Classical Mechanics ala Manifolds

In this drama of mathematics and physics, which fertilize each other in the dark, but which prefer to deny and misconstrue each other face to face—I cannot, however, resist playing the role of a messenger, albeit, as I have abundantly learned, often an unwelcome one. ~ Hermann Weyl

In summer of 2023, I learned some modern Differential geometry and started reading the mathematics of Classical Mechaniccs. While I was in the process of properply recalling things and aiming at coherence about the subject to myself, I was ignited by Tanush’s talk on Mathematics of General Relativity, here at NISER and decided to write the things that I managed to put together in summer, i.e the mathematics of Classical Mechanics in the form of a website, since it gives me a better control of aesthetics and also helps in communication, all while recalling things for myself! So, this page is a result of me indulging in getting creative while seeking coherence. The page needs a little refinement at various places, but the main purpose of it is served.

Introduction

Classical mechanics describes systems with finitely many interacting particles.(A particle is a material body whose dimensions may be neglected in describing its motion.) A system is closed if its particles do not interact with the outside material bodies. The position of a system in space is specified by the position of its particles and defines a point in a smooth, finite-dimensional manifold M, the configuration space of a system. Coordinates on $M$ are called generalized coordinates of a system, and the dimension $n$ = dim $M$ is called the number of degrees of freedom. Systems with infinitely many degrees of freedom are described by Classical Field Theory (which is for another summer to probe into).

The above is quoted directly from Chapter-1 of Takhtajan's Quantum Mechanics for Mathematicians.

The state of a system at any instant of time is described by a point $q \in M$ and by a tangent vector $v \in T_qM$ at this point.

The motion of the system in space is captured by the trajectory $\gamma(t):\mathbb{R}\mapsto M$. The position of the particles in the system is denoted by $q(t) \in M$ given by $\gamma(t)$. The nature makes $\ddot{q}(t)$ depend on the state of the system in a certain way, and it's the aim of classical mechanics to discuss such dependence. This is casted in terms of Principle of least action, leading to the Euler-Lagrange equations.

A Lagrangian system (M,L) on a configuration space M is defined by a smooth function $L:TM\times\mathbb{R}\mapsto \mathbb{R}$ called the Lagrangian function. TM is the tangent bundle to the manifold $M$ and $\mathbb{R}$ in the domain corresponds to the time.

Manifolds

Definition: A set $M$ is a d-dimensional topological manifold $(M,\mathcal{O})$ if it is paracompact, Hausdorff topological space such that for every point $p \in M, \exists$ a neighbourhood $U_p$ and a homeomorphism $x:U_p \mapsto x(U_p) \subseteq \mathbb{R}^d.$

If M is a set, a topology on M is a set $\mathcal{O} \subseteq \mathcal{P(M)}$, i.e a collection of subsets of $M$ such that:

  1. $\varnothing \in \mathcal{O}$ and $M \in \mathcal{O}$.
  2. $U_\alpha \subseteq \mathcal{O},$ any subcollection of $\mathcal{O} \implies \bigcup_\alpha U_\alpha \in \mathcal{O}.$
  3. $U_\alpha \subseteq \mathcal{O},$ a finite subcollection of $\mathcal{O} \implies \bigcap_\alpha U_\alpha \in \mathcal{O}.$

A topological space M, is an ordered pair $(M,\mathcal{O})$, consisting of the set and the topology. We omit $\mathcal{O}$, and simply denote the topological space by its set $M$ from here on. Any set $U \subseteq M$ is called an open set of M, if $U \in \mathcal{O}.$

By a neighbourhood $U_p$ of $p$, we mean $U_p \ni p$ and $U_p \in \mathcal{O}.$

We do not delve into the Hausdorff (a kind of seperation axiom) and Paracompact properties of the manifolds for now, and the basic game can still be understood without probing them.

Charts, Atlasses and Transition functions:

Consider a d-dimensional manifold $\left(M,\mathcal{O}\right)$.

Tangent Spaces

Motivation

We develop the abstraction for the tangent space by looking at a specific function of a vector in a Euclidean space. After a formal definition, we will prove a geometric result that forms the basis of our discussion on Classical Mechanics.

In a Euclidean space $\mathbb{R}^n$, one can define a geometric tangent space at a point $p \in \mathbb{R}^n$ as

\begin{align} \mathbb{R}^n_p := {(a,v)\mid v \in \mathbb{R}^n} \end{align}

Look at a specific function of each element of $\mathbb{R}^n_p$(call it a geometric tangent vector, say $v_p \in \mathbb{R}^n_p$):

Consider $f: \mathbb{R}^n \rightarrow \mathbb{R}$, we have the following linear map, \begin{align} D_v\mid_p:C^\infty(\mathbb{R}^n) \rightarrow &\mathbb{R} \\ f \mapsto &D_vf(p) \\ &=\frac{d}{dt} f(p + tv) \\ &=v^i \frac{\partial f}{\partial x^i} \end{align}

We thus recognize that a geometric tangent vector defines a linear map, more specifially a derivation of a $C^\infty(\mathbb{R}^n)$ function.

Definition: If $p \in \mathbb{R}^n$ a map $w:C^\infty(\mathbb{R}^n) \rightarrow \mathbb{R}$ is called a derivation at $p$ if it is linear over $\mathbb{R}$ and satisfies the product rule: \begin{equation} w(fg) = f(p)wg + g(p)wf \end{equation}

Denote the set of all the derivations at point $p$ by $T_p(\mathbb{R}^n)$, the tangent space at point $p$.

We skip the proofs for above claims and prove similar results for a general smooth manifold $M$.

Definition

With the above motivation, we now define the tangent space at point on a smooth manifold M.

Definition: Let $M$ be a smooth manifold and $p$ a point of M, then a derivation at $p$ is a linear map $w:C^\infty(M)\rightarrow \mathbb{R}$ satisfying the product rule $\eqref{leib}$. Rhe set of all derivations $C^\infty(M)$ at $p$, denoted by $T_pM$; is a vector space called the tangent space to $M$ at $p$. An element of $T_pM$ is called a tangent vector at $p$.

Definition: If M and N are smooth manifolds, and $F:M \rightarrow N$ is a smooth map, then for reach $p \in M$, we define a map \begin{align} dF_p:T_pM\rightarrow &T_{F(p)}N \\ v \mapsto &dF_p(v)(f) := v(f\circ F) \\ \end{align} called the differential of $F$ at $p$. Where $f \in C^\infty(N)$.

Proof: The theorem says that, if two functions agree on a neighborhood around a point then the derivation of the functions at that point regardless of their domains, agree too.
Let $h=f-g$, then $h=0$ in a neighborhood of $p$. Consider $\psi \in C^\infty(M)$ as follows, \begin{align} \psi (q) = \begin{cases} 1, &\textit{if } h(q) \neq 0 \\ \neq 0, & \textit{if }q \in M \setminus \{p\} \\ = 0, & \textit{if } q = p \end{cases} \end{align} So, we have $\psi h = h$, and $h(p) = \psi(p) = 0 \implies vh = v(\psi h) = 0 \implies vf = vg.$
Proof: The differential is a linear map and it suffices to show that the map is a bijection.

Injection: It suffices to show that $ker \ d\imath_p = \{0\}$. Consider $v \in T_pM$, s.t $d\imath_p(v) = 0 \in T_pM.$ Let $B$ be a neigborhood of $p$ s.t $\overline{B} \subseteq M$. Let $f \in C^\infty(U)$ be arbitrary. We can extend $f$ to $\tilde{f} \in C^\infty(M)$ with $f = \tilde{f}$ on $\overline{B}$. From previous theorem, we have \begin{align} vf = v(\tilde{f}\Bigr\vert_U) = v(\tilde{f}\circ \imath) = d\imath_p(v)\tilde{f} = 0 \end{align} Since, this is true for aribitrary $f$, we have $v = 0$. Thus $ker \ d\imath_p = \{0\}$.

Surjection: Consider any arbitatory $w \in T_pM$. Define the following map, \begin{align} v:C^\infty(U) \rightarrow &\mathbb{R} \\ f \mapsto w\tilde{f} \end{align} where $\tilde{f} \in C^\infty(M)$ is the extension of $f$ on whole of $M$, with $\tilde{f} = f$ on $\overline{B}$. By previous theorem we have, $v \tilde{f}\Bigr\vert_U = v(\tilde{f}\circ\imath) = vf$, thus $v$ is well defined and is derivation since $w$ is. And thus for every $w \in T_pM$, we have $v \in T_pU$ s.t (for any $g \in C^\infty(M)$), \begin{align} d\imath_p(v)g = v(g\circ\imath) = w(\widetilde{g\circ\imath}) = wg. \end{align} Thus $d\imath_p$ is surjective.

</p>

Hence, $T_pU \cong T_pM$.

We thus recognize $d\imath_p(v)$ is the same derivation as v, acting on functions defined on larger space but giving the same output due to the locality proved in the previous theorem. </div>

Proof: Let $(U,\psi)$ be a smooth chart of M with $p \in U$. $\psi:U\rightarrow \widehat{U} \subseteq \mathbb{R}^n$ is a diffeomorphism. Thus, $d\psi_p$ is an isomorphism between $T_pM$ and $T_{\psi(p)}\widehat{U}$. Then by previous theorem we have, \begin{align} T_pM \cong T_pU \text{ and } T_{\psi(p)}\widehat{U} \cong T_{\psi(p)}\mathbb{R}^n \end{align} Thus, $$T_pM \cong T_{\psi(p)}\mathbb{R}^n.$$ $\implies dim T_pM = n$.

Coordinate representations

Now for the Geometrical result!
Theorem: Suppose $M$ is a smooth n-dimensional manifold and $p \in M$, then every $v \in T_pM$ is the velocity of some smooth curve in $M$.
Proof:

Consider a chart $U,\varphi$ with $p \in U$ (i.e centered at $p$). Write $v = v^i \frac{\partial}{\partial x^i}\Bigr\vert_p$. Then since $U$ is open, $\exists \epsilon$ such that \begin{align} \gamma:(-\epsilon,\epsilon) \rightarrow &U \\ t \mapsto & (tv^1,...,tv^n) \end{align} is smooth with, $\gamma(0) = p$ and $\gamma'(0) = v^i\frac{\partial}{\partial x^i}\Bigr\vert_{\gamma(0)} = v.$

Theorem: If $F:M\rightarrow N$ be a smooth map, and let $\gamma:J \rightarrow M$ be a smooth curve. For any $t_0 \in J$ , the velocity at $t=t0$ of the composite curve $F\circ \gamma: J \rightarrow N$ is given by \begin{align} (F\circ\gamma)'(t_0) = dF(\gamma'(t_0)). \end{align}
Theorem: If $F:M\rightarrow N$ is a smooth map, $p \in M$, and $v \in T_pM$. Then \begin{align} dF_p(v) = (F\circ\gamma)'(0) \end{align}
Thus it also common to work with the definition of tangent space at a point as the set of all velocity vectors to the curves passing through that point on the manifold!

The Tangent Bundle

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