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)
[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
[a1]	E.T. Whittaker, "Analytical dynamics" , Cambridge Univ. Press (1927) pp. 258