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How to extract physical meaning from differential equations

  1. Jan 5, 2015 #1
    1. The problem statement, all variables and given/known data

    I am given the following coupled differential equations:

    \begin{align}
    (r^2+1)\ddot{θ}+2r\dot{r}\dot{θ} &= u1\\
    \ddot{r}-r\dot{θ}^2&=u2
    \end{align}

    together with the following expression for the kinetic energy:

    \begin{align}
    T &= 0.5(r^2+1)\dot{θ}^2+0.5\dot{r}^2
    \end{align}

    2. Relevant equations

    The problem asks to also take into consideration the potential energy in order to further analyze the system.

    1) How do I analyzed the two equations given in order to identify the potential energy?

    2) Also what is the thinking behind a system of equations in order to extract the physical meaning thus identifying the potential energy (and in general any other physical quantities)?


    3. The attempt at a solution

    My reasoning is as follows: it is clear from the expression of the kinetic energy that the two velocities associated with the system are: ##\hspace{0.3cm} \dot{θ} \hspace{0.3cm} and \hspace{0.3cm} \dot{r} ##

    Also ##\hspace{0.3cm} θ \hspace{0.3cm} and \hspace{0.3cm} r \hspace{0.3cm} ## are displacements.

    Does that means that mass is given by:##\hspace{0.3cm} (r^2+1) \hspace{0.3cm}## ? If that is true, what kind of mass changes as it moves? Also if this is expression is mass then it will become huge if it moves a few hundreds of meters.

    In regard to the potential energy I am thinking about gravitational potential energy. I excluded elastic potential energy since I am not given any spring constants, and electric potential energy is out of the question. Is there any other form of PE I am missing out?

    If gravitational potential energy is the correct form then we have for the first diff. eqtn: ##\hspace{0.3cm} PE = mgh =m(ma)h= (r^2+1)( (r^2+1) \ddot{θ}) θ = (r^2+1)^2 \ddot{θ} θ \hspace{0.3cm}## .
    Notice how I wrote the gravity as a product of the mass times acceleration; I could also use the gravitational constant.

    Similarly for the second equation, the PE comes out: ##\hspace{0.3cm} PE = mgh = 1(1\ddot{r})r = \ddot{r}r \hspace{0.3cm}## .
    From the given expression for the Kinetic Energy I thought that the mass for the second equation dynamics is 1 kg.

    In reality the problem is much more than that; the expression for the Kinetic Energy for the coupled system is a Lyapunov function and we want to prove GAS stability at the origin by taking into account also the Potential Energy. So basically we want to show that zero is the largest invariant set contained in ##\hspace{0.3cm} \dot{V}\hspace{0.3cm}## (Assuming V(x) is pos. def. and radially unbounded;which comes out to be). We are also given the expression for the inputs u1 and u2; I can provide them if someone wants to figure things out further, but my questions are regarding the physical meaning of the two equations and whether there exists any general procedure for determining the actual system.
    Thanks a ton.
     
  2. jcsd
  3. Jan 5, 2015 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    (r^2+1) has some similarity to a mass, it is not a mass. This appears frequently when the actual velocity of an object is not proportional to the time-derivative of a coordinate. If you calculate the kinetic energy of an object in polar coordinates, you get a similar expression.

    I can imagine a system that has this type of kinetic energy. A rotating disk with an additional mass that can move in- or outwards on the disk, but has to rotate together with the disk.
    The system does not have to have such a representation, however. You can work with the equations even without a physical system behind them.
     
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