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Differential of normal vector

  1. Jun 18, 2006 #1
    Are there any facts about the derivative of the normalised normal vector n to a surface embedded in n-dimensional Euclid space? Is it true, for instance, that
    [tex]\frac{\partial n_j}{\partial x^i} = \frac{\partial n_i}{\partial x^j}[/tex]
    The context is as follows. The surface is defined implicitly by a constraint function; there's a Hamiltonian in reduntant coordinates and the canonical Hamiltonian equations of motion for (q,p) ensuring that trajectories lie in the constraint surface. I need to find acceleration [itex]\ddot{q}[/itex]; there the time derivative of n appears. By the way, how could I reformulate this task in the language of differential geometry?
     
  2. jcsd
  3. Jun 19, 2006 #2
    If the surface is given by [itex]\phi (\vec{r})=const.[/itex] then the normal to the surface is [itex]n_i=(\vec{\nabla}\phi)_i[/itex], it's not necessarilly unit normal, but this is easily reconciled. So, in Cartesian coordinates

    [tex]\frac{\partial n_j}{\partial x^i} = \frac{\partial n_i}{\partial x^j}[/tex]

    is correct.
     
    Last edited: Jun 19, 2006
  4. Jun 19, 2006 #3
    [tex]\frac{\partial n_j}{\partial x^i} = \partial_i \frac{\partial_j \phi}{(\nabla \phi \cdot \nabla \phi)^\frac{1}{2}} = \frac{\partial_{ij}\phi}{(\nabla \phi \cdot \nabla \phi)^\frac{1}{2}} - \frac{\partial_j \phi \sum_k \partial_k \phi \partial_{ik}\phi}{(\nabla \phi \cdot \nabla \phi)^{\frac{3}{2}}}[/tex]

    Thanks for the response! But I can't easily reconcile the second term: it doesn't seem to be symmetric.
     
  5. Jun 24, 2006 #4
    It's not, ignore me when I put the bit about "reconcilitation", I didn't mean to put that.
     
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