5. Tangent bundle of uni-sphere in Minkowski space and symplectic or Lagrangian structure

In this section, we show that the space of oriented lines and the tangent bundle of unit sphere in Minkowski space are symplectomorphic.

Let us consider a Minkowski plane <sup>ð</sup>R<sup>2</sup>;F<sup>Þ</sup> first, where <sup>F</sup> is a Finsler metric. The natural symplectic form on T� <sup>R</sup><sup>2</sup> is dx∧d<sup>ξ</sup> <sup>þ</sup> <sup>d</sup>y∧dη, and then the natural symplectic form on <sup>T</sup>R<sup>2</sup> induce by the Finsler metric F is

$$\begin{split} \boldsymbol{\omega} &:= \mathbf{dx} \wedge \mathbf{d} \frac{\partial \boldsymbol{F}}{\partial \boldsymbol{\xi}} + \mathbf{d}y \wedge \mathbf{d} \frac{\partial \boldsymbol{F}}{\partial \eta} \\ = \frac{\partial^2 \boldsymbol{F}}{\partial \boldsymbol{\xi}^2} \mathbf{dx} \wedge \mathbf{d} \boldsymbol{\xi} + \frac{\partial^2 \boldsymbol{F}}{\partial \boldsymbol{\xi} \partial \eta} (\mathbf{dx} \wedge \mathbf{d}\eta + \mathbf{d}y \wedge \mathbf{d}\boldsymbol{\xi}) + \frac{\partial^2 \boldsymbol{F}}{\partial \eta^2} \mathbf{d}y \wedge \mathbf{d}\eta. \end{split} \tag{20}$$

Define a projection <sup>π</sup> : <sup>T</sup>R<sup>2</sup> ! Gr1ðR<sup>2</sup><sup>Þ</sup> by

$$
\pi((\mathbf{x}, \mathbf{y}); (\xi, \eta)) = ((\mathbf{x}, \mathbf{y}) \text{-dF}(\xi, \eta))(\mathbf{x}, \mathbf{y}) (\xi, \eta); (\xi, \eta)). \tag{21}
$$

Let SF be the unit circle in the Minkowski plane and TSF be its tangent bundle. It is a fact that TSF≅Gr1ðR<sup>2</sup>Þ. On the other hand, since TSF is embedded in <sup>T</sup>R2, it inherits a natural symplectic form ω<sup>0</sup> :¼ ωj TSF from <sup>T</sup>R<sup>2</sup> .

Theorem 5.1. π�ω<sup>0</sup> ¼ ωj <sup>S</sup>�R<sup>2</sup> :

Proof. Applying the equality

$$
\frac{\partial F}{\partial \xi} \mathbf{d}\xi + \frac{\partial F}{\partial \eta} \mathbf{d}\eta = 0,\tag{22}
$$

we obtain

$$\begin{split} \pi^\* \omega\_0 &= \frac{\partial^2 F}{\partial \xi^2} \mathbf{d}(\mathbf{x} - \mathbf{d}F(\xi, \eta)((\mathbf{x}, y))\xi) \wedge \mathbf{d}\xi + \frac{\partial^2 F}{\partial \xi \partial \eta} (\mathbf{d}(\mathbf{x} - \mathbf{d}F(\xi, \eta)((\mathbf{x}, y))\xi) \wedge \mathbf{d}\eta \\ &+ \mathbf{d}(y - \mathbf{d}F(\xi, \eta)((\mathbf{x}, y))\eta) \wedge \mathbf{d}\xi) + \frac{\partial^2 F}{\partial \eta^2} \mathbf{d}(y - \mathbf{d}F(\xi, \eta)((\mathbf{x}, y))\eta) \wedge \mathbf{d}\eta \\ &= \frac{\partial^2 F}{\partial \xi^2} \mathbf{d} \mathbf{x} \wedge \mathbf{d}\xi + \frac{\partial^2 F}{\partial \xi \partial \eta} (\mathbf{dx} \wedge \mathbf{d}\eta + \mathbf{d}y \wedge \mathbf{d}\xi) + \frac{\partial^2 F}{\partial \eta^2} \mathbf{d}y \wedge \mathbf{d}\eta \\ &- \mathbf{d}(\mathbf{d}F(\xi, \eta)((\mathbf{x}, y))) \wedge (\frac{\partial^2 F}{\partial \xi^2} \xi \mathbf{d}\xi + \frac{\partial^2 F}{\partial \eta^2} \eta \mathbf{d}\eta + \frac{\partial^2 F}{\partial \xi \partial \eta} (\xi \mathbf{d}\eta + \eta \mathbf{d}\xi)). \end{split} \tag{23}$$

By the positive homogeneity of F, one can get the useful fact that Fðξ;ηÞ ¼ ξ<sup>∂</sup><sup>F</sup> <sup>∂</sup><sup>ξ</sup> þ η<sup>∂</sup><sup>F</sup> <sup>∂</sup>η. Therefore,

$$
\xi \frac{\partial F}{\partial \xi} + \eta \frac{\partial F}{\partial \eta} = 1. \tag{24}
$$

By differentiating (24), we get

linear transformation, so it is brought down to Imðz2z<sup>1</sup> Þ ¼ Imðw2w<sup>1</sup> Þ ¼ 0. Thus, we have <sup>P</sup>∈T<sup>2</sup>

5. Tangent bundle of uni-sphere in Minkowski space and symplectic or

In this section, we show that the space of oriented lines and the tangent bundle of unit sphere

<sup>ω</sup> :<sup>¼</sup> <sup>d</sup><sup>x</sup> <sup>∧</sup><sup>d</sup> <sup>∂</sup><sup>F</sup>

∂<sup>2</sup>F

∂F <sup>∂</sup><sup>ξ</sup> <sup>d</sup><sup>ξ</sup> <sup>þ</sup>

<sup>∂</sup>ξ<sup>2</sup> <sup>d</sup>ðx−dFðξ;ηÞððx;yÞÞξÞ∧d<sup>ξ</sup> <sup>þ</sup>

<sup>þ</sup> <sup>d</sup>ðy−dFðξ;ηÞððx;yÞÞηÞ∧dξÞ þ <sup>∂</sup><sup>2</sup><sup>F</sup>

∂<sup>2</sup>F

∂<sup>2</sup>F <sup>∂</sup>ξ<sup>2</sup> <sup>ξ</sup>d<sup>ξ</sup> <sup>þ</sup>

<sup>∂</sup>ξ<sup>2</sup> <sup>d</sup>x∧d<sup>ξ</sup> <sup>þ</sup>

;FÞ first, where F is a Finsler metric. The natural

<sup>∂</sup>η<sup>2</sup> <sup>d</sup>y∧dη:

dη ¼ 0; (22)

<sup>∂</sup>ξ∂<sup>η</sup> <sup>ð</sup>dðx−dFðξ;ηÞððx;yÞÞξÞ∧d<sup>η</sup>

<sup>∂</sup>η<sup>2</sup> <sup>d</sup>y∧d<sup>η</sup>

<sup>∂</sup>ξ∂<sup>η</sup> <sup>ð</sup>ξd<sup>η</sup> <sup>þ</sup> <sup>η</sup>dξÞÞ:

<sup>∂</sup>η<sup>2</sup> <sup>d</sup>ðy−dFðξ;ηÞððx;yÞÞηÞ∧d<sup>η</sup>

∂<sup>2</sup>F

(20)

(23)

<sup>R</sup><sup>2</sup> is dx∧d<sup>ξ</sup> <sup>þ</sup> <sup>d</sup>y∧dη, and then the natural symplectic form on <sup>T</sup>R<sup>2</sup>

∂η

πððx;yÞ;ðξ;ηÞÞ ¼ ððx;yÞ−dFðξ;ηÞððx;yÞÞðξ;ηÞ;ðξ;ηÞÞ: (21)

<sup>∂</sup><sup>ξ</sup> <sup>þ</sup> <sup>d</sup>y∧<sup>d</sup> <sup>∂</sup><sup>F</sup>

<sup>∂</sup>ξ∂<sup>η</sup> <sup>ð</sup>dx∧d<sup>η</sup> <sup>þ</sup> <sup>d</sup>y∧dξÞ þ <sup>∂</sup><sup>2</sup><sup>F</sup>

Let SF be the unit circle in the Minkowski plane and TSF be its tangent bundle. It is a fact that TSF≅Gr1ðR<sup>2</sup>Þ. On the other hand, since TSF is embedded in <sup>T</sup>R2, it inherits a natural symplectic

> ∂F ∂η

> > ∂<sup>2</sup>F

<sup>∂</sup>ξ∂<sup>η</sup> <sup>ð</sup>dx∧d<sup>η</sup> <sup>þ</sup> <sup>d</sup>y∧dξÞ þ <sup>∂</sup><sup>2</sup><sup>F</sup>

∂<sup>2</sup>F <sup>∂</sup>η<sup>2</sup> <sup>η</sup>d<sup>η</sup> <sup>þ</sup>

by the second case, and that concludes the proof.

in Minkowski space are symplectomorphic.

Let us consider a Minkowski plane <sup>ð</sup>R<sup>2</sup>

<sup>¼</sup> <sup>∂</sup><sup>2</sup><sup>F</sup>

Define a projection <sup>π</sup> : <sup>T</sup>R<sup>2</sup> ! Gr1ðR<sup>2</sup><sup>Þ</sup> by

TSF from <sup>T</sup>R<sup>2</sup>

.

<sup>S</sup>�R<sup>2</sup> :

<sup>¼</sup> <sup>∂</sup><sup>2</sup><sup>F</sup>

−dðdFðξ;ηÞððx;yÞÞÞ∧ð

<sup>∂</sup>ξ<sup>2</sup> <sup>d</sup><sup>x</sup> <sup>∧</sup>d<sup>ξ</sup> <sup>þ</sup>

Lagrangian structure

34 Lagrangian Mechanics

symplectic form on T�

form ω<sup>0</sup> :¼ ωj

we obtain

Theorem 5.1. π�ω<sup>0</sup> ¼ ωj

Proof. Applying the equality

<sup>π</sup>�ω<sup>0</sup> <sup>¼</sup> <sup>∂</sup><sup>2</sup><sup>F</sup>

induce by the Finsler metric F is

$$
\frac{
\partial^2 F
}{
\partial \xi^2
} \xi \mathbf{d} \xi + \frac{
\partial^2 F
}{
\partial \eta^2
} \eta \mathbf{d} \eta + \frac{
\partial^2 F
}{
\partial \xi \partial \eta
} (\xi \mathbf{d} \eta + \eta \mathbf{d} \xi) + \frac{
\partial F
}{
\partial \xi
} \mathbf{d} \xi + \frac{
\partial F
}{
\partial \eta
} \mathbf{d} \eta = 0. \tag{25}
$$

Applying (22) again, we have

$$
\frac{
\partial^2 F}{
\partial \xi^2} \xi \mathbf{d} \xi + \frac{
\partial^2 F}{
\partial \eta^2} \eta \mathbf{d} \eta + \frac{
\partial^2 F}{
\partial \xi \partial \eta} (\xi \mathbf{d} \eta + \eta \mathbf{d} \xi) = 0. \tag{26}
$$

Thus, the claim follows.

Remark 5.2. For a <sup>n</sup>-dimensional Minkowski space <sup>ð</sup>R<sup>n</sup>;FÞ, we just need to add more indices, then the theorem above is also true for <sup>ð</sup>R<sup>n</sup>;FÞ.

Therefore, letting F be a Finsler metric on R<sup>n</sup> and SF be the unit sphere in the Minkowski space <sup>ð</sup>R<sup>n</sup>;FÞ, we obtain the following general theorem:

Theorem 5.3. The symplectic form on the space of lines in a Minkowski space <sup>ð</sup>R<sup>n</sup>;F<sup>Þ</sup> is the canonical symplectic form on the tangent bundle TSF as imbedded in TR<sup>n</sup>.

We have the following remarks:

Remark 5.4. Theorem 5.3 provides a perspective that we can transform calculus on Gr1ðR<sup>2</sup><sup>Þ</sup> to ones on TSF.

and

Remark 5.5. We can analyze the differential structure of the Minkowski space by considering its symplectic form or Lagrangian structure. The Lagrangian structure of tangent spaces of Minkowski space gives the symplectic structure of the space of geodesics in the Minkowski space, and in general, the measures on a space or manifold in integral geometry depend on the differential structures of the space or manifold. Holmes-Thompson volumes are defined based on Lagrangian structure (see, for instance, Refs. [12, 20]), so, as an application, the symplectic structure determined by the symplectic form can be used to determine the symplectic form of the Holmes-Thompson volumes restricted on lines in integral geometry of Minkowski space, about which one can see Refs. [21–23].

Another remark from the proof of Theorem 5.1 is that

Remark 5.6. A combination of (26) and Gelfand transform (see Ref. [6]) may be used to provide a short proof of the general Crofton formula for Minkowski space.
