \dot{q} _ {i} , t ) = T + U $
 +
is called the Lagrange function (formerly, the kinetic potential).
  
<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/l/l057/l057150/l05715061.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>
+
If  $  \partial  L / \partial  t \equiv 0 $,
 +
or  $  \partial  L / \partial  q _  \alpha  \equiv 0 $,
 +
then equations (6) have a generalized energy integral
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715062.png" /> is called the Lagrange function (formerly, the kinetic potential).
+
$$
 +
\sum _ { i= } 1 ^ { n }
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715063.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715064.png" />, then equations (6) have a generalized energy integral
+
\frac{\partial  L }{\partial  \dot{q} _ {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/l/l057/l057150/l05715065.png" /></td> </tr></table>
+
\dot{q} _ {i} - L  = T _ {2} - T _ {0} - U  = \textrm{ const } ,
 +
$$
  
 
or a cyclic integral
 
or a cyclic integral
  
<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/l/l057/l057150/l05715066.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{\partial  L }{\partial  \dot{q} _  \alpha  }
 +
  = \
 +
\beta _  \alpha  = \textrm{ const } ,
 +
$$
  
corresponding to the cyclic coordinate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057150/l05715067.png" />.
+
corresponding to the cyclic coordinate $  q _  \alpha  $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Lagrange,  "Mécanique analytique" , '''1–2''' , Blanchard, reprint , Paris  (1965)  ((Also: Oeuvres, Vol. 11.))</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Lagrange,  "Mécanique analytique" , '''1–2''' , Blanchard, reprint , Paris  (1965)  ((Also: Oeuvres, Vol. 11.))</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.T. Whittaker,  "Analytical dynamics of particles and rigid bodies" , Dover, reprint  (1944)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.I. Arnol'd,  "Mathematical methods of classical mechanics" , Springer  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F.R. [F.R. Gantmakher] Gantmacher,  "Lectures in analytical mechanics" , MIR  (1975)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.T. Whittaker,  "Analytical dynamics of particles and rigid bodies" , Dover, reprint  (1944)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.I. Arnol'd,  "Mathematical methods of classical mechanics" , Springer  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F.R. [F.R. Gantmakher] Gantmacher,  "Lectures in analytical mechanics" , MIR  (1975)  (Translated from Russian)</TD></TR></table>

Revision as of 22:15, 5 June 2020


Ordinary second-order differential equations which describe the motions of mechanical systems under the action of forces applied to them. The equations were established by J.L. Lagrange [1] in two forms: Lagrange's equations of the first kind, or equations in Cartesian coordinates with undetermined Lagrange multipliers, and of the second kind, or equations in generalized Lagrange coordinates.

Lagrange's equations of the first kind describe motions of both holonomic systems, constrained only by geometrical relations of the form

$$ \tag{1 } f _ {s} ( x _ {1} \dots x _ {3N} , t ) = 0 ,\ \ s = 1 \dots k ,\ f _ {s} ( x , t ) \in C ^ {2} , $$

and non-holonomic systems, on which one imposes, as well as relations (1), kinematic relations of the form

$$ \tag{2 } \phi _ {r} ( x _ {1} \dots x _ {3N} , \dot{x} _ {1} \dots \dot{x} _ {3N} , t ) = 0 , $$

$$ r = 1 \dots m ,\ \phi _ {r} ( x , \dot{x} , t ) \in C ^ {1} , $$

where $ x _ \nu $ and $ \dot{x} _ \nu = d x _ \nu / d t $ are the Cartesian coordinates and the velocities of the points, $ N $ is the number of points of the system, $ t $ is the time, and $ m _ {3p-} 2 = m _ {3p-} 1 = m _ {3p} $ is the mass of the $ p $- th point, which has coordinates $ x _ {3p-} 2 $, $ x _ {3p-} 1 $, $ x _ {3p} $.

The relations (1) and (2) are assumed to be independent, that is, the ranks of the matrices $ \| \partial f _ {s} / \partial x _ \nu \| $ and $ \| \partial \phi _ {r} / \partial \dot{x} _ \nu \| $ are equal to $ k $ and $ m $, respectively. Lagrange's equations of the first kind have the form

$$ \tag{3 } m _ \nu \dot{x} dot {} _ \nu = X _ \nu + \sum _ { s= } 1 ^ { k } \lambda _ {s} \frac{\partial f _ {s} }{\partial x _ \nu } + \sum _ { r= } 1 ^ { m } \mu _ {r} \frac{\partial \phi _ {r} }{\partial \dot{x} _ \nu } , $$

where $ \lambda _ {s} $ and $ \mu _ {r} $ are undetermined Lagrange multipliers, proportional to the reactions of the constraints, $ X _ \nu $ are the projections on the coordinate axes of the given active forces, and the force $ F _ {p} $, acting on the $ p $- th point, has the projections $ X _ {3p-} 2 $, $ X _ {3p-} 1 $, $ X _ {3p} $; $ \dot{x} dot {} _ \nu = d \dot{x} _ \nu / d t $.

To the differential equations (3) one must adjoin the $ k + m $ equations (1) and (2), as a result of which one obtains a system of $ 3 N + k + m $ equations in the same number of variables $ x _ \nu $, $ \lambda _ {s} $, $ \mu _ {r} $. In practice, Lagrange's equations of the first kind are usually applied to systems with a small number of unknowns.

Lagrange's equations of the second kind describe only motions of holonomic systems restricted by constraints of the form (1). By introducing $ n = 3 N - k $ independent generalized Lagrange coordinates $ q _ {i} $, by means of which any possible position of the system can be obtained for certain values of the $ q _ {i} $ from the equalities

$$ \tag{4 } x _ \nu = x _ \nu ( q _ {1} \dots q _ {n} , t ) ,\ \ x _ \nu ( q _ {i} , t ) \in C ^ {2} , $$

that convert equations (1) into identities, one can establish for every $ t $ a one-to-one correspondence between the possible positions of the system and the points of some region of the $ n $- dimensional configuration space $ ( q _ {1} \dots q _ {n} ) $. In the case of stationary constraints (1) it is always possible to choose the variables $ q _ {i} $ so that the time $ t $ does not occur in (4). Also, by means of (4) one can write down expressions for the sum of the elementary works of all the active forces $ F _ {p} $ corresponding to all possible displacements of the system:

$$ \sum _ { p= } 1 ^ { N } F _ {p} \cdot \delta r _ {p} = \ \sum _ {\nu = 1 } ^ { 3N } X _ \nu \delta x _ \nu = \ \sum _ { i= } 1 ^ { n } Q _ {i} \delta {q _ {i} } , $$

and the kinetic energy of the system:

$$ T ( q _ {i} , \dot{q} _ {i} , t ) = \frac{1}{2} \sum _ {\nu = 1 } ^ { 3N } m _ \nu \dot{x} _ \nu ^ {2} = \ T _ {2} + T _ {1} + T _ {0} . $$

Here

$$ Q _ {i} = \sum _ {\nu = 1 } ^ { 3N } X _ \nu \frac{\partial x _ \nu }{\partial q _ {i} } $$

is the generalized force corresponding to the coordinate $ q _ {i} $, the $ T _ {s} ( q _ {i} , \dot{q} _ {i} , t ) $ are homogeneous forms of degree $ s $ in the generalized velocities $ \dot{q} _ {i} $, and

$$ T _ {2} = \frac{1}{2} \sum _ { i,j= } 1 ^ { n } a _ {ij} \dot{q} _ {i} \dot{q} _ {j} ,\ \ T _ {1} = \sum _ { i= } 1 ^ { n } a _ {i} \dot{q} _ {i} , $$

$$ T _ {0} = \frac{1}{2} \sum _ {\nu = 1 } ^ { 3N } m _ \nu \left ( \frac{\partial x _ \nu }{\partial t } \right ) ^ {2} , $$

$$ a _ {ij} = \sum _ {\nu = 1 } ^ { 3N } m _ \nu \frac{\partial x _ \nu }{\partial q _ {i} } \frac{\partial x _ \nu }{\partial q _ {j} } , $$

$$ a _ {i} = \sum _ {\nu = 1 } ^ { 3N } m _ \nu \frac{\partial x _ \nu }{\partial q _ {i} } \frac{\partial x _ \nu }{\partial t } . $$

In the case of stationary constraints $ T = T _ {2} $.

Lagrange's equations of the second kind have the form

$$ \tag{5 } \frac{d}{dt} \frac{\partial T }{\partial \dot{q} _ {i} } - \frac{\partial T }{\partial q _ {i} } = \ Q _ {i} ,\ i = 1 \dots n . $$

Equations (5) form a system of $ n $ ordinary second-order differential equations with unknowns $ q _ {i} $. Their form is invariant with respect to the choice of Lagrange coordinates. This system of equations of motion has least possible order $ 2n $. In this respect, and also in the absence of reactions of the constraints in (5), equations (5) have a great advantage compared to Lagrange's equations of the first kind (3). After integrating (5) one can determine the reactions of the constraints from the equations that express Newton's second law for the points of the system (cf. also Newton laws of mechanics).

In the case of potential generalized forces, when there is a force function $ U ( q _ {1} \dots q _ {n} , t ) $ such that $ Q _ {i} = \partial U / \partial q _ {i} $, $ i = 1 \dots n $, equations (5) take the form

$$ \tag{6 } \frac{d}{dt} \frac{\partial L }{\partial \dot{q} _ {i} } - \frac{\partial L }{\partial q _ {i} } = \ 0 ,\ i = 1 \dots n , $$

where $ L ( q _ {i} , \dot{q} _ {i} , t ) = T + U $ is called the Lagrange function (formerly, the kinetic potential).

If $ \partial L / \partial t \equiv 0 $, or $ \partial L / \partial q _ \alpha \equiv 0 $, then equations (6) have a generalized energy integral

$$ \sum _ { i= } 1 ^ { n } \frac{\partial L }{\partial \dot{q} _ {i} } \dot{q} _ {i} - L = T _ {2} - T _ {0} - U = \textrm{ const } , $$

or a cyclic integral

$$ \frac{\partial L }{\partial \dot{q} _ \alpha } = \ \beta _ \alpha = \textrm{ const } , $$

corresponding to the cyclic coordinate $ q _ \alpha $.

References

[1] J.L. Lagrange, "Mécanique analytique" , 1–2 , Blanchard, reprint , Paris (1965) ((Also: Oeuvres, Vol. 11.))

Comments

References

[a1] E.T. Whittaker, "Analytical dynamics of particles and rigid bodies" , Dover, reprint (1944)
[a2] V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian)
[a3] F.R. [F.R. Gantmakher] Gantmacher, "Lectures in analytical mechanics" , MIR (1975) (Translated from Russian)
How to Cite This Entry:
Lagrange equations (in mechanics). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lagrange_equations_(in_mechanics)&oldid=17218
This article was adapted from an original article by V.V. Rumyantsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article