How Do Holonomic Constraints Affect Degrees of Freedom in Lagrangian Mechanics?

[aaaa202]

I'm starting on lagrangian mechanics and is a little puzzled by the use of generalized coordinates. Shortly, what is a degree of freedom?
And what I find harder to understand, why is it that a holonomic constraint allows you to remove a degree of freedom? Consider for instance two particles between which the distance is fixed. This gives 5 degrees of freedom, at least so I heard. Because that is kind of weird to me. As far as I see it the particles can still move anywhere on the x,y and z axis can't they? I can see that in terms of rotations you can only make 2 different ones, and then you can translate the two particles in 3 different directions. But when is it that rotations comes into the picture, because for a collection of N particles you would just have 3N dof, which correspond to movement in three different directions in a euclidean coordinate system.

Talking about the problem with 2 particles with constant distance between them, is it then possible directly, mathematically from the constrain r=c to show that only 5 dof are needed? And can anyone do it?

 

[Aimless]

An easy way to see that this particular picture has five degrees of freedom is to view things from the perspective of the midpoint between the two particles (the center of mass if we assume the particles are equal mass).

In an unconstrained system, you can write the six total degrees of freedom as the position $(x,y,z)$ of the center point, and the length and orientation $(l , \theta , \phi )$ of the line between the two particles. This description is isomorphic to simply describing the system by the position of the two particles $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$, and it is obvious that they both have the same number of degrees of freedom (6).

However, now let's add back in the constraint. If we define the constraint based on the position of the two particles, the constraint equation is $\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2 + (z_1-z_2)^2} = L$, where $L$ is the constant separation between the two particles. To see how this constraint reduces the number of degrees of freedom of the system, you can solve for one of the coordinates in terms of the other five; this constrains the system to a given five-dimensional hypersurface within the six-dimensional space defined by $(x_1,y_1,z_1,x_2,y_2,z_2)$. However, exactly what that surface is can be a little hard to see.

To make it clearer, let's switch back to the other coordinate system, $(x,y,z,l , \theta , \phi )$. Here, the constraint is much easier to visualize: $l=L$, so our hypersurface is given by $(x,y,z,L , \theta , \phi )$. There are five independent coordinates which describe this hypersurface, $(x,y,z, \theta , \phi )$, so therefore there are five degrees of freedom.

At the end of the day, that's what we mean by degrees of freedom. If you define the phase space of your system (that is, the space of allowed parameters) including your constraints, then the degrees of freedom of the system are the number of independent basis vectors which span the constraint surface within the phase space - which is a fancy way of saying "the number of independent ways in which the system can evolve."

 

[voko]

If you define the phase space of your system (that is, the space of allowed parameters)

This is not the phase space. This is the configuration space. The phase space includes (generalized) momenta in addition to coordinates.

 

[Aimless]

This is not the phase space. This is the configuration space. The phase space includes (generalized) momenta in addition to coordinates.

Right. I should be more careful with terminology. Oops.

That said, the statement still applies.

 

[PJC01]

Degrees of freedom refer to the number of independent variables that are required to fully describe the state of a physical system. In other words, it is the minimum number of coordinates or parameters needed to specify the position and orientation of a system at a given time.

In the context of Lagrangian mechanics, generalized coordinates are used to describe the position and orientation of a system. These coordinates are chosen in such a way that they are independent of each other and are easier to work with mathematically. The number of generalized coordinates needed to describe a system is equal to the number of degrees of freedom.

In the example of two particles with a fixed distance between them, there are indeed 5 degrees of freedom. This is because the distance between the particles can be described by a single coordinate, while the positions of the two particles themselves require 3 coordinates each. This results in a total of 5 degrees of freedom.

A holonomic constraint is a mathematical relationship between the coordinates of a system that restricts its motion. In this case, the constraint r=c (where r is the distance between the particles and c is a constant) means that the particles must always maintain a fixed distance between them. This constraint reduces the number of independent coordinates needed to describe the system from 6 (3 for each particle) to 5, as one coordinate is now dependent on the other.

To directly show mathematically that only 5 degrees of freedom are needed, one can use the Lagrange multiplier method to incorporate the constraint into the equations of motion. This results in a set of equations with only 5 independent variables, further showing that 5 degrees of freedom are sufficient to describe the system.

In summary, degrees of freedom represent the minimum number of independent coordinates needed to describe the state of a physical system. A holonomic constraint reduces the number of degrees of freedom by imposing a mathematical relationship between the coordinates. And in the case of two particles with a fixed distance between them, there are 5 degrees of freedom, with one coordinate being dependent on the other due to the constraint.