Articles of gradient flows

Gradient vector of parametric curve

I have ellipse $$(\frac{x}{a})^2 + (\frac{y}{b})^2 = 1$$ Gradient is $$(\frac{2x}{a^2}, \frac{2y}{b^2})$$ How I can obtain this vector from parametrization of my curve? Let I know only $$(x, y) = (a \cdot cos \phi, b \cdot sin \phi)$$ I want have vector-function by φ, and if I choosed value, for example, φ=0, and set in […]

Hamiltonian for Geodesic Flow

I’m trying to prove that geodesic flow on the cotangent bundle $T^* M$ is generated by the Hamiltonian vector field $X_H$ where $$H = \frac{1}{2}g^{ij}p_i p_j$$ but I am stuck. Could somebody show me how to complete the calculation, or where I’ve made a mistake? Cheers! I know that vector field for geodesic flow is […]

When does gradient flow not converge?

I’ve been thinking about gradient flows in the context of Morse theory, where we take a differentiable-enough function $f$ on some space (for now let’s say a compact Riemannian manifold $M$) and use the trajectories of the gradient flow $x'(t) = – \operatorname{grad} f(x(t))$ to analyse the space. In particular the (un)stable manifolds $$W^\pm(p) = […]

Why is gradient the direction of steepest ascent?

$$f(x_1,x_2,…x_n):\mathbb{R}^n \rightarrow \mathbb{R}$$ The definition of the gradient is $$ \frac{\partial f}{\partial x_1}\hat{e}_1 +\ … +\frac{\partial f}{\partial x_n}\hat{e}_n$$ which is a vector. Reading this definition makes me consider that each component of the gradient corresponds to the rate of change with respect to my objective function if I go along with the direction $\hat{e}_i$. But […]