# Motion with constraints

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The equation of motion for a pendulum is y''=-(g/L)sin(y), where y represents the angle of the stiff rod with respect to the vertical, g is the acceleration of gravity, L is the length of the rod. y also varies with respect to time. I've seen how this particular equation is derived, but I would like to know more about the general problem of modeling mechanical systems with constraints. Does anyone have some insight to offer? How could I view this problem from the mechanical perspective (finding positions, energies, momentums, etc.).

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One of the tricks is to set up your coordinate system to coincide with the constraint. The pendulum problem is solved by using spherical coordinates in 2-D; because r is fixed, the only variable is now the angle. That simplifies the problem. I imagine engineering texts do this a lot more than physics.

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It's not always as simple as a pendulum, though. How could I analyze a more complicated constraint?

An example I can think of is motion along a fixed path under the influence of 3 charges (kind of like the 3-body problem).

Is there a general method that could work for analyzing all sorts of constraints?

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It's not always as simple as a pendulum, though. How could I analyze a more complicated constraint?

An example I can think of is motion along a fixed path under the influence of 3 charges (kind of like the 3-body problem).

Is there a general method that could work for analyzing all sorts of constraints?

Look at Lagrangian and Hamiltonian mechanics in any book with a title like "Classical Mechanics" or Classical Dynamics". The books by Goldstein or Marion would do nicely.

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