Find out the answer to "From the equation(Fx+pFy)dx+Fpdp=0I do not understand how we..."

ICESSRAIPCELOtblt

Answered question

2022-03-25

From the equation

(F_{x} + p F_{y}) d x + F_{p} d p = 0

I do not understand how we can get

\dot{x} = F_{p}, \dot{p} = - (F_{x} + p F_{y})

Deon Frost

Beginner2022-03-26Added 6 answers

Step 1
In equation you are looking for vector fields

X (u, v)

with

M u + N v = 0,

using

d x (X) = u, d p (X) = v

. This can be interpreted as an orthogonality condition. Thus one trivial solution is to rotate

(M, N)

90^{\circ}

to get

(u, v) = (N, - M)

(u, v) = (- N, M)

. You can also scale by any other non-zero factor (function).
The resulting ODE system, with

\dot{y} = p \dot{x}

, 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.

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