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Let’s say I have a particle, and I know all the forces acting on it at every position. (Let’s say the particle is in an electric/gravitational field to simplify the mathematics involved.) Now, is there a way to find the trajectory of the particle? (i.e. Its position at every instant in time?)

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    $\begingroup$ Do you understand calculus? And what a differential equation is? $\endgroup$
    – Ghoster
    Commented May 8 at 4:30
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    $\begingroup$ I know all the forces acting on it at every instant in time. Usually you won’t know $\mathbf F(t)$. Instead you’ll know $\mathbf F(\mathbf x, \mathbf v)$. Think about what the laws for gravitational force and electromagnetic force look like. They involve position and velocity, not time. $\endgroup$
    – Ghoster
    Commented May 8 at 5:04
  • $\begingroup$ This is a huge topic. Sometimes you can find an analytic solution. But generally you need to use a numerical method. See en.wikipedia.org/wiki/… $\endgroup$
    – PM 2Ring
    Commented May 8 at 5:05
  • $\begingroup$ $m\ddot{\mathbf x}=\mathbf F(\mathbf x, \dot{\mathbf x})$ is a second-order ordinary differential equation whose solution gives $\mathbf x(t)$. $\endgroup$
    – Ghoster
    Commented May 8 at 5:10
  • $\begingroup$ In principle you just solve a second order differential equation. In practice this can be difficult to do exactly, or even approximately. $\endgroup$
    – gandalf61
    Commented May 8 at 6:22

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Certainly. Choose a suitable co-ordinate system and write the components of acceleration along the chosen axes. Integrate the acceleration expression twice with respect to time to obtain the position function of the particle as a function of time.

It is, however, important to note that most of the times it is not so simple to obtain the "trajectory" equation as you describe it. It involves solving differential equations which can become really hard, really fast.

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    $\begingroup$ In a gravitational or electromagnetic field, the acceleration will be some known function of position and velocity, not a known function of time (despite what the question states). So you can’t just integrate it twice with respect to time. Instead you have to solve a differential equation. $\endgroup$
    – Ghoster
    Commented May 8 at 5:00
  • $\begingroup$ @Ghoster Apologies for the confusion. I did indeed mean a function of position. (Also, why is the force a function of velocity as well?) $\endgroup$ Commented May 8 at 10:19
  • $\begingroup$ @VTNaveenMugundh: The force doesn't have to be a function of velocity, but it can be (e.g., drag forces, magnetic forces), and the mathematical techniques don't change all that much if you allow for the force to depend on both position & velocity. $\endgroup$ Commented May 8 at 11:28
  • $\begingroup$ @Ghoster I don't see how that makes my answer wrong. Sure you usually have to solve a differential equation but I took the simplest case here. I don't see why my answer was downvoted. $\endgroup$
    – Nothing
    Commented May 9 at 6:12
  • $\begingroup$ I didn’t say it was actually wrong; I just thought it was incomplete. (And I still do.) But since you’re new here, and since it does at least cover a case like a uniform time-dependent electric field, where you would know $\mathbf F(t)$, so I’ll reverse my downvote. However, the system won’t let me do that until you make some minor edit. Please notify when when you do. If you decide to explain the differential equation case properly, I’ll upvote. $\endgroup$
    – Ghoster
    Commented May 9 at 6:31

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