关于变分定义、可微性及阿诺德经典力学教材中相关曲线空间与泛函可微性的技术疑问

Hey Sam, great questions—Arnold's Mathematical Methods of Classical Mechanics is a total masterpiece, but it’s famous for leaning hard on geometric intuition instead of formal functional analysis rigor, which can leave gaps like this that make you go "wait, hold on". Let’s unpack your questions clearly:

1. How exactly is the "space of curves" $C$ defined? #

Arnold doesn’t spell this out explicitly, but in the context of classical mechanics (where we’re dealing with trajectories in a configuration space), $C$ almost always refers to a space of smooth (or piecewise smooth) curves mapping a closed interval $[a,b]$ to the configuration manifold $M$ (for most basic problems, $M$ is $\mathbb{R}^n$, like the space of positions of a system of particles).

Formally, this is usually written as $C^\infty([a,b], M)$ (the space of infinitely differentiable curves) or $C^1([a,b], M)$ (first-order continuously differentiable curves)—since Lagrangians depend on positions and velocities (first derivatives of trajectories), $C^1$ is often the minimal regularity needed.

The key thing to note is that Arnold frames this geometrically: curves are just paths in the configuration space, and variations are "infinitesimal deformations" of these paths, rather than formal elements of a function space.

2. Is $C$ a normed vector space? Can differentiability of a functional be stated in terms of the total derivative? #

This splits into two cases depending on the configuration space:

Case 1: Configuration space is $\mathbb{R}^n$ (flat Euclidean space) #

• Yes, $C^k([a,b], \mathbb{R}^n)$ (for $k \geq 0$) is a normed vector space: We can define a $C^k$-norm like this:
  $$
  |\gamma|{C^k} = \max{0 \leq i \leq k} \max_{t \in [a,b]} |\gamma^{{(i)}(t)|_{\mathbb{R}}n}
  $$
  where $\gamma^{(i)}(t)$ is the $i$-th derivative of $\gamma$ at $t$, and $|\cdot|_{\mathbb{R}^n}$ is the standard Euclidean norm on $\mathbb{R}^n$. This makes $C^k([a,b], \mathbb{R}^n)$ a Banach space (complete normed vector space).
• Functional differentiability via total derivative: If we take a functional $F: C^1([a,b], \mathbb{R}^n) \to \mathbb{R}$ (like the action functional), its differentiability can be formalized using the Fréchet derivative (the "total derivative" in functional analysis terms). The Fréchet derivative $DF(\gamma)$ at a curve $\gamma$ is a bounded linear map from $C^1([a,b], \mathbb{R}^n)$ to $\mathbb{R}$, satisfying:
  $$
  \lim_{|\eta|{C^1} \to 0} \frac{|F(\gamma + \eta) - F(\gamma) - DF(\gamma)(\eta)|}{|\eta|{C^1}} = 0
  $$
  The variation $\delta F(\gamma)[\eta]$ that Arnold talks about is exactly the value of this Fréchet derivative at the "variation curve" $\eta$—so $\delta F(\gamma)[\eta] = DF(\gamma)(\eta)$.

Case 2: Configuration space is a general smooth manifold (e.g., a sphere, rotation group $SO(3)$) #

• No, $C^\infty([a,b], M)$ is not a vector space: You can’t add two curves in $M$ and get another curve in $M$ (since $M$ isn’t a linear space). Instead, we talk about the tangent space to the space of curves at a given $\gamma$—this consists of vector fields along $\gamma$ (i.e., maps $t \mapsto v(t) \in T_{\gamma(t)}M$, where $T_pM$ is the tangent space to $M$ at point $p$). These vector fields are exactly what Arnold calls "variations" of $\gamma$.
• Differentiability here is Gateaux differentiability: Instead of a total Fréchet derivative, we use directional derivatives (Gateaux derivatives) along these tangent vectors. The variation $\delta F(\gamma)[v]$ is defined as:
  $$
  \delta F(\gamma)[v] = \frac{d}{d\varepsilon}\bigg|{\varepsilon=0} F(\gamma\varepsilon)
  $$
  where $\gamma_\varepsilon$ is a smooth family of curves with $\gamma_0 = \gamma$ and $\frac{d}{d\varepsilon}\bigg|{\varepsilon=0} \gamma\varepsilon(t) = v(t)$. This is the framework Arnold implicitly uses, since he focuses on the geometric meaning of variations rather than abstract functional analysis.

备注：内容来源于stack exchange，提问作者Sam