Appell's equation of motion

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In classical mechanics, Appell's equation of motion is an alternative general formulation of classical mechanics described by Paul Émile Appell in 1900[1]


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

Here, \alpha_r 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 nonholonomic constraints are involved. Appell’s formulation is an application of Gauss' principle of least constraint.

Example: Euler's equations[edit]

Euler'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 \boldsymbol\omega, and the corresponding angular acceleration vector


\boldsymbol\alpha = \frac{d\boldsymbol\omega}{dt}

The generalized force for a rotation is the torque N, since the work done for an infinitesimal rotation \delta \boldsymbol\phi is dW = \mathbf{N} \cdot \delta \boldsymbol\phi. The velocity of the kth particle is given by


\mathbf{v}_{k} = \boldsymbol\omega \times \mathbf{r}_{k}

where rk is the particle's position in Cartesian coordinates; its corresponding acceleration is


\mathbf{a}_{k} = \frac{d\mathbf{v}_{k}}{dt} = 
\boldsymbol\alpha \times \mathbf{r}_{k} + \boldsymbol\omega \times \mathbf{v}_{k}

Therefore, the function S may be written as


S = \frac{1}{2} \sum_{k=1}^{N} m_{k} \left( \mathbf{a}_{k} \cdot \mathbf{a}_{k} \right)
= \frac{1}{2} \sum_{k=1}^{N} m_{k} \left\{ \left(\boldsymbol\alpha \times \mathbf{r}_{k} \right)^{2} 
+ \left( \boldsymbol\omega \times \mathbf{v}_{k} \right)^{2} 
+ 2 \left( \boldsymbol\alpha \times \mathbf{r}_{k} \right) \cdot \left(\boldsymbol\omega \times \mathbf{v}_{k}\right) \right\}

Setting the derivative of S with respect to \boldsymbol\alpha equal to the torque yields Euler's equations


I_{xx} \alpha_{x} - \left( I_{yy} - I_{zz} \right)\omega_{y} \omega_{z} = N_{x}

I_{yy} \alpha_{y} - \left( I_{zz} - I_{xx} \right)\omega_{z} \omega_{x} = N_{y}

I_{zz} \alpha_{z} - \left( I_{xx} - I_{yy} \right)\omega_{x} \omega_{y} = N_{z}

Derivation[edit]

The change in the particle positions rk for an infinitesimal change in the D generalized coordinates is


d\mathbf{r}_{k} = \sum_{r=1}^{D} dq_{r} \frac{\partial \mathbf{r}_{k}}{\partial q_{r}}

Taking two derivatives with respect to time yields an equivalent equation for the accelerations


\frac{\partial \mathbf{a}_{k}}{\partial \alpha_{r}} = \frac{\partial \mathbf{r}_{k}}{\partial q_{r}}

The work done by an infinitesimal change dqr in the generalized coordinates is


dW = \sum_{r=1}^{D} Q_{r} dq_{r} = \sum_{k=1}^{N} \mathbf{F}_{k} \cdot d\mathbf{r}_{k} = \sum_{k=1}^{N} m_{k} \mathbf{a}_{k} \cdot d\mathbf{r}_{k}

Substituting the formula for drk and swapping the order of the two summations yields the formulae


dW = \sum_{r=1}^{D} Q_{r} dq_{r} = \sum_{k=1}^{N} m_{k} \mathbf{a}_{k} \cdot \sum_{r=1}^{D} dq_{r} \left( \frac{\partial \mathbf{r}_{k}}{\partial q_{r}} \right) = 
\sum_{r=1}^{D} dq_{r} \sum_{k=1}^{N} m_{k} \mathbf{a}_{k} \cdot \left( \frac{\partial \mathbf{r}_{k}}{\partial q_{r}} \right)

Therefore, the generalized forces are


Q_{r} = 
\sum_{k=1}^{N} m_{k} \mathbf{a}_{k} \cdot \left( \frac{\partial \mathbf{r}_{k}}{\partial q_{r}} \right) =
\sum_{k=1}^{N} m_{k} \mathbf{a}_{k} \cdot \left( \frac{\partial \mathbf{a}_{k}}{\partial \alpha_{r}} \right)

This equals the derivative of S with respect to the generalized accelerations


\frac{\partial S}{\partial \alpha_{r}} = 
\frac{\partial}{\partial \alpha_{r}} \frac{1}{2} \sum_{k=1}^{N}  m_{k} \left| \mathbf{a}_{k} \right|^{2} = 
\sum_{k=1}^{N} m_{k} \mathbf{a}_{k} \cdot \left( \frac{\partial \mathbf{a}_{k}}{\partial \alpha_{r}} \right)

yielding Appell’s equation of motion


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

See also[edit]

References[edit]

  1. ^ Appell, P (1900). "Sur une forme générale des équations de la dynamique.". Journal für die reine und angewandte Mathematik 121: 310–?. 

Further reading[edit]

  • Whittaker, ET (1937). A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, with an Introduction to the Problem of Three Bodies (4th ed.). New York: Dover Publications. ISBN. 
  • Seeger (1930). "Unknown title". Journal of the Washington Academy of Science 20: 481–?.