Lagrangian Mechanics: Degree of Freedom & Constraints

[roshan2004]

I am self trying to understand Lagrangian mechanics and I have come across with Degree of freedom and constraints which I think I understood in bits. So please try to explain these terms to me. I use Goldstein's Classical Mechanics.

 

[elibj123]

Degree of freedom is the number of independent coordinates with which you can entirely describe the state of a system.

For example a particle in three dimensions has 3 degrees of freedom. He can be described either by Cartesian coordinates x,y,z or Spherical coordinates r,theta,phi but you will always need three numbers to describe its location.
Two particles in three dimensions have 6 degrees of freedom (3 coordinates per each)

A particle that is constrained to move on a shell of a sphere has only two degrees of freedom. One way to see it is that in Polar coordinates you have r set to a fixed number, and you only have "control" over the angles. Another way to see it is to use Cartesian coordinates x,y,z. Apparently you have 3 coordinates, but you have to include the equation x^2+y^2+z^2=R^2. So if you know 2 coordinates, you'll immediately know the third one. Meaning, you still have 2 degrees of freedom.

A constraint is an equation (or an inequality) that describes a relation between the coordinates, usually of the form F(q1,q2,q3,...)=0.

Notice that this a constraint on the coordinates, and that this is an equation. This is the basic constraint you're dealing with in the Lagrangian formulation, and that usually the constraint that you are referring to when talking about degrees of freedom.

For example another constraint might be x^2+y^2+z^2<=R^2 for a single particle. The particle is trapped inside a sphere, but you still require 3 coordinates to describe him.

 

[roshan2004]

So, if there are five particles moving freely in a plane, do we have to consider constraints there or we can simply write the number of degree of freedoms as 3*5=15

 

[roshan2004]

Can I have some help here?

 

[somitra]

you are right. If there are 5 particles independent of each other. degrees of freedom=3*5=15