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Mathematics
Differential Geometry
Differential movement along a curved surface
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[QUOTE="DEvens, post: 6290337, member: 475460"] If you are interested specifically in the differential geometry, then my answer won't be helpful. If you are interested in getting an equation of motion for an object constrained to a surface, then that is "textbook" stuff. [URL]https://en.wikipedia.org/wiki/Lagrangian_mechanics[/URL] I will give an extremely brief outline. Details are on the wiki page. You start with a Lagrangian. For motion with no constraints you get the Lagrange equations of motion. These can be shown to be equivalent to Newtonian mechanics. One very cool aspect of Lagrangian mechanics is you can express things in terms of any convenient coordinate system. You get equations in terms of those other coordinates "automatically." So if you expressed things in terms of (r, theta, phi), spherical polar coordinates, you automatically get the correct equations of motion in terms of these coordinates. You wind up with canonical momentum parameters that correspond to each of the coordinates you use. When you have a constraint you can express as F(x,y,z) = 0, with F a suitably smooth function, then you can add the constraint to the Lagrangian, with a Lagrange undetermined multiplier ##\lambda##. Then, you treat the multiplier as another coordinate. The result is, when you derive the new field equations with the extra coordinate, you get a system that includes the constraint as an equation of motion. This is a very powerful method, provided your constraints can be expressed in this form. For smooth surfaces this works out very well. For example, a sphere can be defined as x^2+y^2+z^2 - R^2 = 0. Lagrange mechanics and Hamiltonian mechanics are very useful methods. For example, when you move into quantum mechanics, the Lagrangian for a quantum field is a very usual way to express the nature of and interactions with a quantum field. And it's the starting point for the Feynman path integral. Also, when you have conservation constraints or gauge symmetry constraints, you can express them as undetermined multipliers, and so reduce the effective complexity of your system. Much fun. [/QUOTE]
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Differential movement along a curved surface
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