Скачать с ютуб Gradient System Defined by Gradient Vector of Potential Function (also consider Hamiltonian System) в хорошем качестве

Gradient System Defined by Gradient Vector of Potential Function (also consider Hamiltonian System)

The potential function V(x,y)=x^3+y^3-12x*y is used to construct a gradient system of nonlinear differential equations dx/dt = ∂V/∂x = 3x^2 - 12y, dy/dt = ∂V/∂y = 3y^2 - 12x. The equilibrium points and nullclines are found and the phase portrait is drawn. The Jacobian matrix of the vector field is also the Hessian matrix of V, which is a symmetric matrix with real eigenvalues. Linearization near the hyperbolic equilibria using the Hartman-Grobman theorem confirms the phase portrait. We also use the level curves of V to see that the trajectories of the system are orthogonal to the level curves. This is visualized with StreamPlot in Wolfram Mathematica. Then we consider the corresponding Hamiltonian system where V becomes a Hamiltonian function. Here the solution curves lie on the level curves (total mechanical energy is conserved). We get orthogonal trajectories to the gradient system (the dot product of the vector fields is always zero). Links and resources =============================== 🔴 Subscribe to Bill Kinney Math: https://www.youtube.com/user/billkinn... 🔴 Subscribe to my Math Blog, Infinity is Really Big: https://infinityisreallybig.com/ 🔴 Follow me on Twitter:   / billkinneymath   🔴 Follow me on Instagram:   / billkinneymath   🔴 You can support me by buying "Infinite Powers, How Calculus Reveals the Secrets of the Universe", by Steven Strogatz, or anything else you want to buy, starting from this link: https://amzn.to/3eXEmuA. 🔴 Check out my artist son Tyler Kinney's website: https://www.tylertkinney.co/ 🔴 Desiring God website: https://www.desiringgod.org/ AMAZON ASSOCIATE As an Amazon Associate I earn from qualifying purchases.