(Lagrangian mechanics) Determining generalized coordinates/constraints.

[Lavabug]

Does anyone have any tips on how to properly determine the degrees of freedom in simple mechanical systems? I've done many problems but I often encounter a new one (or make one up myself) where I can't seem to get the proper number of generalized coordinates down right. Things like coupled beads on wires, hoops, thompson-tait pendulum...Done pretty much all the classic Lagrangian kinematics problems found in several textbooks but I know my prof is more creative than that. ;) I need something to hone my skills.

I often find 3-D problems easier than 2-D since you're given the holonomous constraints straight away(ie some relation between cylindrical coordinates or the equation of a paraboloid), so you know how many generalized coordinates you have via s = 3n -j, (n particles, j equations of constraint)

 

[LightHero]

The best way to determine the degrees of freedom is to draw a diagram of the system, and then use the Rule of thumb: 1 degree of freedom per independent moving part. In other words, for each part that can move independently of the other parts, you need one degree of freedom. This is even true for parts that are connected together, such as a chain or a rod. Even though those parts are connected, they still have the ability to move independently from one another.So if you have a system with two rods connected by a joint, that would result in three degrees of freedom (two for each rod, plus one for the joint).Once you have determined the number of degrees of freedom, you can then use this information to determine the number of generalized coordinates you need. Generally speaking, the number of generalized coordinates is equal to the number of degrees of freedom. However, if the system is constrained (such as a Thompson-Tait pendulum), the number of generalized coordinates may be less than the number of degrees of freedom. Hope this helps!