In
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
, a spray is a
vector field ''H'' on the
tangent bundle ''TM'' that encodes a
quasilinear second order system of ordinary differential equations on the base manifold ''M''. Usually a spray is required to be homogeneous in the sense that its integral curves ''t''→Φ
Ht(ξ)∈''TM'' obey the rule Φ
Ht(λξ)=Φ
Hλt(ξ) in positive reparameterizations. If this requirement is dropped, ''H'' is called a semispray.
Sprays arise naturally in
Riemannian and
Finsler geometry as the
geodesic spray
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
s whose
integral curves are precisely the tangent curves of locally length minimizing curves.
Semisprays arise naturally as the extremal curves of action integrals in
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- ...
. Generalizing all these examples, any (possibly nonlinear) connection on ''M'' induces a semispray ''H'', and conversely, any semispray ''H'' induces a torsion-free nonlinear connection on ''M''. If the original connection is torsion-free it coincides with the connection induced by ''H'', and homogeneous torsion-free connections are in one-to-one correspondence with full sprays.
[I. Bucataru, R. Miron, ''Finsler-Lagrange Geometry'', Editura Academiei Române, 2007.]
Formal definitions
Let ''M'' be a
differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
and (''TM'',π
''TM'',''M'') its tangent bundle. Then a vector field ''H'' on ''TM'' (that is, a
section of the
double tangent bundle In mathematics, particularly differential topology, the double tangent bundle or the second tangent bundle refers to the tangent bundle of the total space ''TM'' of the tangent bundle of a smooth manifold ''M''
. A note on notation: in this ar ...
''TTM'') is a semispray on ''M'', if any of the three following equivalent conditions holds:
* (π
''TM'')
*''H''
ξ = ξ.
* ''JH''=''V'', where ''J'' is the
tangent structure on ''TM'' and ''V'' is the canonical vector field on ''TM''\0.
* ''j''∘''H''=''H'', where ''j'':''TTM''→''TTM'' is the
canonical flip and ''H'' is seen as a mapping ''TM''→''TTM''.
A semispray ''H'' on ''M'' is a (full) spray if any of the following equivalent conditions hold:
* ''H''
λξ = λ
*(λ''H''
ξ), where λ
*:''TTM''→''TTM'' is the push-forward of the multiplication λ:''TM''→''TM'' by a positive scalar λ>0.
* The Lie-derivative of ''H'' along the canonical vector field ''V'' satisfies
'V'',''H''''H''.
* The integral curves ''t''→Φ
Ht(ξ)∈''TM''\0 of ''H'' satisfy Φ
Ht(λξ)=λΦ
Hλt(ξ) for any λ>0.
Let
be the local coordinates on
associated with the local coordinates
) on
using the coordinate basis on each tangent space. Then
is a semispray on
if it has a local representation of the form
:
on each associated coordinate system on ''TM''. The semispray ''H'' is a (full) spray, if and only if the spray coefficients ''G''
''i'' satisfy
:
Semisprays in Lagrangian mechanics
A physical system is modeled in Lagrangian mechanics by a Lagrangian function ''L'':''TM''→R on the tangent bundle of some configuration space ''M''. The dynamical law is obtained from the Hamiltonian principle, which states that the time evolution γ:
'a'',''b''��''M'' of the state of the system is stationary for the action integral
:
.
In the associated coordinates on ''TM'' the first variation of the action integral reads as
:
where ''X'':
'a'',''b''��R is the variation vector field associated with the variation γ
''s'':
'a'',''b''��''M'' around γ(''t'') = γ
0(''t''). This first variation formula can be recast in a more informative form by introducing the following concepts:
* The covector
with
is the conjugate momentum of
.
* The corresponding one-form
with
is the Hilbert-form associated with the Lagrangian.
* The bilinear form
with
is the fundamental tensor of the Lagrangian at
.
* The Lagrangian satisfies the Legendre condition if the fundamental tensor
is non-degenerate at every
. Then the inverse matrix of
is denoted by
.
* The Energy associated with the Lagrangian is
.
If the Legendre condition is satisfied, then ''d''α∈Ω
2(''TM'') is a
symplectic form, and there exists a unique
Hamiltonian vector field ''H'' on ''TM'' corresponding to the Hamiltonian function ''E'' such that
:
.
Let (''X''
''i'',''Y''
''i'') be the components of the Hamiltonian vector field ''H'' in the associated coordinates on ''TM''. Then
:
and
:
so we see that the Hamiltonian vector field ''H'' is a semispray on the configuration space ''M'' with the spray coefficients
:
Now the first variational formula can be rewritten as
:
and we see γ
'a'',''b''��''M'' is stationary for the action integral with fixed end points if and only if its tangent curve γ':
'a'',''b''��''TM'' is an integral curve for the Hamiltonian vector field ''H''. Hence the dynamics of mechanical systems are described by semisprays arising from action integrals.
Geodesic spray
The locally length minimizing curves of
Riemannian and
Finsler manifold
In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold where a (possibly asymmetric) Minkowski functional is provided on each tangent space , that enables one to define the length of any smooth curve ...
s are called
geodesics
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
. Using the framework of Lagrangian mechanics one can describe these curves with spray structures. Define a Lagrangian function on ''TM'' by
:
where ''F'':''TM''→R is the
Finsler function. In the Riemannian case one uses ''F''
2(''x'',ξ) = ''g''
''ij''(''x'')ξ
''i''ξ
''j''. Now introduce the concepts from the section above. In the Riemannian case it turns out that the fundamental tensor ''g''
''ij''(''x'',ξ) is simply the Riemannian metric ''g''
''ij''(''x''). In the general case the homogeneity condition
:
of the Finsler-function implies the following formulae:
:
In terms of classical mechanical the last equation states that all the energy in the system (''M'',''L'') is in the kinetic form. Furthermore, one obtains the homogeneity properties
:
of which the last one says that the Hamiltonian vector field ''H'' for this mechanical system is a full spray. The constant speed geodesics of the underlying Finsler (or Riemannian) manifold are described by this spray for the following reasons:
* Since ''g''
ξ is positive definite for Finsler spaces, every short enough stationary curve for the length functional is length minimizing.
* Every stationary curve for the action integral is of constant speed
, since the energy is automatically a constant of motion.
* For any curve
of constant speed the action integral and the length functional are related by
:
Therefore, a curve
is stationary to the action integral if and only if it is of constant speed and stationary to the length functional. The Hamiltonian vector field ''H'' is called the ''geodesic spray'' of the Finsler manifold (''M'',''F'') and the corresponding flow Φ
''H''t(ξ) is called the ''geodesic flow''.
Correspondence with nonlinear connections
A semispray
on a smooth manifold
defines an Ehresmann-connection
on the slit tangent bundle through its horizontal and vertical projections
:
:
This connection on ''TM''\0 always has a vanishing torsion tensor, which is defined as the Frölicher-Nijenhuis bracket
''T''=
'J'',''v'' In more elementary terms the torsion can be defined as
:
Introducing the canonical vector field ''V'' on ''TM''\0 and the adjoint structure Θ of the induced connection the horizontal part of the semispray can be written as ''hH''=Θ''V''. The vertical part ε=''vH'' of the semispray is known as the first spray invariant, and the semispray ''H'' itself decomposes into
:
The first spray invariant is related to the tension
:
of the induced non-linear connection through the ordinary differential equation
:
Therefore, the first spray invariant ε (and hence the whole semi-spray ''H'') can be recovered from the non-linear connection by
:
From this relation one also sees that the induced connection is homogeneous if and only if ''H'' is a full spray.
Jacobi fields of sprays and semisprays
A good source for Jacobi fields of semisprays is Section 4.4, ''Jacobi equations of a semispray'' of the publicly available book ''Finsler-Lagrange Geometry'' by Bucătaru and Miron. Of particular note is their concept of a dynamical covariant derivative. I
another paperBucătaru, Constantinescu and Dahl relate this concept to that of the
Kosambi biderivative operator
Kosambi (Pali) or Kaushambi (Sanskrit) was an important city in ancient India. It was the capital of the Vatsa kingdom, one of the sixteen mahajanapadas. It was located on the Yamuna River about southwest of its confluence with the Ganges at ...
.
For a good introduction to
Kosambi
Kosambi (Pali) or Kaushambi ( Sanskrit) was an important city in ancient India. It was the capital of the Vatsa kingdom, one of the sixteen mahajanapadas. It was located on the Yamuna River about southwest of its confluence with the Ganges ...
's methods, see the article, '
What is Kosambi-Cartan-Chern theory?''.
References
* .
* .
{{DEFAULTSORT:Spray (Mathematics)
Differential geometry
Finsler geometry