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Math Help - How do I write something as a first order system, or vector field, on the phase plane

  1. #1
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    How do I write something as a first order system, or vector field, on the phase plane

    I have been asked to write d2s/dt^2 = −s as first order system, or vector field, on the phase plane.

    and then compute its potential, I have no idea how to do either.

    I have looked it up, I am currently on this website Pauls Online Notes : Differential Equations - Systems of Differential Equations and still don't understand. It's not in my course notes either.

    Please help if you have any idea, I have an exam in a couple of week and NEED to know this.
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  2. #2
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    Quote Originally Posted by supaman5 View Post
    I have been asked to write d2s/dt^2 = −s as first order system, or vector field, on the phase plane.

    and then compute its potential, I have no idea how to do either.

    I have looked it up, I am currently on this website Pauls Online Notes : Differential Equations - Systems of Differential Equations and still don't understand. It's not in my course notes either.

    Please help if you have any idea, I have an exam in a couple of week and NEED to know this.
    You need to make the substitution

    x=\frac{ds}{dt} \implies \frac{dx}{dt}=\frac{d^2s}{dt^2}

    Now using the 2nd equation and the ODE we get that

    \frac{dx}{dt}=\frac{d^2}{dt^2}=-s

    So now we have the first order linear system of ODE's

    \dot{s}=x

    \dot{x}=-s

    \mathbf{v}(s,x)=<x,-s>

    I am not sure what you want for the 2nd part. Are you looking for a scalar function

    \phi

    such that

    \nabla \phi =\mathbf{v}
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  3. #3
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    I think I need to find the potential energy so that the equation 0.5mv^2+V=E(total energy)

    and thanks so much for the first part
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  4. #4
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    Re: How do I write something as a first order system, or vector field, on the phase p

    would there not be a constant of integration
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  5. #5
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    Re: How do I write something as a first order system, or vector field, on the phase p

    would it not be \frac{1}{2} \dot{s}^2+\frac{s^2}{2}= constant=E for some E

    to compute its potential
    compare with \ddot{s}=-\nabla(s)
    so
    \nabla(s)=s
    so
    V(s)=\frac{s^2}{2}+K
    then choose K to be equal to zero
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