A simple choreography finder was implemented. It minimises the action functionnal wrt the Fourrier coefficients of the (n!) cycles.
Many questions arise :
[ ] What is the best conditionned coefficient space
[ ] What does it mean to (linearly / non-linearly) reparametrize the coefficient space ? In particular, how does this affect the iterative solver procedures ?
[ ] The "real" physical principle (namely the "principle of least action") does not ask for the action to be either minimal or locally minimal, but to have a zero gradient. So we should miss a lot of potentially interesting choreographies. It would be interesting to implement a method to solve the first condition equations instead of the minimisation.
[ ] It could be interesting to link the choreography program and the simulation to automatically check for stability. => How to characterize and unstable behaviour ?
[ ] Is stability linked with the minimum property (It seems that the answer is no, see G. Minton's thesis "Computer-assisted proofs in geometry and physics") ?
A simple choreography finder was implemented. It minimises the action functionnal wrt the Fourrier coefficients of the (n!) cycles.
Many questions arise : [ ] What is the best conditionned coefficient space [ ] What does it mean to (linearly / non-linearly) reparametrize the coefficient space ? In particular, how does this affect the iterative solver procedures ? [ ] The "real" physical principle (namely the "principle of least action") does not ask for the action to be either minimal or locally minimal, but to have a zero gradient. So we should miss a lot of potentially interesting choreographies. It would be interesting to implement a method to solve the first condition equations instead of the minimisation. [ ] It could be interesting to link the choreography program and the simulation to automatically check for stability. => How to characterize and unstable behaviour ? [ ] Is stability linked with the minimum property (It seems that the answer is no, see G. Minton's thesis "Computer-assisted proofs in geometry and physics") ?