Appell's equation of motion

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]

${\displaystyle Q_{r}={\frac {\partial S}{\partial \alpha _{r}}}}$

Here, ${\displaystyle \alpha _{r}={\ddot {q_{r}}}}$ 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

${\displaystyle dW=\sum _{r=1}^{D}Q_{r}dq_{r}}$

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,

${\displaystyle S={\frac {1}{2}}\sum _{k=1}^{N}m_{k}\mathbf {a} _{k}^{2}\,,}$

where the index k runs over the N particles, and

${\displaystyle \mathbf {a} _{k}={\ddot {\mathbf {r} }}_{k}={\frac {d^{2}\mathbf {r} _{k}}{dt^{2}}}}$

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.

Derivation

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

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

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

${\displaystyle {\frac {\partial \mathbf {a} _{k}}{\partial \alpha _{r}}}={\frac {\partial \mathbf {r} _{k}}{\partial q_{r}}}}$

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

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

where Newton's second law for the kth particle

${\displaystyle \mathbf {F} _{k}=m_{k}\mathbf {a} _{k}}$

has been used. Substituting the formula for drk and swapping the order of the two summations yields the formulae

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

Therefore, the generalized forces are

${\displaystyle Q_{r}=\sum _{k=1}^{N}m_{k}\mathbf {a} _{k}\cdot \left({\frac {\partial \mathbf {r} _{k}}{\partial q_{r}}}\right)=\sum _{k=1}^{N}m_{k}\mathbf {a} _{k}\cdot \left({\frac {\partial \mathbf {a} _{k}}{\partial \alpha _{r}}}\right)}$

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

${\displaystyle {\frac {\partial S}{\partial \alpha _{r}}}={\frac {\partial }{\partial \alpha _{r}}}{\frac {1}{2}}\sum _{k=1}^{N}m_{k}\left|\mathbf {a} _{k}\right|^{2}=\sum _{k=1}^{N}m_{k}\mathbf {a} _{k}\cdot \left({\frac {\partial \mathbf {a} _{k}}{\partial \alpha _{r}}}\right)}$

yielding Appell's equation of motion

${\displaystyle {\frac {\partial S}{\partial \alpha _{r}}}=Q_{r}}$

Examples

Euler's equations

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 ${\displaystyle {\boldsymbol {\omega }}}$ , and the corresponding angular acceleration vector

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

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

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

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

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

Therefore, the function S may be written as

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

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

${\displaystyle I_{xx}\alpha _{x}-\left(I_{yy}-I_{zz}\right)\omega _{y}\omega _{z}=N_{x}}$
${\displaystyle I_{yy}\alpha _{y}-\left(I_{zz}-I_{xx}\right)\omega _{z}\omega _{x}=N_{y}}$
${\displaystyle I_{zz}\alpha _{z}-\left(I_{xx}-I_{yy}\right)\omega _{x}\omega _{y}=N_{z}}$

References

1. ^ Appell, P (1900). "Sur une forme générale des équations de la dynamique". Journal für die reine und angewandte Mathematik. 121: 310–?.
2. ^ Gibbs, JW (1879). "On the Fundamental Formulae of Dynamics". American Journal of Mathematics. 2: 49–64. doi:10.2307/2369196.