# Edward Hughes

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bio website edwardfhughes.wordpress.com location age member for 1 year, 1 month seen 1 hour ago profile views 281

I'm currently studying Part III Mathematics at the University of Cambridge, and hope to pursue a PhD based broadly on the applications of differential geometry in theoretical physics.

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 May20 accepted Vector Field Generating Variation Along Curve May20 comment Vector Field Generating Variation Along Curvethanks I clearly see that's true now. However intuitively I don't understand why! The Lie derivative $L_X Y$ at $p$ is defined as the rate of change of the pullback of $Y$ along the integral curve of $X$ starting at $p$. Is it because the pullback operation depends on the extension of $X$ around this integral curve? In fact that does seem very plausible to me. I'd be very grateful if you could confirm this, then I'll certainly accept your answer! Cheers! May19 comment Vector Field Generating Variation Along CurveI don't quite get your argument. Indeed the commutator of $N$ and $\gamma'$ is precisely the Lie derivative of $N$ with respect to $\gamma'$ which depends only on the value $N$ along $\gamma$. So I don't see how the extension of $N$ has anything to do with the Lie bracket! May19 comment Vector Field Generating Variation Along CurveWell - he's at least assuming that it is linearly independent I guess, but that doesn't seem strong enough to me. Have you got a reference for a differential forms based proof? May19 revised Vector Field Generating Variation Along Curveadded 15 characters in body May19 comment Vector Field Generating Variation Along CurveI don't quite get what you are saying. He seems to be claiming that an arbitrary vector field $N$ along $\gamma$ is automatically a pushforward of a coordinate vector field. This is what I don't believe. Since by what you say that would imply that every vector field along $\gamma$ commutes with the tangent vector field... May19 asked Vector Field Generating Variation Along Curve May19 accepted Hamiltonian for Geodesic Flow May18 awarded Constituent May17 revised Hamiltonian for Geodesic Flowcorrected formulae May17 answered Hamiltonian for Geodesic Flow May17 comment Hamiltonian for Geodesic Flow@Alex - I really don't see how this would work. The Christoffel formula gives three derivatives of $g$ with respect to different combinations of indices, and I have only one in the expression for $dH$. If you can get it to work, could you let me know how?! May16 accepted Symplectic Form Preserved by Orthogonal Transformation May16 comment Symplectic Form Preserved by Orthogonal TransformationAh okay that makes sense! So for a neat solution I'm better off just computing the relevant Lie derivatives then? May16 asked Hamiltonian for Geodesic Flow May16 comment Symplectic Form Preserved by Orthogonal TransformationAh right - so I need another transformation because SO(3) is 3 dimensional. I can't work out what it is though. Is there some nice formula that you know? I know that I could solve this problem by finding the relevant vector fields and showing that they Lie derive $\omega$, but I'd prefer to do a direct solution if possible! Many thanks! May16 revised Symplectic Form Preserved by Orthogonal Transformationedited title May16 asked Symplectic Form Preserved by Orthogonal Transformation May13 asked Levi-Civita Connection for 2-dimensional Riemannian manifold May7 awarded Caucus