Topics in Geometric Mechanics: Week 9

Navigation
MTH-696a
Course outline
Assignments & Solutions
Tutorials & Solutions
Schedule
Lecture Notes
Sources
Home Page of Leo Butler

Week 9

We left off with:

• the symplectic form on the cotangent bundle;
• the Hamiltonian;
• Hamilton's equations are equivalent to Euler-Lagrange.

Properties of the Symplectic Form

Recall the canonical Liouville $1$-form $\theta$ on $T^*M$, and its exterior derivative: \begin{align*} \theta &= \sum_i p_i \D{x_i}, && \Omega = -\D{\theta} = \sum_i \D{x_i} \wedge \D{p_i}. \end{align*}

$\Omega$ is closed and non-degenerate.

Symplectic Manifolds

We say that $(X, \Omega)$ is a symplectic manifold if $\Omega$ is a closed and non-degenerate $2$-form on $X$.

Examples

• $X = T^*M$, $\Omega$ is the canonical symplectic structure
• $X$ is an oriented surface, $\Omega$ is an area form on $X$.
• $X = \Torus{2n}$, $\Omega = \sum_{i, j=1}^{2n} \omega_{ij} \D{\theta_i} \wedge \D{\theta_j}$.

Hamiltonian vector fields

We say a vector field $X$ on $X$ is locally Hamiltonian if $\interiorproduc{X}{\Omega} =: \chi$ is a closed $1$-form; it is Hamiltonian if $\chi = \D{H}$ is exact.

$X = \didi{}{\theta}$ is Hamiltonian on $T^*\sphere{1}$ with $H(\theta, p) = p$.

$Y = \didi{}{p}$ is locally Hamiltonian on $T^*\sphere{1}$ with $\interiorproduc{Y}{\D{\theta} \wedge \D{p}} = -\D{\theta}$.

Hamiltonian vector fields - 2

$X$ is a locally Hamiltonian vector field iff $\lieder{X}{\Omega} = 0$. If $X, Y$ are locally Hamiltonian vector fields, then $\liebracket{X}{Y} =: Z$ is Hamiltonian.

Cartan's magic

Let $X, Y$ be a vector fields, $\theta$ an $n$-form. Then \begin{align*} \lieder{X}{\theta} &= (d\interiorproduc{X} + \interiorproduc{X} d)\theta, \\ \lieder{X}{(\interiorproduc{Y}{\theta})} &= \interiorproduc{\lieder{X}{Y}}{\theta} + \interiorproduc{Y}{\lieder{X}{\theta}}. \end{align*}

By induction on $n \geq 0$. $n=0$: $\theta$ is a $0$-form, or function. Then \begin{align*} \lieder{X}{\theta} &= \ip{\D{\theta}}{X} = \interiorproduc{X}{d \theta} \\ &= \interiorproduc{X}{d \theta} + d\underbrace{\interiorproduc{X}{\theta}}_{=0}. \end{align*} $n \geq 1$: Assume $\theta = \alpha \wedge \beta = (n-1) \text{-form } \wedge 1 \text{-form}$, \begin{align*} & RHS \\ =\ & d( (\interiorproduc{X}{\alpha}) \wedge \beta + (-1)^{n-1} \alpha \wedge (\interiorproduc{X}{\beta})) &+& \interiorproduc{X}{(d \alpha \wedge \beta + (-1)^{n-1} \alpha \wedge d \beta)} \\ =\ & (d\interiorproduc{X}{\alpha}) \wedge \beta &+& (-1)^{n-2} (\interiorproduc{X}{\alpha}) \wedge d\beta \\ & (-1)^{n-1} d \alpha \wedge (\interiorproduc{X}{\beta}) &+& (-1)^{2n-2} \alpha \wedge (d\interiorproduc{X}{\beta}) \\ & (\interiorproduc{X}{d \alpha}) \wedge \beta &+& (-1)^n d \alpha \wedge (\interiorproduc{X}{\beta}) \\ & (-1)^{n-1} (\interiorproduc{X}{\alpha}) \wedge d\beta &+& (-1)^{2n-2} \alpha \wedge (\interiorproduc{X}{d \beta}) \\ =\ & LHS \end{align*}

Volume Preservation

If $X$ is a Hamiltonian vector field with Hamiltonian function $H$, then $H$ is constant along flow lines, the volume form $\Omega^n$ is preserved and the restriction of $V = \alpha \wedge \Omega^{n-1}$ to $H^{-1}(c)$ is preserved, where $\alpha$ is a smooth $1$-form on a nhd of $H^{-1}(c)$ such that $\interiorproduc{X}{\alpha}=1$.

$H$ is constant along flow lines: $$ \lieder{X}{H} = \ip{\D{H}}{X} = \Omega(X, X) = 0, $$ since $\Omega$ is skew symmetric. $\Omega^k$ is preserved since $$\lieder{X}{\Omega^k} = (\lieder{X}{\Omega^{k-1}}) \wedge \Omega + (-1)^{2k-2} \Omega^{k-1} \wedge \lieder{X}{\Omega}.$$ Finally, since $\interiorproduc{X}{\alpha} = 1$ near $H^{-1}(c)$, $\lieder{X}{\alpha}$ annihilates $X$ so \begin{align*} \lieder{X}{V} &= (\lieder{X}{\alpha}) \wedge \Omega^{n-1} && \text{since $\Omega^{n-1}$ is preserved}\\ &= f \D{H} \wedge \Omega^{n-1}. \end{align*} Note that $V$ is independent of the choice of $\alpha$.