- #1
Blazejr
- 23
- 2
Hello
I'm struggling with well-known problem of finding shortest path between two points on a sphere using calculus of variations. I managed to find correct differential equations of great circles, but I'm not confident about validity of methods I used. Below I describe my approach.
In solutions of this problem I found online it was assumed that curve we want to find can be parametrized by angle [itex]\theta[/itex] (of spherical coordinates, which I'm going to use from now on). This is correct - geodesics on sphere are great circles and we can always choose coordinates on sphere such that both points lie on one meridian, say [itex]\phi=0[/itex]. However, I do not want to use my knowledge about those geodesics before I actually find them. Therefore, I have no way of knowing that [itex]\theta[/itex] is correct global parametrization of this curve - it could go along some line of latitude (theta=const.) for some time. Therefore I need to assume [itex]\theta , \phi[/itex] to be independent variables parametrized by some arbitrary parameter along the curve (call it time [itex]t[/itex]).
Problem comes down to finding functions [itex]\theta (t), \phi (t)[/itex] that minimize integral:
[tex]\int L \mathrm{d}t = \int \sqrt{\dot{\theta}^2+\sin ^2 \theta \dot { \phi } ^2} \mathrm{d}t[/tex]
It follows imidietely from Euler-Lagrange equation that quantity [itex]\frac{\sin ^2 \theta \dot{ \phi}}{L}[/itex] is conserved. E-L equation for second variable produce complicated and messy equation.
Here comes my idea. Variable [itex]t[/itex] is arbitrary parametrization of the curve. We can parametrize it by any other parameter that grows monotonically with time. It is easy to check that function under the integral is unaffected by this change of variables. Therefore I can use length of curve [itex]s[/itex] as parameter. If I do that we have that [itex]L(s)[/itex] is actually constant (as it is just rate of change of length w.r.t length) and E-L equations yield:
[tex]\sin ^2 \theta \dot {\phi}=const., \qquad 2 \ddot {\theta} = \sin ( 2 \theta ) \dot {\phi}^2 [/tex]
I've read on some website that those are correct equations for great circles. I can't check, because I don't know how to solve them anyways. If anyone can give me a hint on that I would be grateful, but that's not what I wanted to ask.
Problem is that I am not sure if this approach is correct. Can I really use E-L equations when I parametrize by arclength? This is after all equivalent to imposing (in language of classical mechanics) non-holonomic constraint: [itex]\frac{\mathrm{d}L}{\mathrm{d}s}=0[/itex]. Therefore [itex]\dot{\theta},\dot{\phi}[/itex] are no longer independent.
Is this approach completely wrong and I got good equations "by accident" or am I missing something? If it is, how can I fix my reasoning to find those equations (without simplifying assumption that the curve can be parametrized by [itex]\theta[/itex]?). Thanks in advance.
I'm struggling with well-known problem of finding shortest path between two points on a sphere using calculus of variations. I managed to find correct differential equations of great circles, but I'm not confident about validity of methods I used. Below I describe my approach.
In solutions of this problem I found online it was assumed that curve we want to find can be parametrized by angle [itex]\theta[/itex] (of spherical coordinates, which I'm going to use from now on). This is correct - geodesics on sphere are great circles and we can always choose coordinates on sphere such that both points lie on one meridian, say [itex]\phi=0[/itex]. However, I do not want to use my knowledge about those geodesics before I actually find them. Therefore, I have no way of knowing that [itex]\theta[/itex] is correct global parametrization of this curve - it could go along some line of latitude (theta=const.) for some time. Therefore I need to assume [itex]\theta , \phi[/itex] to be independent variables parametrized by some arbitrary parameter along the curve (call it time [itex]t[/itex]).
Problem comes down to finding functions [itex]\theta (t), \phi (t)[/itex] that minimize integral:
[tex]\int L \mathrm{d}t = \int \sqrt{\dot{\theta}^2+\sin ^2 \theta \dot { \phi } ^2} \mathrm{d}t[/tex]
It follows imidietely from Euler-Lagrange equation that quantity [itex]\frac{\sin ^2 \theta \dot{ \phi}}{L}[/itex] is conserved. E-L equation for second variable produce complicated and messy equation.
Here comes my idea. Variable [itex]t[/itex] is arbitrary parametrization of the curve. We can parametrize it by any other parameter that grows monotonically with time. It is easy to check that function under the integral is unaffected by this change of variables. Therefore I can use length of curve [itex]s[/itex] as parameter. If I do that we have that [itex]L(s)[/itex] is actually constant (as it is just rate of change of length w.r.t length) and E-L equations yield:
[tex]\sin ^2 \theta \dot {\phi}=const., \qquad 2 \ddot {\theta} = \sin ( 2 \theta ) \dot {\phi}^2 [/tex]
I've read on some website that those are correct equations for great circles. I can't check, because I don't know how to solve them anyways. If anyone can give me a hint on that I would be grateful, but that's not what I wanted to ask.
Problem is that I am not sure if this approach is correct. Can I really use E-L equations when I parametrize by arclength? This is after all equivalent to imposing (in language of classical mechanics) non-holonomic constraint: [itex]\frac{\mathrm{d}L}{\mathrm{d}s}=0[/itex]. Therefore [itex]\dot{\theta},\dot{\phi}[/itex] are no longer independent.
Is this approach completely wrong and I got good equations "by accident" or am I missing something? If it is, how can I fix my reasoning to find those equations (without simplifying assumption that the curve can be parametrized by [itex]\theta[/itex]?). Thanks in advance.