How are constraint forces represented in Lagrangian mechanics?

Question

Suppose we try to obtain the movement equation for a particle sliding on a sphere (no friction, ideal bodies...). The only forces acting on the particle are its weight and - here's my problem - a force that keeps the particle attached to the sphere*. How I am supposed to represent mathematically this kind of forces? As a restriction on the coordinates of the particle?

As a side note: My teacher refers to this, in Spanish, as a "fuerza de ligadura". I don't have a clue on its correct translation; does ligature force make any sense?

Answer 1

Constraints are handled in Lagranian mechanics through either of two approaches:

1) The constraint equation is used to reduce the degrees of freedom of the system. For example, if a particle is constrained to the surface of a sphere, then the Lagrangian can be written entirely in terms of two generalized coordinates and their associated momenta (typically, one chooses the polar and azimuthal angles of spherical coordinates to be the generalized coordinates). One cannot find the constraint forces this way--they are tacitly ignored as a result of reducing the complexity of the problem.

2) Alternatively, one adds a Lagrange multiplier as an extra degree of freedom. If one has a Lagrangian $L$ and constraint function $f(q_i, t)$ that is constrained to some value $C$ and is a function of the generalized coordinates and time, then one modifies the Lagrangian as follows:

$$L \mapsto L' = L + \lambda [f(q_i, t) - C]$$

Applying the Euler-Lagrange equations to this new Lagrangian $L'$ reproduces all the relevant dynamics of the system. In particular, it allows you to solve for the Lagrange multiplier $\lambda$. Doing so makes it possible to find the constraint forces, which are

$$F_i = \lambda \frac{\partial f}{\partial q_i}$$

It's important to note that such a constraint function $f$ must be a function of only time and the generalized coordinates. The term for this kind of constraint is holonomic. Problems with non-holonomic constraints cannot be treated in this way.

Answer 2

Yes, as a restriction on the particle coordinates, like $r = const$. It will give you two equations for two angles $\theta$ and $\phi$, with no equation for the radius $r$.

If you like, the restriction $r = const$ is a solution for such a radial force that manages to keep always your particle on the sphere.

Answer 3

The usual English name is constraint forces. The usual way of dealing with them is by not dealing with them, i.e. imposing restrictions to the coordinates, as you mention, and projecting external forces onto the movement path. Once you have solved for the motion, you can figure out the constraint forces from the acceleration.

In Lagrangian mechanics, the equations of motion are written based on generalized coordinates, which should be chosen to take those constraints into account.

Answer 4

In the Lagrangian formulation, it is usually called constraint force. Constraint force is a general terms that apply on a particle to constraint its motion on a particular trajectory.

In you case, there are two forces acting on the particle: Weight of the particle which is pointing down, and a radial normal force pushing outward on the particle. The resulting of these two forces are the constraint force causing the particle moving along the surface.

Mathematically, you just need to add these two forces vector together to obtain the constraint force. But you should notice that weight is constant, but normal force is not. When the normal force is too weak, the ball will leave the surface and the constraint force is 0.

Answer 5

They are called constraint forces in English. To constrain is synonymous with restrict.

Answer 6

In a sphere, the restriction equation $\phi$ is $R-r=0$, that means that at all times the spherical coordinate $r$ will have a constant value $R$. (or any other letter)