Killing symmetry and

hidden conserved quantity

on Finsler manifold

Takayoshi OotsukaOchanomizu University

Symposium on Gravity and Light, 30th Sep.-3rd Oct. 2013

§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)

§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.

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)

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[ ] [ ]

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

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!

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

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

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

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.

§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).

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

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

For example, we will calculate a Finsler non-linear connection.

Covariant Euler-Lagrange equations are

a=1,2,3)

c*

Auto-parallel forms are

If we take a parameter , then,

§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,

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)

Let us think

This has famous conserved quantities; Runge-Lentz vector:

for generalized symmetries:

§Non-trivial example (planet motion)

We can check K1 is a quasi-Killing symmetry of F,

is conserved.

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.

§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.

If is Killing vector such as

Therefore

Proof of

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