Institut für Theoretische Physik,
Freie Universität Berlin
Arnimallee 14, D - 14195 Berlin
P. Fiziev
and H. Kleinert
July 10, 1998
A recently discovered action principle for the motion of particles in spaces with curvature and torsion states that the classical trajectories of spinless point particles are straightest paths (autoparallels), not shortest paths (geodesics), as commonly believed. To illustrate the correctness of the new action principle, it is applied to the motion of a rigid body in the body-fixed coordinate system where it yields the correct Euler equations of motion as autoparallels rather than geodesics.
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1) Introduction
In the literature on gravity with curvature and torsion
[1],
there
is a widespread belief that in spaces with
torsion (for the geometry of
such spaces see
[2]), spinless particles move
on shortest paths [3].
However, it was discovered
in Ref. [4] (in a completely different context,
when solving the path integral of the hydrogen atom)
that the correct trajectories are the straightest paths
in a given geometry, thus calling for a revision of the field-theoretic
arguments in [3].
In Ref. [5],
a classical action principle was found
to comply with this physical fact.
The key to the new action principle
is the observation, that
a space 
with nonzero Riemann
curvature and torsion may be mapped locally
into a euclidean space 
by an anholonomic transformation
[6], [4].
For the particle with mass m in , the equations
of motion 
yield straight-line
trajectories. Under an anholonomic transformation, these
lines go over into autoparallels satisfying the equation
with 
being the Cartan connection (
is the inverse of
, i.e.,
).
The cartan connection can be written as
,
where
is the
Levi-Cevita connection,
the
Christoffel symbol
which depends only on the
metric tensor 
, and
is a
combination of torsion tensors, the so-called
contortion tensor
Although the torsion tensor
is the
antisymmetric part of the connection,
, it does
contribute to the symmetric part of the affine connection:
Thus it contributes the equation
for the autoparallel (1).
It is a well-known fact of classical mechanics, that equations of motion can be transformed nonholonomically without changing the physical trajectories.
This is in contrast to the behavior of the action principle under
nonholonomic transformations.
In a flat space,
the orbits in the euclidean space may be derived
from Hamilton's principle 
applied
to the classical action 
.
Under the anholonomic transformation
, this action goes into
the curvilinear form 
.
In the space , a naive application of Hamilton's principle
would produce wrong equations of motion
in the space .
One would find 
which are the
equations for the geodesics rather than the autoparallels;
they lack
torsion force 
contained in 
via (2).
The problem of describing the dynamics of a rigid body within the
body-fixed frame may be used as a case to find out which
of the trajectories is correct.
In the usual parameter space of Euler angles, there is
no torsion and a constant Riemann curvature [7].
Thus the trajectories are geodesics.
This changes when describing the system
within the body-fixed frame.
The transformation between the coordinates is
non-holonomic and the new
space possesses torsion.
We shall apply the variational principle of Ref. [5]
and derive,
within the body-fixed reference system
both the Euler equations for the angular momentum
and the equations
for the translational motion and show, that the rigid body moves along
autoparallel trajectories
in the body-fixed reference system.
2) The two coordinate frames of the Rigid Body
Since their discovery more than two centuries ago,
the Euler equations
have played a key role
in understanding the rotations
of a rigid body around a fixed point O.
(see for example [8] - [12] and the references therein).
The vectors
refer to the body-fixed system, 
being the
angular momentum,
the instantaneous angular velocity, and
the moments of the external forces. The dot
denotes differentiation with respect to the time t.
The unit basis vectors
in B may be assumed to point along the eigenvectors of
the inertia tensor with respect to O.
Then the angular moment has the components
(no sum over i).
To describe also the translational
motion of the rigid body one usually chooses the point O
to coincide with the center of mass
moving through space.
The motion satisfies the
additional
equations:
where 
is the linear momentum, M the body's
total mass, 
the velocity
of the center of mass,
and 
the driving force.
Presently, the equations (3) and (4) play
an important role in missile dynamics analysis [12].
Usually, the two sets of
equations are derived as the Newton equations
of motion in the stationary reference system S:
Let 
and 
be
the angular and linear velocities
in S and
those in B.
They are related by
a linear transformation
(see [8] - [10]):
The 
matrix elements 
and 
depend on the coordinates
in the system S. They satisfy
,
,
making the transformation (7)
and (8)
anholonomic
coordinate
transformations (see
[8]-[14] for the general
theory of the dynamics in
anholonomic coordinates).
Under the transformation (7), the equations
(5) go over into
(1). The term 
describes the moment of the gyroscopic force.
The additional term
in equation (4) arise
similarly.
Both terms are a consequence of the anholonomy
of the transformation (7), (8).
Within the stationary system S, the equations of motion (5) and (6) can be derived via Hamilton's action principle. If one transforms the classical action to the system B, however, the coordinates become nonholonomic and a naive application of Hamilton's principle would produce wrong equations of motion lacking the additional gyroscopic moments.
In 1901, Poincaré showed [13] how to vary an action expressed in terms of nonholonomic coordinates. Following his treatment, one certainly recovers the gyroscopic forces. His treatment is reviewed in the textbook [14].
We shall now first demonstrate that Poincaré's treatment may be viewed geometrically as an application of a recently proposed action principle in spaces with curvature and torsion.
Let us
rederive the equations of motion in the appropriate
geometric language.
3) The 
geometry in
a rotating
body-fixed system produced by an anholonomic
transformation from the rest system
Consider first a rigid body rotating
around a fixed point.
Within the body-fixed system B,
we
introduce anholonomic coordinates
corresponding to the transformation (7).
They are the components of the body's
rotation vector in the axis-angle parametrization.
The anholonomic transformation (7)
defines their infinitesimal increments 
.
For a precise specification, let us go to the system
S where
the standard Euler angles 
[10]
parametrize (holonomically)
the body's configuration space 
. The
components of the
angular velocity in the system B
are 
(with the short notation 
).
The basic relation
has coefficients which form the matrix
The symbol 
stands for increments do not
belong to an integrable function; they do not
satisfy the Schwarz integrability condition
which would read,
with the coefficients of (7),
.
The metric 
and its inverse 
have the matrices
where 
is the determinant of g.
The space of nonholonomic coordinates is flat. The space of Euler angles has a
constant Riemann
curvature. From g we find
the Christoffel symbol:
The associated Riemannian Ricci tensor is
with the constant contraction
.
On the other hand, the space of Euler angles is free of torsion.
Since the mapping is nonholonomic, the space
can carry torsion.
To calculate this we observe
that the only nonzero components of the affine-flat Cartan connection are
Using the metric 
we find
the antisymmetric part is the torsion tensor 
,
where 
denotes the Levi-Civita antisymmetric
tensor 
[2].
Since torsion involves a single derivative of the transformation
matrices between two sets of coordinates, the chain rule of differentiation
makes it an additive quantity.
With the 
-space being
free of torsion, the -space must have a torsion
which is the negative of 
after transforming the indices appropriately:
Indeed, inverting (7) to 
the torsion arising when going from 
- to
-coordinates is by definition
which due to 
and
coincides with (13).
Explicitly, we find from the above using the metric 
:
4) Action
principle
for the rotational motion in the body-fixed 
Pure rotations around
an arbitrary fixed point of the body are governed
by the Euler equations (3).
The kinetic energy is 
(
with an extended Einstein's summation
convention to include also
diagonal components of the inertia tensor).
An application of Hamilton's action principle 
to
the classical action 
in the system S
certainly yields
the correct equations
of motion. Under the transformation (7),
they become the Euler equations (3).
Let us now transform the action
via (7) to the body-fixed system B.
The result is very simple:
By applying naively
Hamilton's action principle we would find the equations
for each
i which are not the correct Euler
equations (3) -- the gyroscopic moments being missed.
The contradiction is caused by the fact that the variations 
in the space of Euler angles 
and the variations
in the space of anholonomic coordinates
,
are related with each other
in a
path dependent
way (``nonlocal" on the time axis) [5].
Explicitly, there is the functional equation
Its most important consequence is that
closed paths in the space are not
mapped into closed paths in the space ,
due to a nonvanishing Burgers vector 
.
The usual Hamilton action principle in the system
S proceeds by considering a variation 
between two paths
with common ends: 
. Together, they form a closed
path in the space .
The above naive
application of Hamilton's action principle in the system B
employed analogous
closed-path
variations 
between two paths
,
with also common ends: 
. This, however, cannot be correct. Under
a transformation (7), variations 
do not
produce closed-path variations 
in the
space .
The nonzero Burgers vector 
of the anholonomy causes
a closure failure. This has to be accounted for
in a correct
derivation of the equations of motion.
The anholonomy of the transformation (7)
requires distinguishing two types of variations of the paths
in the space :
closed path
variations of the tpye described above, and
anholonomic variations 
,
which are images of the
variations in the space . Only the latter will
produce
the correct equations of motion from an action principle
in the system B.
To calculate the variation of the classical
action in the system B we derive the following simple
formula for the anholonomic variation 
of
the angular velocity 
:
(corresponding to formula (9) in Ref. [5]).
Indeed, the transformation (7) and the definition
(9) lead to the equation
which by (13) becomes
leading to
(18) via (15).
The variations are
still performed
in the system S. They can be mapped locally into
holonomic closed-path variations
in
the system B,
, which
have
the usual fixed-endpoint property 
reflecting the closed-path condition
in the system S.
Let us emphasize that our
nonholonomic variations (15)
are completely intrinsic to the system B.
By applying these
to the action 
,
an integration by parts
with the boundary conditions leads to
Since
are now ordinary (holonomic) variations,
we find the Euler equations (1).
Thus the anholonomic action principle
produces the correct
equations of motion
in the
body-fixed system B.
5) Autoparallels
Let us now verify that
these body-fixed equations of motion correspond to the equations for
autoparalles rather than geodesics.
The Lagrangian
of a symmetric rigid body in the body-fixed coordinate system
has the general form
with the generalized coordinates 
and the
metric of the kinetic energy 
.
The equations of motion for autoparallels in this space
are independent of I and read
.
Now, since 
is a constant,
the space has obviously
a vanishing Riemann connection.
Because of (15), it has a nonvanishing torsion.
Using the decomposition (2), the equation for the autoparallels
reduce to
.
The antisymmetry of the torsion tensor (15)
causes the torsion force to vanish,
i.e., the autoparallels happen to coincide with the geodesics.
Let us now go over to the asymmetric rigid body
with the metric
(no sum over i).
This is done by an additional holonomic transformation
. Since the transformation coefficients
are independent of the coordinates, this does not change the torsion
of the space.
Since
the equations of motion contain
the torsion tensor with the first index raised by 
and the second lowered by 
[see Eq. (2)],
the torsion tensor
changes into 
and the
equation of motion becomes
. The autoparallel
is now different from the geodesic
by containing the effect of
the gyroscopic force.
6) Action principle for general motion
in the body-fixed
To extend the action principle
to the general motion of a rigid
body one usually
chooses the point O in B to be the center of mass of the body and
studies the movements of O and rotations around O. Then
are the body's principal moments of inertia.
The configuration space is a 6-dimensional manifold
and the kinetic
energy
consists of two terms:
The first is the translational kinetic
energy 
in the system B
the second is
the rotational kinetic
energy
in the system B.
In the system S, the position and orientation of the body
are parametrized by the (holonomic) coordinates
,
with 
being the mass center
coordinates and the Euler angles.
The body-fixed
basis vectors in B (see Fig. 1) are some function of the
Euler angles 
.
Their components in the cartesian basis
of the system S
are 
.
They form a orthogonal matrix
.
The components of the velocity of the center of mass
in the system S are 
. The
components of 
with respect to the basis 
are 
.
The last relation permits us to introduce a new
set of anholonomic coordinates 
describing
the center of mass motion in the system S as seen from the system B.
By analogy with formula (9), we define the infinitesimal
increments 
as:
The set 
gives a complete
set of an anholonomic coordinates for the configuration
space 
of the moving body.
Let us write down the anholonomic geometry of this space.
Introducing 
matrices,
whose rows and columns are labeled by capital Latin and Greek letters,
the geometry possesses
the affine connection 
with the torsion tensor
and a vanishing Cartan curvature tensor
, implying
an nonvanishing Riemann curvature tensor 
.
This geometry is a simple extension of the 3-dimensional anholonomic
geometry on described in Section 2. It reflects the structure
of the present configuration space
, i.e., the
space of configurations of the rigid body
comprising translations of the
center of mass and rotations around it.
As a direct consequence of the definition (23),
we have the relation 
which, together with (9),
leads to the following simple form of the rigid body's
Lagrangian in the system B:
As before, a naive application of
Hamiltons principle in the system B would produce a wrong
equation 
for the motion of the center-of-mass.
The correct anholonomic variation of
the velocity
Indeed, the transformation (8)
and the definition (23) lead to
The variations 
may be mapped locally
into holonomic closed-path variations 
in
the system B with the property 
reflecting the closed path condition 
.
Formula (23) follows using
the equations
,
and the relation 
.
By applying the anholonomic variation (23)
to the translational energy, we obtain
An integration by parts using of the fixed-end
conditions 
leads
to the following expression for the total
anholonomic variation of the action 
:
From the second term, we find the equations (2)
for the translational motion.
By an extension of the geometric arguments
in Sec. 2, the extended equations of motion describe again autoparallels, not
geodesics.
5) Conclusion
By subjecting the action in the body-fixed system to
the new anholonomic variations with respect to
translational
and rotational
degrees of freedom
of the rigid body
we have been
able to derive both
the correct
Euler
equations (3) and the equations (4)
completely within the body-fixed system
without reference to the stationary
systems.
Geometrically, the equations of motion
in the body-fixed coordinates correspond to autoparralels rather than
geodesics.
This has important implications on the classical field theory of gravity in spaces with curvature and torsion. It calls for a revision of the customary variational derivations of the field equations [1]. We have just learned that these variations require the knowledge of the torsion in the "superspace" inhabited by the metric tensor and the torsion field. DeWitt has shown us how to construct the "supermetric" in a the space of metrics with Riemannian geometry. His construction will require an extension to spaces with torsion, before it will be possible to derive correct field equations for metric and the torsion fields from a given field action.
Apart from that, the
above action principle
for the rigid body which is
intrinsic to the body-fixed system
may
have practical consequences
for variational calculations of the mechanical motion
of rigid bodies in accelerated frames of references,
for instance encounter maneuvers
in satellite physics.
Acknowledgement:
We are greateful to Dr. S.V. Shabanov for useful discussions.
