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Math Help - Newton's Second Law --> Conservation of Energy

  1. #1
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    Newton's Second Law --> Conservation of Energy

    Derive the Principle of Conservation of Energy from Newton’s Second Law for a particle of fixed mass with position vector x(t) which varies with time t, subject to a conservative force given by a potential V (x).

    I have no idea how to get started on this question, I get the feeling it is fairly straightforward but the wording of this question seems foreign to me.

    Anyone help me out at all?
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    So Newton's Second law states that

    \displaystyle{\sum_{k}\vec{F_{k}}=\dot{\vec{p}}.}

    You're asked to derive the conservation of energy equation by using Newton's Second law. The conservation of energy states that

    \displaystyle{\frac{d}{dt}(T+V)=0,} right? Here T is the kinetic energy, and V the potential energy. The total energy is constant.

    Here are a few equations that might be useful to you:

    \displaystyle{T=\frac{\vec{p}^{\;2}}{2m}}, and

    \displaystyle{\vec{F}=-\nabla V.}

    I would start by differentiating T+V, and using various equations and identities to show that you get zero.
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  3. #3
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    Thanks man, having looked at that I don't really feel as though I understand what your saying.

    Perhaps I am underestimating this question somewhat, but I do not understand what you have done there or are suggesting!

    Is there anyway you could simplify it even further for me? Sorry !
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  4. #4
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    Ok, let's take a closer look. Do you understand that we need to show

    \displaystyle{\frac{d}{dt}(T+V)=0}?
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    I don't understand where the T value comes from, which may just be down to my coursework having something represented differently.

    I also don't appreciate the role of the 0, why does this equal 0? I would have imagined that the conservation of energy simple states that an initial energy is the same as the final energy?
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  6. #6
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    The T is kinetic energy. Think of the one-dimensional version as

    T=\frac{1}{2}\,mv^{2}=\frac{p^{2}}{2m}, where p=mv is the momentum. Different books call it different things.

    Conservation of energy means that the total energy doesn't change. If my condition that the time derivative of the total energy is zero is true, then I can integrate:

    \displaystyle{\frac{d}{dt}(T+V)=0} implies

    T+V=\text{constant}.

    Because of this, I can say that initial energy equals final energy, because it doesn't change:

    T_{i}+V_{i}=T_{f}+V_{f}.

    That's exactly what you said. The two conditions are really equivalent, because I could just as easily differentiate this second equation to get back to the first one. Make sense?
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  7. #7
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    Yup I am with ya now, it was mostly down to different notations that I wasn't with you so far!

    However I am still sketchy as to how we go from Newtons law to the conservation rules, I just still dont see the steps that tie them together?
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  8. #8
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    Well, let's take the indicated derivative and see where that takes us. We have

    \displaystyle{\frac{dT}{dt}+\frac{dU}{dt}}, that we want to evaluate. Our goal is to show that it is zero. Now we know that

    \displaystyle{T=\frac{m}{2}\,\vec{v}^{\;2}}, and hence its derivative is what?

    Also, we can just apply the usual Calc III rules of partial differentiation to find the derivative of the potential energy:

    \displaystyle{\frac{dU}{dt}=\frac{\partial U}{\partial x}\,\frac{dx}{dt}+\frac{\partial U}{\partial y}\,\frac{dy}{dt}+\frac{\partial U}{\partial z}\,\frac{dz}{dt}+\frac{\partial U}{\partial t}=\nabla U\cdot\dot{\vec{x}}+\frac{\partial U}{\partial t}}.

    But because U is conservative, what can you tell me about

    \displaystyle{\frac{\partial U}{\partial t}}?
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  9. #9
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    Im not sure I understand what you mean by conservative? Im sorry if I dont seem to be catching on very quickly here, I do appreciate your help greatly!

    The derivative of T is =mv also I assume?
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  10. #10
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    Go here for some info on conservative potential energy functions. (Conservative potential and conservative force are mutually dependent).

    Your derivative of T is incorrect. You've computed dT/dv. But we need to compute dT/dt: the time derivative of the kinetic energy. What do you get for that? Also, keep in mind that we have vectors floating around everywhere.
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