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Killing symmetry and
hidden conserved quantity
on Finsler manifold
Takayoshi OotsukaOchanomizu University
Symposium on Gravity and Light, 30th Sep.-3rd Oct. 2013
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§Content
1. Short review of Finsler manifold (p.3~7)
2. covariant Lagrange formulation (p.8,9)
3. Noether’s theorem in the Finsler-Lagrange formulation
(p.10,11)
4. Finsler non-linear connection (p.12~16)
5. Killing vector and hidden conserved quantity (p.17)
6. Application to planet motion (example) (p.18~21)
7. Summary (p.22)
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§Introduction
In our presentation, we call Finsler metric
such that is a function of and , and satisfy the
following homogeneity condition,
We don’t assume the other conditions:
positivity,
regularity,
for application to wide area of physics. With this minimal
definitions, Finsler geometries can be applied to particle
Lagrange systems, thermodynamics, gauge theories,
path-integral quantization and so on.
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Indicatrix:surface of the unit distance from x measured by F
x
Pictorial image of Finsler manifold
Finsler manifold (M, F )Finsler structure gives norm to vector
x
v
xc
Finsler manifold is a set of M and F
Indicatrix also defines area, angle,
(Busemann, L. Tamassy)
and non-linear connection.
(with L. Kozma)
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It depends on the orientation of the curve, and
it is independent on the choice of parameter.
Geometric distance = The length of the oriented curve
cM[ ] [ ]
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ex.3 Minkowski spacetime x
ex.2 Riemann’s example
ex.1 Riemannian manifold
xRiemannian’s indicatrix
are quadratic curves
Finsler’s indicatrix
are “convex” curve
absolutely homogeneity
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1min
1min
x
1h
climbing time ≠ downhill time
ex.4 Matsumoto metric
We usually use maps which are written by walking time distance.
So we frequently use Finsler distance!
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is not a geometric space, but can be
endowed with a Finsler structure,
constructed from the Lagrangian , and
becomes Finsler manifold. The action functional is given by
and it derives covariant Euler-Lagrange equation;
This form is re-parametrization invariant, therefore we can
define a time parameter later for convenient use of calculation.
§Finsler-Lagrange formulation
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ex.1 pendulum
(M, F )Trajectory becomes geodesics.
Euler-Lagrange eq. becomes covariant
(time parameter independent).
ex.2 charged particle
(M,F )
ex.3 Kerr-Randers optics
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Noether’s theorem becomes very simple by Finsler-Lagrange
formulation. We define a prolongation of a vector field,
and define a Lie derivative of F(x,dx) by X,
Then Noether’s theorem becomes
If holds, we will call K a Killing vector on (M,F),
and if holds, we will call K a quasi-Killing vector.
§Noether’s theorem
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These Killing vectors correspond to variational symmetries.
We can check this by taking a conventional parameterization: ,
Therefore this Noether’s theorem is a geometrization of variational
symmetry on Finsler manifold. Furthermore, Noether’s theorem also
holds by which are generators of generalized
symmetries.
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§Finsler non-linear connection
Many mathematician think Finsler geometry from tangent
bundle viewpoint, but it is easy and
intuitive for physicist to consider it from point-Finsler view.
In Riemannian case, the corresponding Finsler manifold
becomes , and we will consider
F (x,dx) as a geometric object on M, a non-linear form, and
call it as “Finsler 1-form”. Point-Finsler viewpoint stands on
the indicatrix of F , the picture of Finsler 1-form, so we will
introduce a connection which preserve the indicatrix,F (x,dx).
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x
In Riemannian manifold, holonomy gauge group becomes
finite group, but in Finsler case, it is necessary for infinite
dimensional group for preserving their indicatrix.
Though instead of thinking infinite dimensional groups,
we generalize coefficients of linear connection;
This is very similar to Berwald’s connection, but we think
the connection on M not on TM.
x
quadraticquadratic
RiemannFinsler
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The Finsler non-linear connection is defined by,
Using this connection, we can rewrite E-L equation as auto-parallel
form, and furthermore we can easily generalize this formula to
singular Lagrange systems! (with L. Kozma)
We can prove
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For example, we will calculate a Finsler non-linear connection.
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Covariant Euler-Lagrange equations are
a=1,2,3)
c*
Auto-parallel forms are
If we take a parameter , then,
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§Another definition of Killing vector
Here we will give another definition of Killing vector on
Finsler manifold by using the Finsler non-linear connection.
Given a quasi Killing vector
we define what we call a “Killing 1-form”
then we can prove where
So we can define Killing (co)vector by and this is
useful for finding hidden conserved quantity,
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If the Finsler 1-form is Riemannian,
then the Finsler non-linear connection becomes Riemann
connection,
Furthermore if we assume that is a usual 1-form,
then becomes
and if we assume that is non-linear form such as
then becomes
Therefore such an ansatz corresponds to Killing tensor.
Trivial example (Riemannian case)
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Let us think
This has famous conserved quantities; Runge-Lentz vector:
for generalized symmetries:
§Non-trivial example (planet motion)
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We can check K1 is a quasi-Killing symmetry of F,
is conserved.
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Finding these Killing vectors is quite difficult at first definition,
but if we assume and we think
with Finsler non-linear connection:
then it becomes
We can get simple equations for Runge-Lentz vector.
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§Summary and Further work
• Lagrange systems can be formulated in Finsler geometry.
• Noether’s theorem becomes simple form.
• Using Finsler connection, we can discover hidden
conserved quantities.
• We can recognize a generalization of Killing vector and
Killing tensor on Finsler manifold.
• This technique can be applied to singular Lagrange case.
• Field Lagrange system is also possible to think same way.
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If is Killing vector such as
Therefore
Proof of
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We can also rewrite field Lagrange systems by Kawaguchi
manifold , where and Kawaguchi n-form
constructed from the field Lagrangian.
The action functional is given by
and it derives covariant Euler-Lagrange equation;
This form is also re-parametrization invariant.
cf.) Kawaguchi-Lagrange formulation