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Homework Help: Derive lagrangian: finding potential energy of a particle constrained to a surface

  1. Jan 25, 2009 #1
    Theres a question in which I need to derive a langrangian, and then the equations of motion for a particle constrained onto a surface. I think I'll be able to do it, except for the fact that I have no idea how to formulate the POTENTIAL ENERGY of the particle. I'd like to have a go at the rest myself, but can someone help me with how to just find the potential energy?
    The particle is constrained to a surface:
    z(x,y)= x^2/a^2 + y^2/b^2,
    and its moving in presence of a gravitational force in -ve z direction.

    Obviously the gravitational force is F=-mg, and potential will be some function of x and y??

    I'd really appreciate some help!
     
  2. jcsd
  3. Jan 25, 2009 #2

    gabbagabbahey

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    Re: derive lagrangian: finding potential energy of a particle constrained to a surfac

    Start with the gravitational potential; what is that?
     
  4. Jan 25, 2009 #3
    Re: derive lagrangian: finding potential energy of a particle constrained to a surfac

    Thanks for replying. Gravitational potential is the potential energy per unit mass? Sorry, I still don't know where to go from there
     
  5. Jan 25, 2009 #4

    gabbagabbahey

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    Re: derive lagrangian: finding potential energy of a particle constrained to a surfac

    The gravitional potential is defined by [tex]\vec{F}_g=-\vec{\nabla}U_g[/tex], so for [tex]\vec{F}_g=-mg\hat{z}[/tex], one usually writes [tex]U_g=mgz[/tex]

    ...Surely you've seen this before?
     
  6. Jan 25, 2009 #5
    Re: derive lagrangian: finding potential energy of a particle constrained to a surfac

    Yes, I have. Sorry, I knew it would be something obvious like that; sometimes I just overcomplicate things in my head and I miss what I should really get straight away. Thankyou for your help, I should be able to do it now.
     
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