Problem posed and solved by Daniel Bernoulli in 1732 and by Euler en 1781. See this paper pages 111-114 for history and resolution. Web :  -  scipython.com/blog/the-hanging-chain/ -  proofwiki.org/wiki/Bernoulli%27s_Hanging_Chain_Problem -  for any oscillations : math.arizona.edu/~gabitov/teaching/181/math_485/Final_Report/Spinning_chain_final_report.pdf -  video of the actual movements of a chain : Oscillation modes in a Hanging Chain

 

Cartesian equation :  where  is the first Bessel function of order 0, solution of . For a chain length , time equation: for  ;  is defined by , so  ; we put then  where  is the nth positive zero of. As the pulsation of the simple pendulum is , .

 

Animation pour n = 1	Animation pour n = 2	Animation pour n = 3

The curve of the suspended chain, or, more precisely, the curve of small oscillations of the chain suspended at one end, is the shape taken by a heavy wire
suspended at one end and placed in a uniform gravitational field.

For these small oscillations, the motion is periodic, but it becomes chaotic for any oscillations: the chain can then be approximated by a multiple pendulum, généralisation of the double pendulum.
See this impressive video showing the chaotic oscillations of such a pendulum.

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© Robert FERRÉOL  2025