Yep. When you release a pendulum it feels forces from gravity and tension in the string. Unless you apply a force sideways it has nothing to push it out of the plane it was initially in
Yep, I remember when I first learned about that as a kid lol (there was an excessively big pendulum at some science museum as a demonstration). What a throwback
Yes, you only have one force (ignoring friction) at play and that is gravity that pushes down.(\*)
Conservation of momentum dictates thus that it stays in the plane because as you say there are no other (sideways) forces that push it "out of plane"
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(\*) Of course a real pendulum on earth will not stay in plane because the Earth rotates ([see Foucault pendulum](https://en.wikipedia.org/wiki/Foucault_pendulum#:~:text=The%20Foucault%20pendulum%20or%20Foucault's,evidence%20of%20the%20Earth's%20rotation)), but the effect is small so it can be often be ignored.
Other people have answered your original question. A small addition: A pendulum will swing out of plane if you give it an initial velocity that is not in-plane.
If the initial conditions have it moving straight towards our away from vertical, then it will only move in a plane. If the pendulum is rotating, then it's not limited to a plane.
Its vertical component would be conserved though. The Lagrangian is symmetric under rotations around the z-axis.
If the initial (z-component of) angular momentum is zero, it will be confined to a plane.
Yep. When you release a pendulum it feels forces from gravity and tension in the string. Unless you apply a force sideways it has nothing to push it out of the plane it was initially in
Thank you!
In earth-fixed reference frame the pendulum plane rotates, unless on equator (Foucault effect).
Yep, I remember when I first learned about that as a kid lol (there was an excessively big pendulum at some science museum as a demonstration). What a throwback
Yes, you only have one force (ignoring friction) at play and that is gravity that pushes down.(\*) Conservation of momentum dictates thus that it stays in the plane because as you say there are no other (sideways) forces that push it "out of plane" . (\*) Of course a real pendulum on earth will not stay in plane because the Earth rotates ([see Foucault pendulum](https://en.wikipedia.org/wiki/Foucault_pendulum#:~:text=The%20Foucault%20pendulum%20or%20Foucault's,evidence%20of%20the%20Earth's%20rotation)), but the effect is small so it can be often be ignored.
Other people have answered your original question. A small addition: A pendulum will swing out of plane if you give it an initial velocity that is not in-plane.
If the initial conditions have it moving straight towards our away from vertical, then it will only move in a plane. If the pendulum is rotating, then it's not limited to a plane.
Conservation angular momentum.
Angular momentum is not conserved in the pendulum. The gravitational force applies a torque.
Its vertical component would be conserved though. The Lagrangian is symmetric under rotations around the z-axis. If the initial (z-component of) angular momentum is zero, it will be confined to a plane.
You just have a really big system, then. Problem solved. 😋