Search Results

Suppose I have a smooth vector field that has the form $$ X(y) = \sum_j \lambda_j y^j \partial_j + \text{higher order terms}$$ for $\lambda_j>0$. Let $\Phi_t$ be the flow of $X$. Then it follows that …

I am pretty sure this should be text book material, but I couldn't find this anywhere; maybe I just don't know where to look. Problem: Suppose we have a smooth vector field $X = a_i x^i \partial_i + …

Setup: Suppose we are given a smooth function $\phi$ that has a nondegenerate minimum at $x=0$. Then we can choose a coordinate system $x$ such that the gradient is given by $$X = \mathrm{grad} \phi = …

Let $X$ be a vector field on a compact manifold $M$ that has the form $$ X = \lambda_1 x^1 \partial_1 + \dots + \lambda_n x^n \partial_n + \dots$$ with respect to some chart $x$ around a point $p$. Al …

Let $\Omega$ be a domain in $\mathbb{R}^2$ with smooth boundary. A billiard trajectory is a continuous curve $c: \mathbb{R}\supseteq I \longrightarrow \overline{\Omega}$ such that $c(t) \in \partia …

A hands down proof not using the theory of Hamiltonian systems can be done by just proving that the Jacobian determinant of the transformation is zero. We have $$ TSM \cong \pi^* TM \oplus VSM,$$ wh …

The linearization of the vector field $X$ at the singular point zero is $$DX|_0 = \begin{pmatrix} 1 & 1\\ -1 & 0\end{pmatrix},$$ the eigenvalues of which are $$ \lambda_{1, 2} = \frac{1}{2} \pm \frac{ …