Differential movement along a curved surface

[kairama15]

TL;DR Summary

I'd like to understand movement along the surface of a 3 dimensional graph.

I'd like to understand the movement of a particle along the surface of a three dimensional graph. For example, if there is a flat two dimensional plane (z=2 for all x and y), and a unit vector describes its initial direction of movement (<sqrt(2)/2i+sqrt(2)/2j> for example), then the vector will just move in a straight line along the plane. However, if the surface of the function is curved like the surface of the sphere, the particle will move in a curved path. If the function is z=sqrt(1-x^2-y^2) and the initial direction of movement is <1i+0j> at point (0,0,1), the particle will move around the sphere in a circle. I imagine this somewhat like a coin going around the surface of one of those donation things at the mall. The coin moves around the curved surface and spirals into the middle where it falls down the hole (I know this isn't nearly a perfect example because gravity is involved, but the idea is similar.)

To make it easy and to get familiar with this, I first tried to make this happen on a two dimensional graph on function like y=sin(x). Ideally, I'd want to start at a point on the graph and move along the surface of the graph, tracing out the function. If I start at any point, (x,y) and a unit vector describing the initial slope (direction) of the movement along the graph <ai+bj>, the unit vector will move a differential distance "ds" and will rotate a differential angle "dtheta". I've worked out that the next x value will be [x+cos(atan(dy/dx)+k*ds)] and the next y value will be [y+sin(atan(dy/dx)+k*ds)] where k is the curvature of the function at that point. The unit vector's angle will rotate dtheta degrees (dtheta = ds/r = k*ds) , and the new unit vector describing the new direction of movement (the slope at the next point) is <(cos(atan(b/a)+k*ds)i + (sin(atan(b/a)+k*ds)j>. If I iterate this on a simple program like excel, the points generated trace out the function (like sin(x)).

So the question is, how do I extend this idea to a 3d graph like z=sin(x)+sin(y) ? Id like to know what the new point is after moving a differential distance "ds" along the surface, and how does the unit vector describing the direction of movement change?

 

[BvU]

I know this isn't nearly a perfect example because gravity is involved

Does that mean gravity is not involved in the other examples ?
In the first example gravity in the $z$ direction does not matter
The second example is a half sphere, and starting at $(0,0,1)$, i.e. the top, with $\vec v = (1, 0, 0)\$, I would expect a simple drop along the surface (until it separates from the sphere at some point), not your circle.

I'd like to understand the movement of a particle along the surface of a three dimensional graph.

So what is the equation of motion ?

 

[romsofia]

Look up the concept of "Cartan connection", or simply the concept of connection in differential geometry.

 

[DEvens]

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.

https://en.wikipedia.org/wiki/Lagrangian_mechanics

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.