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Ordinary differential equations which describe the motions of both holonomic and non-holonomic systems, established by P.E. Appell [[#References|[1]]]. They are sometimes referred to as Gibbs–Appell equations, since they were first proposed by J.W. Gibbs [[#References|[3]]] for holonomic systems. The Appell equations in independent Lagrange coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a0127901.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a0127902.png" />) have the form of second-order equations
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<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/a/a012/a012790/a0127903.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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Ordinary differential equations which describe the motions of both holonomic and non-holonomic systems, established by P.E. Appell [[#References|[1]]]. They are sometimes referred to as Gibbs–Appell equations, since they were first proposed by J.W. Gibbs [[#References|[3]]] for holonomic systems. The Appell equations in independent Lagrange coordinates  $  q _ {s} $(
 +
$  s =1 \dots n $)
 +
have the form of second-order equations
 +
 
 +
$$ \tag{1 }
 +
 
 +
\frac{\partial  S }{\partial  \dot{q} dot _ {i} }
 +
  = Q _ {i}  ^ {*} ,\ \
 +
i = 1 \dots k \leq  n .
 +
$$
  
 
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/a/a012/a012790/a0127904.png" /></td> </tr></table>
+
$$
 +
=
 +
\frac{1}{2}
 +
\sum _ {\nu = 1 } ^ { N }
 +
m _  \nu  w _  \nu  ^ {2}
 +
$$
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a0127905.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a0127906.png" /> are the masses and the accelerations of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a0127907.png" /> points of the system) is the energy of acceleration of the system, which is so expressed that it contains the second derivatives of the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a0127908.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a0127909.png" />, only, the variations of which are considered as independent; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279010.png" /> are the generalized forces corresponding to the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279011.png" />, obtained as coefficients in front of the independent variations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279012.png" /> in the expression for the work of the given active forces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279013.png" /> corresponding to virtual displacements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279014.png" />:
+
( $  m _  \nu  $
 +
and $  w _  \nu  $
 +
are the masses and the accelerations of the $  N $
 +
points of the system) is the energy of acceleration of the system, which is so expressed that it contains the second derivatives of the coordinates $  q _ {i} $,  
 +
$  i = 1 \dots k $,  
 +
only, the variations of which are considered as independent; $  Q _ {i}  ^ {*} $
 +
are the generalized forces corresponding to the coordinates $  q _ {i} $,  
 +
obtained as coefficients in front of the independent variations $  \delta q _ {i} $
 +
in the expression for the work of the given active forces $  F _  \nu  $
 +
corresponding to virtual displacements $  \delta r _  \nu  $:
  
<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/a/a012/a012790/a01279015.png" /></td> </tr></table>
+
$$
 +
\sum _ {\nu = 1 } ^ { N }
 +
F _  \nu  \delta r _  \nu  = \sum _ {i = 1 } ^ { k }
 +
Q _ {1}  ^ {*} \delta q _ {i} .
 +
$$
  
In evaluating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279017.png" /> the dependent variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279018.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279019.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279020.png" /> are expressed in terms of the independent velocities (variations) by solving the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279021.png" /> non-holonomic constraint equations (cf. [[Non-holonomic systems|Non-holonomic systems]]), expressed in the generalized coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279022.png" /> (and by solving the equations for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279023.png" /> obtained from them). Differentiation with respect to the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279024.png" /> of the expressions found for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279025.png" /> yields expressions for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279026.png" /> in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279027.png" />.
+
In evaluating $  S $
 +
and $  Q _ {i}  ^ {*} $
 +
the dependent variables $  \dot{q} _ {j} $(
 +
$  \delta q _ {j} $)
 +
(j = k+1 \dots n) $
 +
are expressed in terms of the independent velocities (variations) by solving the $  n - k $
 +
non-holonomic constraint equations (cf. [[Non-holonomic systems|Non-holonomic systems]]), expressed in the generalized coordinates $  q _ {s} $(
 +
and by solving the equations for $  \delta q _ {s} $
 +
obtained from them). Differentiation with respect to the time $  t $
 +
of the expressions found for $  \dot{q} _ {j} $
 +
yields expressions for $  \dot{q} dot _ {j} $
 +
in terms of $  \dot{q} dot _ {i} $.
  
Equations (1), together with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279028.png" /> equations of the non-integrable constraints, form a system (of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279029.png" />) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279030.png" /> differential equations involving the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279031.png" /> unknowns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279032.png" />.
+
Equations (1), together with the $  n - k $
 +
equations of the non-integrable constraints, form a system (of order $  n + k $)  
 +
of $  n $
 +
differential equations involving the $  n $
 +
unknowns $  q _ {s} $.
  
For a holonomic system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279033.png" />, all velocities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279034.png" /> and variations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279035.png" /> are independent, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279036.png" />, and equations (1) are a different notation for the [[Lagrange equations (in mechanics)|Lagrange equations (in mechanics)]]) of the second kind.
+
For a holonomic system $  k = n $,  
 +
all velocities $  q _ {s} $
 +
and variations $  \delta q _ {s} $
 +
are independent, $  Q _ {i}  ^ {*} = Q _ {i} $,  
 +
and equations (1) are a different notation for the [[Lagrange equations (in mechanics)|Lagrange equations (in mechanics)]]) of the second kind.
  
Appell's equations in quasi-coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279037.png" />, where
+
Appell's equations in quasi-coordinates $  \pi _ {r} $,  
 +
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/a/a012/a012790/a01279038.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\dot \pi  _ {r}  = \sum _ {i = 1 } ^ { n }
 +
a _ {r _ {i}  } \dot{q} _ {i} ,\ \
 +
r = 1 \dots k,
 +
$$
  
 
have the form
 
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/a/a012/a012790/a01279039.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279040.png" /> is the energy of acceleration, expressed in terms of the second  "derivatives"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279041.png" /> (with respect to the time) of the quasi-coordinates, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279042.png" /> are the generalized forces corresponding to the quasi-coordinates. Equations (3), together with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279043.png" /> equations of the non-integrable constraints and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279044.png" /> equations (2), form a system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279045.png" /> differential equations of the first order with the same number of unknowns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279047.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012790/a01279049.png" />.
+
\frac{\partial  S }{\partial  \dot \pi  dot _ {r} }
 +
  = \Pi _ {r} ,\ \
 +
r = 1 \dots k \leq  n .
 +
$$
 +
 
 +
Here  $  S $
 +
is the energy of acceleration, expressed in terms of the second  "derivatives"   $ \dot \pi  dot _ {r} $(
 +
with respect to the time) of the quasi-coordinates, and $  \Pi _ {r} $
 +
are the generalized forces corresponding to the quasi-coordinates. Equations (3), together with the $  n - k $
 +
equations of the non-integrable constraints and the $  k $
 +
equations (2), form a system of $  n + k $
 +
differential equations of the first order with the same number of unknowns $  q _ {s} $,  
 +
$  s = 1 \dots n $,  
 +
and $  \dot \pi  _ {r} $,  
 +
$  r = 1 \dots k $.
  
 
Appell's equations are the most general equations of motion of mechanical systems.
 
Appell's equations are the most general equations of motion of mechanical systems.
Line 31: Line 106:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.E. Appell,  "Sur une forme génerale des équations de la dynamique"  ''C.R. Acad. Sci. Paris Sér. I Math.'' , '''129'''  (1899)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.E. Appell,  "Sur une forme générale des équations de la dynamique et sur le principe de Gauss"  ''J. Reine Angew. Math.'' , '''122'''  (1900)  pp. 205–208</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.W. Gibbs,  "On the fundamental formula of dynamics"  ''Amer. J. Math.'' , '''2'''  (1879)  pp. 49–64</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.E. Appell,  "Sur une forme génerale des équations de la dynamique"  ''C.R. Acad. Sci. Paris Sér. I Math.'' , '''129'''  (1899)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.E. Appell,  "Sur une forme générale des équations de la dynamique et sur le principe de Gauss"  ''J. Reine Angew. Math.'' , '''122'''  (1900)  pp. 205–208</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.W. Gibbs,  "On the fundamental formula of dynamics"  ''Amer. J. Math.'' , '''2'''  (1879)  pp. 49–64</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.T. Whittaker,  "Analytical dynamics" , Cambridge Univ. Press  (1927)  pp. 258</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.T. Whittaker,  "Analytical dynamics" , Cambridge Univ. Press  (1927)  pp. 258</TD></TR></table>

Revision as of 18:47, 5 April 2020


Ordinary differential equations which describe the motions of both holonomic and non-holonomic systems, established by P.E. Appell [1]. They are sometimes referred to as Gibbs–Appell equations, since they were first proposed by J.W. Gibbs [3] for holonomic systems. The Appell equations in independent Lagrange coordinates $ q _ {s} $( $ s =1 \dots n $) have the form of second-order equations

$$ \tag{1 } \frac{\partial S }{\partial \dot{q} dot _ {i} } = Q _ {i} ^ {*} ,\ \ i = 1 \dots k \leq n . $$

Here

$$ S = \frac{1}{2} \sum _ {\nu = 1 } ^ { N } m _ \nu w _ \nu ^ {2} $$

( $ m _ \nu $ and $ w _ \nu $ are the masses and the accelerations of the $ N $ points of the system) is the energy of acceleration of the system, which is so expressed that it contains the second derivatives of the coordinates $ q _ {i} $, $ i = 1 \dots k $, only, the variations of which are considered as independent; $ Q _ {i} ^ {*} $ are the generalized forces corresponding to the coordinates $ q _ {i} $, obtained as coefficients in front of the independent variations $ \delta q _ {i} $ in the expression for the work of the given active forces $ F _ \nu $ corresponding to virtual displacements $ \delta r _ \nu $:

$$ \sum _ {\nu = 1 } ^ { N } F _ \nu \delta r _ \nu = \sum _ {i = 1 } ^ { k } Q _ {1} ^ {*} \delta q _ {i} . $$

In evaluating $ S $ and $ Q _ {i} ^ {*} $ the dependent variables $ \dot{q} _ {j} $( $ \delta q _ {j} $) $ (j = k+1 \dots n) $ are expressed in terms of the independent velocities (variations) by solving the $ n - k $ non-holonomic constraint equations (cf. Non-holonomic systems), expressed in the generalized coordinates $ q _ {s} $( and by solving the equations for $ \delta q _ {s} $ obtained from them). Differentiation with respect to the time $ t $ of the expressions found for $ \dot{q} _ {j} $ yields expressions for $ \dot{q} dot _ {j} $ in terms of $ \dot{q} dot _ {i} $.

Equations (1), together with the $ n - k $ equations of the non-integrable constraints, form a system (of order $ n + k $) of $ n $ differential equations involving the $ n $ unknowns $ q _ {s} $.

For a holonomic system $ k = n $, all velocities $ q _ {s} $ and variations $ \delta q _ {s} $ are independent, $ Q _ {i} ^ {*} = Q _ {i} $, and equations (1) are a different notation for the Lagrange equations (in mechanics)) of the second kind.

Appell's equations in quasi-coordinates $ \pi _ {r} $, where

$$ \tag{2 } \dot \pi _ {r} = \sum _ {i = 1 } ^ { n } a _ {r _ {i} } \dot{q} _ {i} ,\ \ r = 1 \dots k, $$

have the form

$$ \tag{3 } \frac{\partial S }{\partial \dot \pi dot _ {r} } = \Pi _ {r} ,\ \ r = 1 \dots k \leq n . $$

Here $ S $ is the energy of acceleration, expressed in terms of the second "derivatives" $ \dot \pi dot _ {r} $( with respect to the time) of the quasi-coordinates, and $ \Pi _ {r} $ are the generalized forces corresponding to the quasi-coordinates. Equations (3), together with the $ n - k $ equations of the non-integrable constraints and the $ k $ equations (2), form a system of $ n + k $ differential equations of the first order with the same number of unknowns $ q _ {s} $, $ s = 1 \dots n $, and $ \dot \pi _ {r} $, $ r = 1 \dots k $.

Appell's equations are the most general equations of motion of mechanical systems.

References

[1] P.E. Appell, "Sur une forme génerale des équations de la dynamique" C.R. Acad. Sci. Paris Sér. I Math. , 129 (1899)
[2] P.E. Appell, "Sur une forme générale des équations de la dynamique et sur le principe de Gauss" J. Reine Angew. Math. , 122 (1900) pp. 205–208
[3] J.W. Gibbs, "On the fundamental formula of dynamics" Amer. J. Math. , 2 (1879) pp. 49–64

Comments

References

[a1] E.T. Whittaker, "Analytical dynamics" , Cambridge Univ. Press (1927) pp. 258
How to Cite This Entry:
Appell equations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Appell_equations&oldid=45195
This article was adapted from an original article by V.V. Rumyantsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article