Tangent and Cotangent Spaces#
Definition 32 (Manifolds via Constraints)
Let \(Q \subseteq \mathbb{R}^n\). \(Q\) is called a k-dimensional manifold iff, there is an open set \(U \subseteq \mathbb{R}^n\) and a function \(F \in C^\infty(U, \mathbb{R}^{n-k})\) such that:
with
The matrix \(\partial F(x)\) is the Jacobi matrix of \(F\) at \(x\):
and condition (77) requires that the \(n-k\) gradients of \(F\) be linearily independent on \(Q\). The idea is that the constraint (76) removes \(n-k\) degrees of freedom, leaving \(k\) over.
Definition Definition 33 is a simplified version of the general definition which allows for different constraint functions \(F\) depending on the position \(x\).
Remark 12 (Manifolds via Curves, Generalized)
Let \(Q \subseteq \mathbb{R}^n\). \(Q\) is called a k-dimensional manifold iff, for all \(x \in Q\), there is an open environment \(U(x)\) of \(x\) such that the following holds:
with
Definition 33 (Manifolds via Curves)
Let \(Q \subseteq \mathbb{R}^n\). \(Q\) is called a k-dimensional manifold iff, there is an open set \(U \subseteq \mathbb{R}^n\) and a \(F \in C^\infty(U, \mathbb{R}^{n-k})\) such that:
with
The matrix \(\partial F(x)\) is the Jacobi matrix of \(F\) at \(x\):
and condition (81) requires that the \(n-k\) gradients of \(F\) be linearily independent on \(Q\). The idea is that \(F\) removes \(n-k\) degrees of freedom, leaving \(k\) over.
Definition Definition 33 is a simplified version of the general definition given below which allows for different constraints \(F\) depending on the position \(x\).
Definition 34 (Tangent Vectors, Tangent Space)
Let \(Q\) be a k-dimensional manifold in \(\mathbb{R}^n\).
(a) A vector \(v \in \mathbb{R}^n\) is called a tangent vector of \(Q\) at \(x \in Q\) iff there is a function \(\psi \in C^\infty(I_0, Q)\) such that:
wher \(I_0\) is an open intervall containing \(0\).
(b) The tangent space of \(Q\) at \(x\), called \(T_xQ\), is the set of all tangent vectors of \(Q\) at \(x\).
Definition 35 (Parameterized Manifolds)
Let \(Q\) be a k-dimensional manifold in \(\mathbb{R}^n\), \(U \subset \mathbb{R}^k\) and \(\phi \in C^\infty(U, \mathbb{R}^n)\). \(\phi\) is called a parameterization of \(Q\), iff
and
for all \(u \in U\).
Theorem 43 (Tangent Space)
Let \(Q\) be a k-dimensional manifold in \(\mathbb{R}^n\). We define:
Let Q be an n-dimensional manifold with a smooth parameterization: \(\phi: U \to V\) where \(U \subseteq \mathbb{R}^n\) is an open set, \(V\) is a vector space, and \(\phi \in C^\infty(U, V)\).
Tangent Space#
At each point \(\phi(u_0) \in Q\) where \(u_0 = (u^1_0, \ldots, u^n_0) \in U\), we define the directional derivative operators \(\partial_{u^i}|_{u_0}\) for \(i = 1, \ldots, n\) by their action on smooth functions:
\( \partial_{u^i}|_{u_0}: C^{\infty}(Q) \to \mathbb{R}, \quad f \mapsto \frac{\partial}{\partial u^i}\bigg|_{u=u_0} f(\phi(u)) \)
The tangent space at \(\phi(u_0)\) is:
\( T_{\phi(u_0)} Q = \text{span}\{\partial_{u^1}|_{u_0}, \ldots, \partial_{u^n}|_{u_0}\} \)
This is an n-dimensional vector space. A general tangent vector has the form \(v = \sum_{i=1}^n v^i \partial_{u^i}|_{u_0}\) for \(v^i \in \mathbb{R}\).
Cotangent Space#
The cotangent space \(T^*_{\phi(u_0)}Q\) is the dual space of \(T_{\phi(u_0)}Q\). It is also n-dimensional, with basis elements \(du^i|_{u_0}\) for \(i = 1, \ldots, n\) defined by:
\( du^i|_{u_0}: T_{\phi(u_0)}Q \to \mathbb{R}, \quad \text{where } du^i|_{u_0}(\partial_{u^j}|_{u_0}) = \delta^i_j \)
For a general tangent vector \(v = \sum_{j=1}^n v^j \partial_{u^j}|_{u_0}\), we have \(du^i|_{u_0}(v) = v^i\).
Differential of a Function#
For any smooth function \(f \in C^{\infty}(Q)\), the differential \(df\) is the covector:
\( df: T_{\phi(u_0)}Q \to \mathbb{R}, \quad df(v) = v(f) \)
Explicitly, for \(v = \sum_{i=1}^n v^i \partial_{u^i}|_{u_0}\):
\( df(v) = \sum_{i=1}^n v^i \cdot \partial_{u^i}|_{u_0} f = \sum_{i=1}^n v^i \cdot \frac{\partial}{\partial u^i}\bigg|_{u=u_0} f(\phi(u)) \)
and, for \(f = u^j\):
\( du^j(v) = \sum_{i=1}^n v^i \cdot \partial_{u^i}|_{u_0} u^j = v^j \)
In the basis \(\{du^1, \ldots, du^n\}\), we can write: \(df = \sum_{i=1}^n (\partial_{u^i} f) \cdot du^i\).
Tangent Bundle#
The tangent bundle is the collection of all tangent spaces:
\( TQ = \bigcup_{x \in Q} \{x\} \times T_x Q = \{(x, v) \mid x \in Q, v \in T_x Q\} \)
Using the parameterization \(\phi: U \to Q\), we can also write:
\( TQ = \{(\phi(u), v) \mid u \in U, v \in T_{\phi(u)}Q\} \)
or more explicitly:
\( TQ = \left\{\left(\phi(u), \sum_{i=1}^n v^i \partial_{u^i}|_u\right) \mid u \in U, v^i \in \mathbb{R}\right\} \)
Note that the basis vectors \(\partial_{u^i}|_u\) depend on the point \(\phi(u)\) — they represent the tangent vectors at that specific point. The tangent bundle TQ is a 2n-dimensional manifold.
Cotangent Bundle#
The cotangent bundle is the collection of all cotangent spaces:
\( T^*Q = \bigcup_{x \in Q} \{x\} \times T^*_x Q = \{(x, p) \mid x \in Q, p \in T^*_x Q\} \)
Using the parameterization:
\( T^*Q = \{(\phi(u), p) \mid u \in U, p \in T^*_{\phi(u)}Q\} \)
or more explicitly:
\( T^*Q = \left\{\left(\phi(u), \sum_{i=1}^n p_i \, du^i\right) \mid u \in U, p_i \in \mathbb{R}\right\} \)
The cotangent bundle \(T^*Q\) is also a 2n-dimensional manifold and serves as the phase space for Hamiltonian mechanics.
Key Points#
The tangent space \(T_x Q\) at each point \(x\) is an n-dimensional vector space containing tangent vectors (velocities)
The cotangent space \(T^*_x Q\) at each point \(x\) is the dual n-dimensional vector space containing covectors (momenta)
The basis vectors \(\partial_{u^i}|_u\) and dual basis \(du^i|_u\) depend on the point via the parameters \(u = (u^1, \ldots, u^n)\)
This definition is coordinate-independent in the sense that it depends only on the manifold structure of Q, though we use the parameterization \(\phi(u)\) to make computations explicit