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4. Let V : R? → R be a differentiable function. (i) Suppose r : R → R? satisfies the equation (t) = -VV(x(t)) for all teR. Prove that the function V(r(t)) is non-increasing. (ii) Suppose r : R → R² satisfies the equation 0 1 r(t) = JVV(x(t)), J = for all t e R. Prove that the function V(r(t)) is constant.

4. Let V : R? → R be a differentiable function. (i) Suppose r : R → R? satisfies the equation (t) = -VV(x(t)) for all teR. Prove that the function V(r(t)) is non-increasing. (ii) Suppose r : R → R² satisfies the equation 0 1 r(t) = JVV(x(t)), J = for all t e R. Prove that the function V(r(t)) is constant.

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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4. Let V : R? -→ R be a differentiable function. (i) Suppose a : R → R? satisfies the equation (t) = -VV (x(t) for all teR. Prove that the function V(r(t)) is non-increasing. (ii) Suppose a : R → R? satisfies the equation #*(t) = J VV(r(t}), J=. %3D for all t e R. Prove that the function V(a(t)) is constant.
Transcribed Image Text:4. Let V : R? -→ R be a differentiable function. (i) Suppose a : R → R? satisfies the equation (t) = -VV (x(t) for all teR. Prove that the function V(r(t)) is non-increasing. (ii) Suppose a : R → R? satisfies the equation #*(t) = J VV(r(t}), J=. %3D for all t e R. Prove that the function V(a(t)) is constant.
Expert Solution
Step 1

(i.)

Given:

A differentiable function, V:ℝ2→ℝ

The function x:ℝ→ℝ2 satisfies the equation ddtxt=-∇Vxt for all t∈ℝ

To prove:

The function Vxt is non-increasing

 

(ii.)

Given:

A differentiable function, V:ℝ2→ℝ

The function x:ℝ→ℝ2 satisfies the equation ddtxt=J∇Vxt, J=01-10 for all t∈ℝ

To prove:

The function Vxt is constant

 

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