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Stacked spring carts
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Simulation of two stacked carts both having compactly supported springs on either side. The top cart can interact by springs with the edges of the bottom cart, and the bottom cart with obstacles on either side. All cart wheels are massless. The bottom cart is five times heavier than the top cart.
The separable Hamiltonian system has two degrees of freedom (DOF), but reduces to a single DOF if the obstacles are unreachable. The obstacles are not reachable if the system has zero momentum and insufficient energy to reach the obstacles.
0:00 intro
0:20 The carts do not reach the obstacles. Single DOF systems always exhibit regular motion
0:30 The obstacles are reached yielding chaotic motion
0:56 The obstacles are reached but with a regular pattern with non-chaotic motion
The Hamiltonian system
H = 1/2*(p1 - p2)^2/M + 1/2*p2^2/m + Spring_Potentials(q1,q2)
while simple, has a non-trivial potential function involving the up(x)-function which ramps up the spring potential from 0 to a linear spring with smooth derivatives of all orders, using the up(x) function only for the ramp-up and without application of convolutions.
The phase space was depicted using only p2 and q2
The simulation was performed using high order explicit symplectic integrators and was rendered in real time
🎵 "Coming Home Remix" by "LHS" | CC | not affiliated with/endorsed by.
The separable Hamiltonian system has two degrees of freedom (DOF), but reduces to a single DOF if the obstacles are unreachable. The obstacles are not reachable if the system has zero momentum and insufficient energy to reach the obstacles.
0:00 intro
0:20 The carts do not reach the obstacles. Single DOF systems always exhibit regular motion
0:30 The obstacles are reached yielding chaotic motion
0:56 The obstacles are reached but with a regular pattern with non-chaotic motion
The Hamiltonian system
H = 1/2*(p1 - p2)^2/M + 1/2*p2^2/m + Spring_Potentials(q1,q2)
while simple, has a non-trivial potential function involving the up(x)-function which ramps up the spring potential from 0 to a linear spring with smooth derivatives of all orders, using the up(x) function only for the ramp-up and without application of convolutions.
The phase space was depicted using only p2 and q2
The simulation was performed using high order explicit symplectic integrators and was rendered in real time
🎵 "Coming Home Remix" by "LHS" | CC | not affiliated with/endorsed by.
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