Lagrangian Mechanics: Degree of Freedom & Constraints

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In summary, Degree of freedom is the number of independent coordinates used to describe the state of a system. For example, a particle in three dimensions has 3 degrees of freedom, while two particles in three dimensions have 6 degrees of freedom. Constraints, on the other hand, are equations that describe relationships between the coordinates. In Lagrangian mechanics, the basic constraint is usually an equation that relates the coordinates. For a particle constrained to move on a shell of a sphere, there are only two degrees of freedom. In the case of five particles moving freely in a plane, there are 15 degrees of freedom, as each particle has 3 degrees of freedom.
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roshan2004
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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.
 
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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.
 
  • #3
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
 
  • #4
Can I have some help here?
 
  • #5
you are right. If there are 5 particles independent of each other. degrees of freedom=3*5=15
 

1. What is the purpose of Lagrangian mechanics?

Lagrangian mechanics is a mathematical framework used to describe the motion of systems of particles. It provides a more efficient and elegant approach to solving mechanical problems compared to traditional Newtonian mechanics.

2. What is the meaning of "degree of freedom" in Lagrangian mechanics?

Degree of freedom refers to the number of independent parameters needed to describe the configuration of a system. In Lagrangian mechanics, it is the number of variables required to fully specify the position and orientation of all particles in a system.

3. How are constraints handled in Lagrangian mechanics?

Constraints in a system are represented by equations that restrict the motion of particles. In Lagrangian mechanics, these constraints are incorporated into the equations of motion using Lagrange multipliers.

4. What are holonomic and non-holonomic constraints in Lagrangian mechanics?

Holonomic constraints are those that can be expressed as equations involving only the coordinates and time. Non-holonomic constraints, on the other hand, involve the velocities or higher derivatives of coordinates and cannot be expressed as simple equations.

5. How does Lagrangian mechanics differ from Hamiltonian mechanics?

Lagrangian and Hamiltonian mechanics are two equivalent approaches to solving mechanical problems. In Lagrangian mechanics, the equations of motion are derived from the system's kinetic and potential energies, while in Hamiltonian mechanics, they are derived from the system's total energy.

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