Degrees of Freedom total confusion

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Degrees of Freedom total confusion
What are Degrees of Freedom in the Lagrangian/Hamiltonian formulation of classical mechanics?

I've been getting very confused trying to understand this concept.

Would six degrees of freedom
mean up, down, left, right, back & forward?

I've seen it said that 3 degrees of freedom mean the x, y & z axes, so my above sentence is really only 3 degrees of freedom, is it?

Then, I seen the Susskind classical mechanics lecture claim there are 6 degrees of freedom, x, y & z axes & then position, velocity & acceleration.

But, then I read in a physicsforums.com post the claim that velocity & acceleration are not dimensions ergo they are not degrees of freedom!

Don't degrees of freedom have to be independent of your choice of coordinates? I mean, isn't that one of the fundamental advantages to the use of the variational calculus in describing physical systems?

The wikipedia article is not in the least helpful & I've tried to watch the http://www.youtube.com/watch?v=8X1x9RLaaxc nptelhrd lectures to only get more confused.

What does it mean to have a degree of freedom constricted also?

Thanks it would be really helpful for somebody to shed some light on this idea for me.

antiselfdual

Landau and Lif****z wrote:

The number of independent quantities which must be specified in order to uniquely define the position of any system is called the numbers of degrees of freedom.

So single body moving freely in 3-dimensional space has three degrees of freedom.

What you might be getting mixed up with is that you can view a problem with 3 degrees of freedom as a 6 dimensional problem, because you have to solve for the 3 velocity components and 3 position components (or in Hamilonian mechanics you have the momenta and position vectors as your independent coodinates instead of velocity and position).

Podge_irl

Degrees of freedom essentially correspond to coordinates in configuration space. To entirely define a particle you need to know both it's position and it's velocity. You have one degree of freedom for a particles position in a particular dimension and one for its velocity in that dimension. Two particles may occupy the same position and be travelling at different speeds after all.

The configuration space (or phase space) of a particle is twice the size of the physical space it is moving in as you need to also define the velocity at which it is travelling (interesting, this is why phase spaces are symplectic spaces, but that is a rather different discussion).

antiselfdual wrote: »

So single body moving freely in 3-dimensional space has three degrees of freedom.

This is possibly obscuring the issue. A single body moving freely has three degrees of freedom (and one parameter). A freely moving particle is a particle experiencing no external forces. However, in a physical system with forces you will have to account for both position and velocity so your phase space will be 2n dimensional (n=3 in this scenario).

antiselfdual

Hmm okay well it seems from a cursory glance at the wikipedia articles that you can define degrees of freedom to be either physical degrees of freedom (3N for a system of N particles, this is what Landau and Goldstein say too), or else phase space dimension, in which case it would be 6N for N particles. Weirdly the wiki articles are for the two separate cases of "Physics" and "Mechanics."

Podge_irl

Saw that, don't understand it. I would always think of degrees of freedom as the latter.

enda1

From the engineering sense, a body has 6 degrees of freedom, 3 displacements and three rotations.

Don't know if this helps to you mathsy guys though