3. 1-Density

The real Finsler spaces, as generalizations of real Riemannian manifolds, were introduced in Ref. [1] about a century ago and have been studied by many researchers (see, for instance, Refs. [2– 4]), and Finsler spaces (see, for instance, Refs. [5, 6]) have become an interest of research for the studies of geometry, including differential geometry and integral geometry, in recent decades. By the way, there are applications of Finsler geometry in physics and engineering, and in particular, Finsler geometry can be applied to engineering dynamical systems, on which one can see Ref. [7]. As a typical Finsler space, L<sup>p</sup> space, 1 < p < ∞, has the main features of a Finsler space. As such, we focus on L<sup>p</sup> space, 1 < p < ∞, in this chapter, but some results can be generalized to general Finsler spaces, on which one can refer to Ref. [8]. The L<sup>p</sup> space, 1 < p < ∞, as a generalization of Euclidean space, has a rich structure in functional analysis (see, for instance, Refs. [9, 10]), and particularly in Banach space. Furthermore, it has broad applications in statistics (see, for instance, Refs. [11, 12]), engineering (see, for instance, Ref. [13, 27]), mechanics (see, for instance, Ref. [14]), computational science (see, for instance, Ref. [15]), biology (see, for instance,

, 1 < p < ∞, also has broad applications, in particular, signal processing in engi-

, <sup>l</sup>ðr;θÞ<sup>∈</sup> Gr1ðR2

ðx;y;r;θÞ : x cos ðθÞ þ y sin ðθÞ ¼ r

π<sup>1</sup> and π<sup>2</sup> are the natural projections of fibers. The Gelfand transform of a 2-density ϕ ¼

GTðϕÞ ¼ π<sup>1</sup>�π�

This chapter is structured as follows: In Section 2, we provide a description on Gelfand transform, which is one of the most fundamental transforms in integral geometry; in Section 3, we introduce density needed for the measure of length of curves; in Section 4, we further study the Lagrangian subspaces of complex L<sup>p</sup> spaces; in Section 5, we work on tangent bundle of unit sphere in Minkowski space and its symplectic or Lagrangian structure; in Section 6, we apply the Lagrangian structure to establish the length formula in integral geometry; and in Section 7, we further apply the Lagrangian structure of a Minkowski space to establish the

, 0 < p ≤ 1, in the sense of conjugacy to the

Gr1ðR<sup>2</sup><sup>Þ</sup> (1)

o

<sup>2</sup>ϕ; (2)

Þ,ðx;yÞ∈ lðr;θÞ

o ;

Ref. [16]), and other areas. Along this direction, Lp

formula for the Holmes-Thompson area in integral geometry.

R2 ← π1 F! π2

ððx;yÞ, <sup>l</sup>ðr;θÞÞ : <sup>ð</sup>x;yÞ<sup>∈</sup> R2

≃ n

.

neering, on which one can see Refs. [17–19].

scenario of Lp

30 Lagrangian Mechanics

2. Gelfand transform

Given a double fibration:

jdr∧dθj is defined as

which is a 1-density R<sup>2</sup>

F ¼ n

where

Lemma 3.1. For any <sup>v</sup> ¼ ðα;βÞ∈Tðx;yÞR<sup>2</sup>;

$$\text{GT}(\varphi)((x,y),\ v) = \mathbf{4}|v|.\tag{3}$$

Proof. For <sup>v</sup> ¼ ðα;βÞ<sup>∈</sup> <sup>T</sup>ðx;yÞR<sup>2</sup> ; there exists

$$\vec{v} = (\alpha\beta, \alpha\cos(\theta) + \beta\sin(\theta), \theta) \in T\_{((x,y), \, l(r,\theta))}\mathbf{F},\tag{4}$$

such that dπ1ð~vÞ ¼ v. Therefore, we have

$$\begin{split} \text{GT}(q)(\upsilon) &= \int \int \pi\_2^\* \varrho(\vec{v}, \bullet) \\ &= \int \sum\_{((x), \flat) \in \mathcal{A} \cap (\partial \mathcal{O}) \times \text{spin}(\partial \mathcal{O})} |(\vec{v}, \bullet)| \\ &= \int |a \cos(\theta) + \beta \sin(\theta)| d\theta \\ &= \int |\upsilon \cdot (\cos(\theta), \sin(\theta))| d\theta \\ &= \int |\upsilon \cdot (\cos(\theta), \sin(\theta))| d\theta \\ &= |\upsilon| \int\_0^{2\pi} \cos(\theta\_0 + \theta) d\theta \text{ where } \alpha = |\upsilon| \cos \left(\theta\_0\right), \beta = |\upsilon| \sin \left(\theta\_0\right) \\ &= 4|\upsilon|. \end{split} \tag{5}$$

Remark 3.2. By Alvarez's Gelfand transform for Crofton type formulas, we know that

$$\int\_{l\in\mathbb{R}^2} \mathfrak{F}(\gamma\mathbb{1}(r,\theta)) \mathrm{d}r \mathrm{d}\theta = \int\_{\mathcal{V}} \mathrm{GT}(\varphi). \tag{6}$$

Thus, we have now proved the Crofton formula: Given a differentiable curve γ in R<sup>2</sup> , the length of γ can be computed in the following formula:

$$\text{Length}(\gamma) = \frac{1}{4} \int\_{l \in \mathbb{R}^2} \#(\gamma \lhd(r, \theta)) \text{d}r \text{d}\theta. \tag{7}$$
