What is Generalized coordinates: Definition and 37 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.
So I think the mass can only move in two "coordinates" the axis of which the mass is connected to ##k_1## and the axis connecting it to ##k_2##. Therefore, the D.O.F is 2. I don't understand what it the meaning of "variables of integration" What does it mean?
Apart from that, I attempted to...
I am looking at a textbook solution to the following problem of finding the equation of motion of a half disk. In the solution, the author considers the half disk has a COM at the black dot, and to find the instantaneous translational velocity of the center of mass (he considers rotational...
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...
Homework Statement
a: In plane polar coordinates, find the scalar product of the vector (0,1) with itself.
b: What would be the r, θ components of the unit vector in the θ direction?
Homework Equations
Scalar product of 2 vectors = AαgαβBβ
The Attempt at a Solution
For part a, I used the...
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...
When discussing about generalized coordinates, Goldstein says the following:
"All sorts of quantities may be impressed to serve as generalized coordinates. Thus, the amplitudes in a Fourier expansion of vector(rj) may be used as generalized coordinates, or we may find it convenient to employ...
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...
I have seen both rk and qj both used to represent generalized coordinates in the Lagrange equations. Are these both the same things? Does it matter which you use?
Thanks!
If suppose only if the velocities are determined for all N particles can the system be completely determined, can we not extend and say that only if acceleration for all particles are provided can the system be completely determined? For instance can there not be two systems of N particles with...
I can start explaining the problem but a more quicker way would be to open this link:
http://onlinelibrary.wiley.com/doi/10.1002/9783527627486.app2/pdf
and check the paragraph resulting in expression (B.5).
Note that I don't really care about the kinetic energy they talk about in this link...
Homework Statement
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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...
When you have a general coordinate chart on spacetime you have a lot of freedom to pick your coordinates, but you are always going to have 4 coordinates and each 4-tuple uniquely (in that chart) identifies one event in the manifold.
When you are choosing generalized coordinates for a...
Hello,
I am currently reading about the topic alluded to in the topic of this thread. In Taylor's Classical Mechanics, the author appears to be making a requirement about any arbitrary coordinate system you employ in solving some particular problem. He says,
"Instead of the Cartesian...
Homework Statement
I need some help understanding a derivation in a textbook. It involves the Lagrangian in generalized coordinates.
Homework Equations
The text states that generalized coordinates {q_1, ..., q_3N} are related to original Cartesian coordinates q_\alpha = f_\alpha(\mathbf r_1...
My question is kinda simple but it has been causing me some trouble for a while. In the problem of the pendulum rotating about an axis, why isn't the angle of rotation about the axis a generalized coordinate? The doubt appears when i try to write the hamiltonian for the system and i don't know...
I am trying to understand what generalized coordinates are but I'm having some trouble. After reading up on them a bit my best understanding of the idea of generalized coordinates is the following:
Because choice of coordinate system is arbitrary when solving physical systems (or anything for...
Hey!
I was reading Goldestein's book on classical mechanics and I came across this (Page 20 3rd Edition):
"Note that in a system of Cartesian coordinates the partial derivative of T with
respect to qj vanishes. Thus, speaking in the language of differential geometry,
this term arises...
In some texts about Lagrangian mechanics,its written that the generalized coordinates need not be length and angles(as is usual in coordinate systems)but they also can be quantities with other dimensions,say,energy,length^2 or even dimensionless.
I want to know how will be the Lagrange's...
The picture for the double pendulum I am referring to is pretty standard, wikipedia for example uses it and so does any other textbook.
I do not completely understand why one uses the second angle measured from the vertical y-axis for the second generalized coordinate. The second angle is not...
I've been wondering what the interpretation of the moment of inertia tensor in generalized coordinates is, and whether there is a way to derive it from first principles, similar to the integration we do in a Cartesian coordinate system. Specifically, I've been given the inertia matrix for a...
Hello,
I wasn't quite sure where to make this topic, so I hope I didn't do wrong by putting it here.
The question I'm having is somewhat difficult to describe and I guess it's more of a mathematical question really, but since I'm learning mechanics now and came up with it, I thought it...
Homework Statement
We two beams of timber, of identical length joined together at the middle, perpendicular forming a "X" in a sense. Underneath the end of each beam we have a spring attached, thus 4 in total. 3 have identical spring constants and the forth is greater than the other 3. We...
Homework Statement
Suppose there is a square plate, of side a and mass M, whose
corners are supported by massless springs, with spring constants K, K, K, and k <= K
(the faulty one). The springs are confined so that they stretch and compress vertically,
with unperturbed length L. The...
Are generalized coordinates, as used in Legrangian mechanics, just a different name for coordinates on a chart in a manifold? The idea of generalized coordinates never quite "clicked" with me, but after reading a paper today, it seems that they are just an implicit way of working with manifolds...
Homework Statement
When I use generalized coordinates how do I know that I can add the kinetic contributions from each to get the total kinetic energy? How do I know that you are not "counting the same KE twice"?
e.g. if you have a double pendulum how do you know that you can just add the...
This is a question that I'm asking myself for my own understanding, not a homework question.
I realize that in most derivations of the Euler-Lagrange equations the coordinate system is assumed to be general. However, just to make sure, I want to apply the "brute force" method (as Shankar...
Homework Statement
Standard double pendulum setup. A string with mass, connected to a string with a mass, mounted to the ceiling. Given is m1,m2,l1,l2
a) choose a suitable set of coordinates and write a lagrangian function, assuming it swings in a single vertical plane (I did this, using L...
I have just started to read Goldstein's classical mechanics, and he got me a bit confused: is it correct to think of polar and spherical coordinates as of generalized coordinates? the way I got it, every coordinate system different from the standard cartesian-one is a set of generalized...
Dear friends,
Well, I’ve got a problem to solve but I’m not going to ask you to do it for me. Instead, what I need is an explanation of what I am doing wrong.
The problem is as follows: we have a rod of mass m and length l hanging of a rail (don’t know how to call it). It moves as the...