In
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
, Appell's equation of motion (aka the Gibbs–Appell equation of motion) is an alternative general formulation of
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
described by
Josiah Willard Gibbs
Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in t ...
in 1879
and
Paul Émile Appell
:''M. P. Appell is the same person: it stands for Monsieur Paul Appell''.
Paul Émile Appell (27 September 1855, in Strasbourg – 24 October 1930, in Paris) was a French mathematician and Rector of the University of Paris. Appell polynomials an ...
in 1900.
Statement
The Gibbs-Appell equation reads
:
where
is an arbitrary generalized acceleration, or the second time derivative of the
generalized coordinate
In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 39 ...
s
, and
is its corresponding
generalized force. The generalized force gives the work done
:
where the index
runs over the
generalized coordinates
, which usually correspond to the
degrees of freedom of the system. The function
is defined as the mass-weighted sum of the particle
acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by t ...
s squared,
:
where the index
runs over the
particles, and
:
is the acceleration of the
-th particle, the second time derivative of its
position vector . Each
is expressed in terms of
generalized coordinates
In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 39 ...
, and
is expressed in terms of the generalized accelerations.
Relations to other formulations of classical mechanics
Appell's formulation does not introduce any new physics to classical mechanics and as such is equivalent to other reformulations of classical mechanics, such as
Lagrangian mechanics
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph- ...
, and
Hamiltonian mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
. All classical mechanics is contained within Newton's laws of motion. In some cases, Appell's equation of motion may be more convenient than the commonly used Lagrangian mechanics, particularly when
nonholonomic constraints are involved. In fact, Appell's equation leads directly to Lagrange's equations of motion. Moreover, it can be used to derive Kane's equations, which are particularly suited for describing the motion of complex spacecraft. Appell's formulation is an application of
Gauss' principle of least constraint.
Derivation
The change in the particle positions r
''k'' 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 ''dq
r'' in the generalized coordinates is
:
where Newton's second law for the ''k''th particle
:
has been used. Substituting the formula for ''d''r
''k'' 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
:
Examples
Euler's equations of rigid body dynamics
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
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
, and the corresponding angular acceleration vector
:
The generalized force for a rotation is the torque
, since the work done for an infinitesimal rotation
is
. The velocity of the
-th particle is given by
:
where
is the particle's position in Cartesian coordinates; its corresponding acceleration is
:
Therefore, the function
may be written as
:
Setting the derivative of ''S'' with respect to
equal to the torque yields Euler's equations
:
:
:
See also
*
Principle of stationary action
*
Analytical mechanics
References
Further reading
*
*
*
* {{cite journal , last = Brell , first = H , year = 1913 , title = Nachweis der Aquivalenz des verallgemeinerten Prinzipes der kleinsten Aktion mit dem Prinzip des kleinsten Zwanges , journal = Wien. Sitz. , volume = 122 , pages = 933–944 Connection of Appell's formulation with the
principle of least action
The stationary-action principle – also known as the principle of least action – is a variational principle that, when applied to the '' action'' of a mechanical system, yields the equations of motion for that system. The principle states tha ...
.
PDF copy of Appell's article at Goettingen University
PDF copy of a second article on Appell's equations and Gauss's principle
Classical mechanics