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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

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

( $  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|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)|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====
<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====

====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>