词条 | Appell's equation of motion |
释义 |
In classical mechanics, Appell's equation of motion (aka Gibbs-Appell equation of motion) is an alternative general formulation of classical mechanics described by Paul Émile Appell in 1900[1] and Josiah Willard Gibbs in 1879[2] Here, is an arbitrary generalized acceleration, the second time derivative of the generalized coordinates qr and Qr is its corresponding generalized force; that is, the work done is given by 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, where the index k runs over the N particles, and is the acceleration of the kth particle, the second time derivative of its position vector rk. Each rk is expressed in terms of generalized coordinates, and ak is expressed in terms of the generalized accelerations. Appells formulation does not introduce any new physics to classical mechanics. It is fully equivalent to the other formulations of classical mechanics such as Newton's second law, Lagrangian mechanics, Hamiltonian mechanics, and the principle of least action. Appell's equation of motion may be more convenient in some cases, particularly when nonholonomic constraints are involved. Appell’s formulation is an application of Gauss' principle of least constraint. DerivationThe change in the particle positions rk for an infinitesimal change in the D generalized coordinates is Taking two derivatives with respect to time yields an equivalent equation for the accelerations The work done by an infinitesimal change dqr in the generalized coordinates is where Newton's second law for the kth particle has been used. Substituting the formula for drk and swapping the order of the two summations yields the formulae Therefore, the generalized forces are This equals the derivative of S with respect to the generalized accelerations yielding Appell’s equation of motion ExamplesEuler's equationsEuler's equations provide an excellent illustration of Appell's formulation. Consider a rigid body of N particles joined by rigid rods. The rotation of the body may be described by an angular velocity vector , and the corresponding angular acceleration vector The generalized force for a rotation is the torque N, since the work done for an infinitesimal rotation is . The velocity of the kth particle is given by where rk is the particle's position in Cartesian coordinates; its corresponding acceleration is Therefore, the function S may be written as Setting the derivative of S with respect to equal to the torque yields Euler's equations See also
References1. ^{{cite journal | last = Appell | first = P | year = 1900 | title = Sur une forme générale des équations de la dynamique. | journal = Journal für die reine und angewandte Mathematik | volume = 121 | pages = 310–? }} 2. ^{{cite journal | last = Gibbs | first = JW | year = 1879 | title = On the Fundamental Formulae of Dynamics. | journal = American Journal of Mathematics | volume = 2 | pages = 49–64 | doi=10.2307/2369196}} Further reading
1 : Classical mechanics |
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