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Simple Symplectic Reduction Example

  1. Apr 20, 2015 #1
    1. The problem statement, all variables and given/known data
    I'm struggling to perform a symplectic reduction and don't really understand the process in general. I have a fairly solid understanding of differential equations but am just starting to explore differential geometry. Hopefully somebody will be able to walk me through this very simple example. Given the following optimization problem:
    \begin{equation}\text{min}\;J = \frac{1}{2}\int_0^{t_f} u^2dt\end{equation}
    \begin{equation}\dot{x_1} = x_2\end{equation}
    \begin{equation}\dot{x_2} = u\end{equation}

    Where u is a control variable and we have arbitrary boundary conditions defined. Our control Hamiltonian has 2 states and 2 co-states, thus has dimension 2n.

    2. Relevant equations
    \begin{equation}H = <\lambda,f> + L = \lambda_1x_2 + \lambda_2u + \frac{1}{2}u^2\end{equation}
    \begin{equation}\frac{\partial H}{\partial u} = 0 = \lambda_2 + u\end{equation}
    \begin{equation}\dot{x} = \frac{\partial H}{\partial \lambda}\end{equation}
    \begin{equation}\dot{\lambda} = -\frac{\partial H}{\partial x}\end{equation}
    Through integration, we know the solution has the following form:
    \begin{equation}x_2 = \frac{1}{2}C_1t^2 - C_2t + C_3\end{equation}
    \begin{equation}x_1 = \frac{1}{6}C_1t^3 - \frac{1}{2}C_2t^2 + C_3t + C_4\end{equation}
    And using given boundary conditions, the problem simplifies to a 4 dimensional search for each of the 4 constants (or 5 depending on if final time is free, but let's ignore that for now).

    3. The attempt at a solution
    Knowing the following is a constant of motion:
    \begin{equation}\int_0^t \dot{\lambda_1} dt = C_1\end{equation}
    because
    \begin{equation}\dot{\lambda_1} = 0\end{equation}
    According to the Marsden-Weinstein reduction theorem, I should be able to reduce the Hamiltonian to dimension 2n-2.
    \begin{equation}H = <\lambda,f> + L = C_1x_2 + \lambda_2u + \frac{1}{2}u^2\end{equation}
    Yet when I rederive the equations of motion, the constant still reappears.
    \begin{equation}u = -\lambda_2\end{equation}
    \begin{equation}\dot{x_2} = \lambda_2\end{equation}
    \begin{equation}\dot{\lambda_2} = -C_1\end{equation}
    According to papers I've read online, I should be able to simplify this optimization problem down in dimensionality, but with the first constant reappearing in the equations of motions, I effectively still have a 4 dimensional search. Keep in mind that I'm assuming I'm using some numerical method to solve this problem, not an analytic procedure. Even though this is an easy problem to solve analytically, I don't care about the solution so much as the method itself. If anybody could help me properly reduce the Hamiltonian, that would be great!
     
  2. jcsd
  3. Apr 26, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
  4. May 1, 2015 #3
    Here's a link to a paper that I'm more or less trying to follow:
    http://www.utdallas.edu/~txo140730/papers/SymRedOptCtrl.pdf
    Example 5.2 from that paper is precisely what I'm trying to do, although probably through a different method. I'd be happy with either one. They begin with a system of 10 differential equations and are able to reduce it to a system of 7 differential equations. My issue is that I can't completely decouple these integrals and constants from my reduced system whereas literature seems to suggest that I should be able to.

    Also the search with the example I gave would only be 2 dimensional, 3 with free final time, instead of 4 and 5 like I mentioned before. This is because we'd have some boundary conditions already defined like a start location, but this is really besides the main point.
     
  5. May 18, 2015 #4
    Bump. Haven't made any progress.
     
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