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Problem 3*: Let V: R²R be a smooth function with a unique, nondegenerate critical point at (0,0). Let y > 0. Now consider the system of ordinary differential equations: av x' = dy-Yox, y'=-or-yor Do the following: I. Show that the only equilibrium point of the system is (0,0). II. Show that for any nonconstant solution, V(x(1), y(t)) is strictly monotonically decreasing. III. Show that the system only Hamiltonian if and only if V satisfies += 0, in which case (0,0) is a saddle point.

Problem 3*: Let V: R²R be a smooth function with a unique, nondegenerate critical point at (0,0). Let y > 0. Now consider the system of ordinary differential equations: av x' = dy-Yox, y'=-or-yor Do the following: I. Show that the only equilibrium point of the system is (0,0). II. Show that for any nonconstant solution, V(x(1), y(t)) is strictly monotonically decreasing. III. Show that the system only Hamiltonian if and only if V satisfies += 0, in which case (0,0) is a saddle point.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Problem 3*: Let V: R²R be a smooth function with a unique, nondegenerate critical point at (0,0). Let y > 0. Now consider the system of ordinary differential equations: av x' = Oy - Yox, y = -x-Yoy Do the following: I. Show that the only equilibrium point of the system is (0,0). II. Show that for any nonconstant solution, V(x(1), y(t)) is strictly monotonically decreasing. III. Show that the system only Hamiltonian if and only if V satisfies + OV = 0, in which case (0,0) is a saddle point. 0x² IV. Compute the Hamiltonian when V(x, y) = x² - y².
Transcribed Image Text:Problem 3*: Let V: R²R be a smooth function with a unique, nondegenerate critical point at (0,0). Let y > 0. Now consider the system of ordinary differential equations: av x' = Oy - Yox, y = -x-Yoy Do the following: I. Show that the only equilibrium point of the system is (0,0). II. Show that for any nonconstant solution, V(x(1), y(t)) is strictly monotonically decreasing. III. Show that the system only Hamiltonian if and only if V satisfies + OV = 0, in which case (0,0) is a saddle point. 0x² IV. Compute the Hamiltonian when V(x, y) = x² - y².
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