Hamiltonian Equation with High curvature

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Hamiltonian equation is defined



$$H(x,y) = U(x)+K(y); fracdxdt = y, fracdydt = -x$$



One example would be define



$$U(x) = fracx^22; K(y) = fracy^22$$



Solution to the differential equation can be



$$x(t) = rcos(a+t); y(t) = -rsin(a+t)$$



In this case; if one draw the graph; it's a circle.



My question is :



how do we construct Hamiltonian system where its graph will have very high curvature (sharp corner such as heart shape) ? Can you give me such example?



I'm experimenting Hamiltonian Monte Carlo algorithms on one dimension space.(by introducing momentum; it will be two dimension). I'm interesting to see how does HMC behave when there is high curvature in the momentum-position space




differential-equations differential-geometry markov-chains




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    up vote
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    Hamiltonian equation is defined



    $$H(x,y) = U(x)+K(y); fracdxdt = y, fracdydt = -x$$



    One example would be define



    $$U(x) = fracx^22; K(y) = fracy^22$$



    Solution to the differential equation can be



    $$x(t) = rcos(a+t); y(t) = -rsin(a+t)$$



    In this case; if one draw the graph; it's a circle.



    My question is :



    how do we construct Hamiltonian system where its graph will have very high curvature (sharp corner such as heart shape) ? Can you give me such example?



    I'm experimenting Hamiltonian Monte Carlo algorithms on one dimension space.(by introducing momentum; it will be two dimension). I'm interesting to see how does HMC behave when there is high curvature in the momentum-position space




    differential-equations differential-geometry markov-chains




    share|cite|improve this question





















      up vote
      1
      down vote

      favorite
      1









      up vote
      1
      down vote

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      1






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      Hamiltonian equation is defined



      $$H(x,y) = U(x)+K(y); fracdxdt = y, fracdydt = -x$$



      One example would be define



      $$U(x) = fracx^22; K(y) = fracy^22$$



      Solution to the differential equation can be



      $$x(t) = rcos(a+t); y(t) = -rsin(a+t)$$



      In this case; if one draw the graph; it's a circle.



      My question is :



      how do we construct Hamiltonian system where its graph will have very high curvature (sharp corner such as heart shape) ? Can you give me such example?



      I'm experimenting Hamiltonian Monte Carlo algorithms on one dimension space.(by introducing momentum; it will be two dimension). I'm interesting to see how does HMC behave when there is high curvature in the momentum-position space




      differential-equations differential-geometry markov-chains




      share|cite|improve this question











      Hamiltonian equation is defined



      $$H(x,y) = U(x)+K(y); fracdxdt = y, fracdydt = -x$$



      One example would be define



      $$U(x) = fracx^22; K(y) = fracy^22$$



      Solution to the differential equation can be



      $$x(t) = rcos(a+t); y(t) = -rsin(a+t)$$



      In this case; if one draw the graph; it's a circle.



      My question is :



      how do we construct Hamiltonian system where its graph will have very high curvature (sharp corner such as heart shape) ? Can you give me such example?



      I'm experimenting Hamiltonian Monte Carlo algorithms on one dimension space.(by introducing momentum; it will be two dimension). I'm interesting to see how does HMC behave when there is high curvature in the momentum-position space





      differential-equations differential-geometry markov-chains





      share|cite|improve this question










      share|cite|improve this question




      share|cite|improve this question









      asked Jul 28 at 15:38









      ElleryL

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