Lagrangian and degrees of freedom

[shounakbhatta]

Hello,

I have a very basic question:

Degrees of freedom for a particle describes the formal state of a physical system. Like a particle in 3 dimension space has 3 co-ordinates and if it moves in 3 velocity components, then it has 6 degrees of freedom.

Lagrangian also measures this, right?

 

[GregoryS]

Lagrangian measures nothing! It is a function which depends on s co-ordinates and s velocities (where s is degrees of freedom). And degrees of freedom for one particle moving in 3D space equals 3 (and not 6)!

 

[UltrafastPED]

You get 6 degrees of freedom for a _rigid body_: three for linear motion, and three for rotational motion.

Ideal particles don't have any size, so you cannot see them rotate!

 

[ashwin garani]

Degrees of freedom is just the number of independent quantities required to specify in a unique way the position of a particle or a system of particles.When you consider a particle 3 unique numbers are enough to specify its position assuming it is free to move in 3-D space. The number of Lagrange's equation that can be written down for a particle will be equal to the number of degrees of freedom for that particle.

 

[BruceW]

I think I understand what shounakbhatta means. In the Lagrangian formalism, a particle has 6 degrees of freedom. It has 3 for the position and 3 for the velocity. When we specify the functional derivative of the Action to equal zero, we consider variations of the Lagrangian due to purely changes in the velocity and due to purely changes in the position. So the velocity and position of the particle are treated as separate degrees of freedom when we are varying the Lagrangian.

 

[shounakbhatta]

Thank you BruceW, I got it. Is Lagrangian in someway related to General Relativity or Quantum mechanics?

 

[BruceW]

they absolutely love the Lagrangian in quantum field theories. I think there is a Lagrangian for General relativity. But I don't think it is very useful. Much more useful is when quantum field theory is introduced as part of our universe. This is how inflation and symmetry breaking e.t.c. are explained. Look up Andrei Linde "particle physics and inflationary cosmology", pretty interesting. It does assume some knowledge of quantum field theory and general relativity though. I am not able to understand all of it.