Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: Eternal Student on 01/03/2024 17:00:25
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Hi.
Consider a Newtonian system of some particles moving in space. Let's use conventional Cartesian co-ordinates to describe locations of the particles and there will also be some velocity for each particle, both of these things (position and velocity) are functions of time.
Is "the configuration space" for the system just the set of possible co-ordinates for the system OR
the set of positions AND velocities (or momenta if you'd rather use that)?
Let's say we have 1 particle in the system. Is "configuration space" R3 = just the possible positions
OR is the configuration space R6 = possible positions and possible velocities.
Wikipedia seems to be suggesting it is just the space of possible co-ordinates:
In classical mechanics, the parameters that define the configuration of a system are called generalized coordinates, and the space defined by these coordinates is called the configuration space of the physical system.
.... ... Example: a particle in 3D space
The position of a single particle moving in ordinary Euclidean 3-space is defined by the vector q = ( x , y , z ) , and therefore its configuration space is Q = R3
See https://en.wikipedia.org/wiki/Configuration_space_(physics) for more info.
I was fairly sure that "configuration space" included both positions and velocities of the particles - but maybe I have just mis-remembered this.
---- End of the main question --- a more advanced or developed question follows -----
Now, let's take a more specific case: Suppose that the state of a system cannot be described adequately just by knowing the positions of the particles. For example, suppose there is some force acting on a particle that isn't just a function of the positions of particles but it is also dependant on the velocities of the particles. [ A fully concrete example: There is a magnetic field in our system in addition to some electrostatic forces and the particles in our system are charged - so the force acting on a particle depends on both its velocity through a magnetic field and also on its position in an electric field ]. Do we now declare that since we must know both positions and velocities of the particles to ascertain the state of the system, the configuration space is now R6n (with n = number of particles) instead of R3n , i.e. that velocities do have to be considered ?
I don't know..... maybe we still don't include the velocities as part of the generalised co-ordinates we would require because, for example, provided we have the position of each particle at each moment of time, then we can derive the velocity anyway just from this. Although if that's what we would assume, then "the configuration space" is a bit more than just a description of all possible generalised co-ordinates - it is no longer enough to know a particle was at (1, 1, 1) and also (2, 2, 2) we must know what times it was actually at these places if we are to derive the velocity.
Just to be clear I need to know what the official definition of "configuration space" is supposed to be. Is it always just the set of possible generalised co-ordinates and never something that includes the set of generalised momenta (or velocities)? Are "the generalised co-ordinates" something which we assess on a sliding scale - it must be possible to describe the state of the system using only the generalised co-ordinates, so that if we HAD to know something about velocity then velocity would have to be included as a generalised co-ordinate?
The Wikipedia definition of "generalised co-ordinate" is less than usefull (to me):
In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.
[ From: https://en.wikipedia.org/wiki/Generalized_coordinates ]
The first sentence makes for a circular or useless definition, we must already know what our configuration space is in order to know what our generalised co-ordinates should be. The second sentence, however, does seem to suggest that our generalised co-ordinates must be sufficient to uniquely define the state of our system.
Best Wishes.
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Under "classical" rules you can know a configuration absolutely at any point in time, but (a) Heisenberg sensibly points out that you can't, (b) if you don't know the velocities of 3 particles in space, you can't predict anything about them and (c) even if you did and there was nothing else in the space, the 3-body problem still defies an analytic solution.
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Hi.
I've probably had enough time to answer my own question.
The formal definition of "configuration" for a system where only classical mechanics will be used is just the position of every particle in the system. It doesn't seem to matter in the slightest if the system is fully described just by positions or not.
So the "configuration space" is just the space of all possible positions of the particles.
The forum was still useful. Just trying to write out the question or problem precisely actually helps to formulate what the resolutions may be. So thank you to everyone.
Definition 2.2.1. The set of all possible (in principle) instantaneous configurations for a
given physical system is known as configuration space. We will denote it by C.
It is important to note that this includes positions, but it does not include velocities.
One informal way of thinking about configuration space is as the space of all distinct
photographs one can take of the system, at least in principle.
[Taken from some lecture notes for Durham University: https://www.maths.dur.ac.uk/users/inaki.garcia-etxebarria/MPII/ch/2.2a.pdf ]
The article does go on to emphasise that configuration and configuration space is not sufficent to describe the dynamics of most systems and therefore is woefully inadequate to fully describe what I would have considered "the state" of the system. Thus, making the definition of "generalised co-ordinates" given in Wikipedia essentially misleading.
From WIki (and given in the earlier post):
In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space.
If you took away the last few words in red, the definition is just plain wrong. Knowing the generalised co-ordinates for a system is simply inadequate to know the state of the system. If you leave the words in red, then it's unhelpful because you have only a circular definition since the term "configuration space" is only defined by reference to generalised co-ordinates in Wiki.
Best Wishes.