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Homework Help: Hamilton function of a free particle/Landau's book

  1. Jul 21, 2011 #1


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    1. The problem statement, all variables and given/known data
    The problem is taken out of Landau's book on classical mechanics. I must find the Hamilton function and the corresponding Hamilton equations for a free particle in Cartesian, cylindrical and spherical coordinates.

    2. Relevant equations
    Hamilton function: [itex]H(p,q,t)= \sum p_i \dot q _i -L[/itex] where L is the Lagrangian.
    Hamilton equations: [itex]\dot q_i = \frac{\partial H}{\partial p_i}[/itex] and [itex]\dot p_i =-\frac{\partial H}{\partial q _i}[/itex].

    3. The attempt at a solution
    I'm stuck on Cartesian coordinates so far. [itex]L=\frac{m}{2}(\dot x ^2 + \dot y^2 +\dot z^2)-U(x,y,z)[/itex].
    [itex]p_i=\frac{\partial L}{\partial \dot q _i}=m \dot q_i \Rightarrow \dot p_i=m \ddot q_i[/itex].
    [itex]H=m \dot x ^2 +m \dot y^2 +m \dot z ^2 - \frac{m}{2} (\dot x^2 + \dot y^2 +\dot z^2)+U(x,y,z)=\frac{m}{2}(\dot x ^2 +\dot y^2 +\dot z^2)+U(x,y,z)[/itex]. I notice that the Hamilton function is the Hamiltonian and that in this case it's worth the total energy of the system (the free particle).
    However the solution given in the book is [itex]H=\frac{1}{2m} (p _x ^2+ p_y^2 +p_z ^2)+U(x,y,z)[/itex]. Why is it expressed under this form? I took Landau's expressions and definitions and land on a different answer... why?!

    Edit: Hmm I guess it's because H should depend only on p, q and t. Never on [itex]\dot q[/itex]?
  2. jcsd
  3. Jul 22, 2011 #2
    Lagrangian is a function of [itex]q[/itex], [itex]\dot{q}[/itex], and [itex]t[/itex]
    Hamiltonian is a function of [itex]q[/itex], [itex]p[/itex], and [itex]t[/itex]

    You might think that this is just semantics since [itex]\dot{q}[/itex] and [itex]p[/itex] are proportional to each other, but it is very important distinction.

    Not sure Landau is the best book to learn these stuff from. It loves elegant solutions and hates verbose explanation.
  4. Jul 22, 2011 #3


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    Ok thanks a lot for the explanation.
    By the way, about the book, do you have any other suggestion? I currently own Goldstein's 1st edition on classical mechanics. I find very few worked examples and lots of theory. So I try to complete with Landau's book.
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