Generalized coordinates Definition and 11 Discussions
In analytical mechanics, the term generalized coordinates refers to the parameters that describe the configuration of the system relative to some reference configuration. These parameters must uniquely define the configuration of the system relative to the reference configuration. This is done assuming that this can be done with a single chart. The generalized velocities are the time derivatives of the generalized coordinates of the system.
An example of a generalized coordinate is the angle that locates a point moving on a circle. The adjective "generalized" distinguishes these parameters from the traditional use of the term coordinate to refer to Cartesian coordinates: for example, describing the location of the point on the circle using x and y coordinates.
Although there may be many choices for generalized coordinates for a physical system, parameters that are convenient are usually selected for the specification of the configuration of the system and which make the solution of its equations of motion easier. If these parameters are independent of one another, the number of independent generalized coordinates is defined by the number of degrees of freedom of the system.Generalized coordinates are paired with generalized momenta to provide canonical coordinates on phase space.
In case of P holonomic constraints and N particles, I have 3N-P degrees of freedom and I have to look for 3N-P generalized coordinates if I want them to vary independently, but what about non-holonomic constraints? I know if I have N particles and P non-holonomic constraints, I still need 3N...
It's a topic that's been giving be a headache for some time. I'm not sure if/why/whether I can always consider velocities and (independent) coordinates to be independent, whether in case of cartesian coordinates and velocities or generalized coordinates and velocities.
Hello all, so I’ve been reading Jennifer Coopersmith’s The Lazy Universe: An Introduction to the Principle of Least Action, and on page 72 it says:
If I understand it right, she’s saying that in our Euler-Lagrange equation ## \frac {\partial L} {\partial q} - \frac {d} {dt} \frac {\partial L}...
Hello,
reading the wiki entry for Langevin observers on rotating disk - Born_coordinates I'm struggling with the following quoted sentence: But as we see from Fig. 1, ideal clocks carried by these ring-riding observers cannot be synchronized.
I do not grasp why, starting from the figure...
From "A Student's Guide to Langrangins and Hamiltonians", Patrick Hamill, Cambridge, 2017 edition.
Apologies: since I do not know how to put dots above a variable in this box, I will put the dots as superscripts. Similarly for the limits in a sum.
On page 6,
"we denote the coordinates by qi...
The form of the Lagrangian is: L = K - U
When cast in terms of generalized coordinates, the kinetic energy (K) can be a function of the rates of generalized coordinates AND the coordinates themselves (velocity and position); a case would be a double pendulum.
However, the potential energy (U)...
Yes, that is a serious title for the thread.
Could someone please define GENERALIZED COORDINATES?
In other words (and with a thread title like that, I damn well better be sure there are other words )
I understand variational methods, Lagrange, Hamilton, (and all that).
I understand the...
Homework Statement
[/B]
A uniform solid ball of mass m rolls without slipping down a right angled wedge of mass M and angle θ from the horizontal, which itself can slide without friction on a horizontal floor. Find the acceleration of the ball relative to the wedge.
2. The attempt at a...
Hi! I've been trying to find the equation of motion for the simple pendulum using x as the generalized coordinate (instead of the angle), but I haven't been able to get the right solution...
Homework Statement
The data is as usual, mass m, length l and gravity g. The X,Y axes origin can be...