I am trying to determine the position of a point mass $P$ moving in 2D space after time $t$. The system is defined as follows at time $t_0$:

  • $P$ has an initial position $\mathbf p_0 =(x_0,y_0)$
  • $P$ has an initial velocity $\mathbf v_0 =(v_{0_x},v_{0_y})$
  • $P$ has a fixed mass $M$
  • $P$ is attached to multiple ideal springs $S_1, S_2, ... S_n$
    • Each spring $S_i$ is attached at the other end to a fixed point $\mathbf q_i = (q_{i_x}, q_{i_y})$
    • Each spring $S_i$ has a different stiffness $k_i$
    • Each spring $S_i$ has a natural length $l_i$
    • Each spring $S_i$ may initially by stretched or compressed at $t_0$ by an arbitrary amount, so their initial lengths may not be equal to $l_i$

Here's an image to illustrate.

$P$ and springs

This is tricky to compute in one go since as $P$ moves, its velocity and acceleration change due to the springs changing in length. However, for any possible position for $P$, the acceleration vector is known and fixed, since the $\mathbf q_i$ and spring parameters are fixed.

I understand that at $t_0$ I can compute the 2D acceleration vector of $P$ by just summing the force vectors produced by each spring $S_i$ towards or away from each $\mathbf q_i$. My intuition tells me that to compute the final position of $P$ at time $t$, I need to integrate this 2D acceleration over time twice. once to get the velocity delta/acceleration, which I can add to $\mathbf v_0$, and once again to get the displacement, which I can add to the initial position.

Assuming this is a viable approach, the problem I'm running into is that the $x$ and $y$ components for the spring lengths (and hence forces) are related. Is this a multi-variable calculus/vector field problem? How can I write an equation for the position of $P$ at time $t$?


closed as off-topic by sammy gerbil, ZeroTheHero, Yashas, Kyle Kanos, Jon Custer Apr 10 '17 at 13:34

This question appears to be off-topic. The users who voted to close gave this specific reason:

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  • $\begingroup$ Does your question mean you are to simulate this in a computer, or are you finding the exact analytical equation to solve the position at time t? $\endgroup$ – philip_0008 Apr 9 '17 at 20:15
  • $\begingroup$ I would like to obtain the exact position for a non-zero, arbitrary time step. In a computer simulation, this could be approximated by assuming the acceleration is constant for a given position for $P$ and computing several small time steps, modifying $\mathbf v$ by the acceleration times time elapsed each time. $\endgroup$ – user1832602 Apr 10 '17 at 0:44
  • $\begingroup$ How familiar are you with Lagrangian formulation of mechanics? $\endgroup$ – Kyle Kanos Apr 10 '17 at 10:03
  • $\begingroup$ Not at all, but perhaps that's a good place to start. To be clear, would this be the only path forward for solving this problem exactly? Also, I see that the question has been marked as off-topic. Fair enough, but I really would like to solve this problem. Did it fail the lack of effort check or is it not specific enough? The problem with the latter, of course, being that I don't know which specific physics concept needs to be used to solve this! $\endgroup$ – user1832602 Apr 11 '17 at 2:44
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    $\begingroup$ Even with only two springs the problem has no analytical solution except for small displacements from equilibrium. Numerical solution is the only way you will do this. $\endgroup$ – sammy gerbil Jun 10 '18 at 21:11