Is there any known way to identify which variable has the most impact in the dynamics of a system given its lagrangian or hamiltonian formulation? Let's say i have a system with 3 variables, two angles and a tension. How can i discriminate if $angle_1$ has more influence in the evolution of the dynamics than $angle_2$ or the tension $T$?
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$\begingroup$ What the geometry? Are you interested in performance changes due to design changes? $\endgroup$– JAlexCommented May 24, 2022 at 17:06
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2 Answers
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A very quick answer, maybe you can compute a mean of the derivatives of the action respect to the variables, and the one which has a higher derivatives mean, is the one that affects most the system. (mean: integrate it around common-sense solutions, or initial conditions for your variables, divided by the variation)
Explanation: If a variable doesn't change the action it means that the equations of motion won't restrict it's movement, so whenever a variable changes more the action it's gonna become more important for the dynamics I assume.
answered May 24, 2022 at 14:35
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The answer is neither. The reason being that dynamic systems are inherently chaotic and tiny changes in the initial conditions have profound changes in the long term. The variation in the results is only bounded by the kinematics of the system.
You can answer deterministically at any instant which variable influences the rate of change of the state, but adding up multiple steps the task becomes impossible. It is not a matter of lack of computing power, it is a mater of long terms results become non-meaningful fairly quickly.
