The Spherical Pendulum - Homework 2
Spherical Pendulum.
The Spherical Pendulum - Homework 2 - correction
We let $A : V \to V$ be a linear transformation, $x, y \in V$. \begin{align*} \exp(tA) &= I + tA + \frac{1}{2} t^2 A^2 + \cdots \\ \exp(tA) \cdot (x \tensorproduct y) &:= (\exp(tA)x) \tensorproduct (\exp(tA) y) \end{align*}
Week 3
Recall:
- we looked at classical Newtonian mechanics;
- we looked at some tensor algebra;
- let's continue with some tensor algebra.
Standing Notational Conventions
- $V, W$ are finite-dimensional vector spaces; $V^*, W^*$ are their duals.
- $\tenalg{*}{V}$ is the tensor algebra over $V$ with tensor product $\tensorproduct$.
- $\symalg{*}{V}$ is the symmetric tensor algebra over $V$ with tensor product $\symmetricproduct$.
- $\extalg{*}{V}$ is the alternating tensor algebra over $V$ with wedge product $\extproduct$.
- $\hom{V}{W}$ is the vector space of linear transformations $V \to W$.
- Given $v \in V, \phi \in V^*$, define \[ \ip{\phi}{v} = \phi(v). \]
Adjoints and Transposes
- A linear transformation
\[ \begin{matrix} V & \stackrel{A}{\longrightarrow} & W \\ & \text{induces} \\ V^* & \stackrel{A^*}{\longleftarrow} & W^* \\ \end{matrix} \]
- Defined by
\begin{align*} \ip{A^* \phi}{v} &:= \ip{\phi}{Av} && \phi \in W^*, v \in V. \end{align*}
- \begin{align*} (\image A)^{\perp} &= \kernel{A^*} && \kernel A = (\image{A^*})^{\perp} \end{align*}
Traces
- For $x \tensorproduct y \in V \tensorproduct V^*$ define \[ \mathrm{Tr}\, (x \tensorproduct y) := \ip{y}{x}. \]
- The $\mathrm{Tr}$ form extends to a linear function $\hom{V}{V} \cong V \tensorproduct V^* \to F$.
- For $A \in \hom{V}{V}$ \[ \mathrm{Tr}\, (A) = \sum_{i=1}^n \ip{\phi_i}{Ae_i} = \sum_{i=1}^n \ip{A^* \phi_i}{e_i} = \mathrm{Tr}\, (A^*) \]
Traces
- The map \[ \begin{matrix} A & \mapsto & \mathrm{Tr}_{A} \\ x \tensorproduct y & \mapsto & \ip{y}{Ax} =: \mathrm{Tr}_{A}(x \tensorproduct y) \end{matrix} \] is an isomorphism \[ \hom{V}{V}^* \cong \hom{V}{V}. \]
Volume forms and determinants
- $V$ is an $n$-dimensional vector space
- $\extalg{n}{V}$ is $1$-dimensional;
- $\extalg{k}{V}$ is $n \choose k$-dimensional for $0 \leq k \leq n$.
Duality
- There is a natural dual \( V^* \) and there are often unnatural duals, too.
Inner products
- $V$ is a vector space with basis $e_1, \ldots, e_n$
- Define $\Omega = e_1 \wedge \cdots \wedge e_n$
- Let $\eta \in \extalg{k}{V}, \rho \in \extalg{n-k}{V}$. Define \begin{align*} \ip{\eta}{\rho} := \eta \wedge \rho / \Omega. \end{align*}
