# Appell equations

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 générale des équations de la dynamique" C.R. Acad. Sci. Paris Sér. I Math. , 129 (1899) Zbl 30.0641.02 [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 Zbl 31.0692.02 [3] J.W. Gibbs, "On the fundamental formula of dynamics" Amer. J. Math. , 2 (1879) pp. 49–64 [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=55631
This article was adapted from an original article by V.V. Rumyantsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article