Lagrangian and degrees of freedom

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Discussion Overview

The discussion revolves around the concept of degrees of freedom in relation to the Lagrangian formalism in physics. Participants explore the definitions and implications of degrees of freedom for particles and rigid bodies, as well as the relationship between the Lagrangian and various physical theories, including quantum mechanics and general relativity.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant asserts that a particle in three-dimensional space has six degrees of freedom, comprising three coordinates and three velocity components.
  • Another participant counters that the degrees of freedom for a single particle in three-dimensional space is three, emphasizing that the Lagrangian is a function dependent on coordinates and velocities.
  • A different viewpoint states that a rigid body has six degrees of freedom, three for linear motion and three for rotational motion, while ideal particles do not possess rotational degrees of freedom due to their lack of size.
  • It is proposed that the number of Lagrange's equations corresponds to the degrees of freedom of a particle, which is three for a free particle in three-dimensional space.
  • One participant expresses understanding of the Lagrangian formalism, suggesting that a particle has six degrees of freedom when considering variations in both position and velocity.
  • A question is raised about the relationship between the Lagrangian and theories such as general relativity and quantum mechanics.
  • Another participant mentions the significance of the Lagrangian in quantum field theories and its application in explaining concepts like inflation and symmetry breaking, while expressing uncertainty about the utility of the Lagrangian in general relativity.

Areas of Agreement / Disagreement

Participants exhibit disagreement regarding the number of degrees of freedom for a particle, with some asserting three and others claiming six. The discussion remains unresolved on this point, and there are varying opinions on the relevance of the Lagrangian in different physical theories.

Contextual Notes

Participants reference the Lagrangian's dependence on coordinates and velocities, but there is no consensus on the interpretation of degrees of freedom in the context of particles versus rigid bodies. The discussion also touches on the complexity of relating the Lagrangian to advanced theories like quantum field theory and general relativity, indicating a need for foundational knowledge in these areas.

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

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