Appell's Equation of Motion

In classical mechanics, Appell's equation of motion is an alternative general formulation of classical mechanics described by Paul Émile Appell in 1900


frac{partial S}{partial alpha_{r}} = Q_{r}

Here, is an arbitrary generalized acceleration and Qr is its corresponding generalized force; that is, the work done is given by


dW = sum_{r=1}^{D} Q_{r} dq_{r}

where the index r runs over the D generalized coordinates qr, which usually correspond to the degrees of freedom of the system. The function S is defined as the mass-weighted sum of the particle accelerations squared, having the dimension of a generalised force for a generalised acceleration:


S = frac{1}{2} sum_{k=1}^{N} m_{k} mathbf{a}_{k}^{2}

where the index k runs over the N particles. Although fully equivalent to the other formulations of classical mechanics such as Newton's second law and the principle of least action, Appell's equation of motion may be more convenient in some cases, particularly when constraints are involved. Appell’s formulation can be viewed as a variation of Gauss' principle of least constraint.

Read more about Appell's Equation Of MotionExample: Euler's Equations, Derivation

Famous quotes containing the words motion and/or equation:

    As I walked on the glacis I heard the sound of a bagpipe from the soldiers’ dwellings in the rock, and was further soothed and affected by the sight of a soldier’s cat walking up a cleated plank in a high loophole designed for mus-catry, as serene as Wisdom herself, and with a gracefully waving motion of her tail, as if her ways were ways of pleasantness and all her paths were peace.
    Henry David Thoreau (1817–1862)

    Jail sentences have many functions, but one is surely to send a message about what our society abhors and what it values. This week, the equation was twofold: female infidelity twice as bad as male abuse, the life of a woman half as valuable as that of a man. The killing of the woman taken in adultery has a long history and survives today in many cultures. One of those is our own.
    Anna Quindlen (b. 1952)