From the equation \(\displaystyle{\left({F}_{{x}}+{p}{F}_{{y}}\right)}{\left.{d}{x}\right.}+{F}_{{p}}{d}{p}={0}\) I do not understand how

ICESSRAIPCELOtblt

ICESSRAIPCELOtblt

Answered question

2022-03-25

From the equation
(Fx+pFy)dx+Fpdp=0
I do not understand how we can get
x˙=Fp, p˙=(Fx+pFy)

Answer & Explanation

Deon Frost

Deon Frost

Beginner2022-03-26Added 6 answers

Step 1
In equation you are looking for vector fields X(u, v) with
Mu+Nv=0,
using dx(X)=u, dp(X)=v. This can be interpreted as an orthogonality condition. Thus one trivial solution is to rotate (M, N) by 90 to get (u, v)=(N, M) or (u, v)=(N, M). You can also scale by any other non-zero factor (function).
The resulting ODE system, with y˙=px˙, gives solutions x(t), y(t), p(t). On segments where x(t) is monotonic it can be inverted, and then also y and p can be obtained as functions of x.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?