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Math Help - extremals no. 1

  1. #1
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    extremals no. 1

    Find the extremals of \int^{2}_{0}[\dot{x}^2 + 2\dot{x}] dt with x(0) = 0 and x(2) = 1 subject to the constraint \int^{2}_{0}x dt = 2
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  2. #2
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    Quote Originally Posted by wik_chick88 View Post
    Find the extremals of \int^{2}_{0}[\dot{x}^2 + 2\dot{x}] dt with x(0) = 0 and x(2) = 1 subject to the constraint \int^{2}_{0}x dt = 2
    \text{A problem of Hamilton's Variational Principle.}

     L(\dot{x},x,t) = \dot{x}^2 + 2\dot{x} +\lambda (x-1)

    -  \frac{d}{dt} \frac{\partial L}{\partial \dot{x}} - \frac{\partial L}{\partial x} =0 \Rightarrow 2\ddot{x} - \lambda = 0 \Rightarrow x(t)=\frac{1}{4} \lambda \, t^2+\mu t + \gamma

    - x(0)=0 \Rightarrow \gamma=0
    x(2)=1 \Rightarrow \lambda+2\mu=1 \Rightarrow \mu=\frac{1-\lambda}{2}
    \int^{2}_{0}x \, dt = 2 \Rightarrow \lambda=-3
     x(t)=-\frac{3}{4} t^2+2 t \Rightarrow \dot{x}(t)=-\frac{3}{2} t+2
    \int^{2}_{0}(\dot{x}^2 + 2\dot{x}) \, dt=\int^{2}_{0}(\tfrac{9}{4}t^2 -9t + 8) \, dt = 4
    Last edited by mr fantastic; September 18th 2009 at 08:18 AM. Reason: Restored original reply
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