iMechanica - Comments for "a point and a particle"
https://imechanica.org/node/5214
Comments for "a point and a particle"enContinuum and Particles
https://imechanica.org/comment/13352#comment-13352
<a id="comment-13352"></a>
<p><em>In reply to <a href="https://imechanica.org/node/5214">a point and a particle</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Dear Mr Rui Huang
</p>
<p>
In point (2) of your blog you have defined a discrete system with a finite volume and a finite mass. I prefer that you call that a Differential Volume dV (or Elementary Volume) and a Differential Mass dm (or Elementary Mass). The Continuous Domain can be obtained from that by Exact Integration or Numerically by Discretization of the Continuous Domain.
</p>
<p>
Mohammed lamine MOUSSAOUI
</p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Sun, 24 Jan 2010 10:32:09 +0000mohammedlaminecomment 13352 at https://imechanica.orgThe Particle and its Properties
https://imechanica.org/comment/13098#comment-13098
<a id="comment-13098"></a>
<p><em>In reply to <a href="https://imechanica.org/node/5214">a point and a particle</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
The density of a particle is one of its properties which defines its mass for a unitary volume. If we need to integrate in the continum domain we can use the definition differential volume dV. Like in fluid mechanics these properties can be applied to solid mechanics. The kinetic energy can then be defined by (1/2)rho.(dot V)^2.
</p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Sat, 19 Dec 2009 14:25:43 +0000mohammedlaminecomment 13098 at https://imechanica.orgThe Particle and its Properties
https://imechanica.org/comment/13097#comment-13097
<a id="comment-13097"></a>
<p><em>In reply to <a href="https://imechanica.org/node/5214">a point and a particle</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
The density of a particle is one of its properties which defines its mass for a unitary volume. If we need to integrate in the continum domain we can use the definition differential volume dV. Like in fluid mechanics these properties can be applied to solid mechanics. The kinetic energy can then be defined by (1/2)rho.(dot V)^2.
</p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Sat, 19 Dec 2009 14:25:41 +0000mohammedlaminecomment 13097 at https://imechanica.orgDear Venkatesan,
https://imechanica.org/comment/10523#comment-10523
<a id="comment-10523"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/10512#comment-10512">Reply to Xu on Delta function:</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
<a href="http://www.imechanica.org/user/1362" title="View user profile.">Dear Venkatesan,</a>
</p>
<p>
From my understanding, the domain of δ(x) is the test function space. The range of
</p>
<p>
δ(x) is always the real line. It is not "dependent on the function it is associated with".
</p>
<p>
Engineers invent δ(x) and mathematicians established its soild mathematical foundation.
</p>
<p>
As I said before, we can have our own understnding of δ(x), but the key point is to use it properly to get right results.
</p>
<p>
Thank you very much for your comments.
</p>
<p>
best regards
</p>
<p>
Xu
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Tue, 14 Apr 2009 07:00:09 +0000Xu Guocomment 10523 at https://imechanica.orgReply to Xu on Delta function:
https://imechanica.org/comment/10512#comment-10512
<a id="comment-10512"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/10500#comment-10500">δ(x) as a functional </a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Dear Xu
</p>
<p>
Thanks for the comments.
</p>
<p>
I wanted to mention only that δ(x) is defined on the domain of x but the range may be dependent on the function it is associated with.
</p>
<p>
you said: <<it is not the mathematical EXPRESSION of a "force function" as you mentioned.>>
</p>
<p>
But I never said that. I only indicated where we associate a delta function to activate the effect of load, so delta function acts like a "on/off" switch. The force function (that I mentioned) is only relevant to that problem, where the product δ(x-ct)P is considered as a force function.
</p>
<p>
The functionals you are talking about comes when delta acts on the function, like δ(y(x)), I believe, and thats when the domain changes to the function space (from the usual x). While I understand that the domain here becomes the function space, I don't know what it's range would be in, (a functional space?)
</p>
<p>
Vgn
</p>
<p>
Graduate Student
</p>
<p>
University of Oklahoma
</p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Mon, 13 Apr 2009 19:02:03 +0000Gopinath Venkatesancomment 10512 at https://imechanica.orgsupport of Dirac's delta
https://imechanica.org/comment/10507#comment-10507
<a id="comment-10507"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/10501#comment-10501">But "function spaces" doesn't really address what I asked</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>Dear Ajit:</p>
<p>I don't see any ambiguity in the definition of Dirac's delta. One first defines it when the support is the origin and then the support can be shifted to any other point in the real line (or R^n for higher dimensional problems). Shifting can be defined for any distribution. Dirac's delta would not be completely specified unless you are given its support, which is a single point. So, one would say "Dirac's delta supported at x = 0" or "Dirac's delta supported at x = a" (these are two different distributions). Support of a distribution is the closure of the set of points (in the standard topology of R^n) for which the distribution is not the "zero distribution", i.e. for those points there is always a nonzero test function on which the distribution is nonzero.</p>
<p>Regards,<br />
Arash </p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Mon, 13 Apr 2009 16:17:08 +0000arash_yavaricomment 10507 at https://imechanica.orgDear Ajit,
https://imechanica.org/comment/10503#comment-10503
<a id="comment-10503"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/10501#comment-10501">But "function spaces" doesn't really address what I asked</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Dear Ajit,
</p>
<p>
I suppose that for mathematicians, when they talk about the " Dirac's delta <em>defined</em> at
</p>
<p>
x= a", maybe they will use the term δa(x) or δ(x-a) to communicate the meaning that
</p>
<p>
a linear continuous functional which returns the value of the test function<strong> at x=a. </strong>Here<strong> a </strong>is
</p>
<p>
a parameter used to identify a specific \delta function. <img src="http://www.imechanica.org/modules/tinymce/includes/jscripts/tiny_mce/plugins/emotions/images/smiley-smile.gif" border="0" alt="Smile" title="Smile" /></p>
<p>
</p>
<p>
Anyway, put the mathematics aside, as mechanicians, we can have our own language.
</p>
<p>
The key point is to use \delta function properly to get right results.
</p>
<p>
</p>
<p>
best regards
</p>
<p>
Xu
</p>
<p>
</p>
<p>
</p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Mon, 13 Apr 2009 12:03:22 +0000Xu Guocomment 10503 at https://imechanica.orgBut "function spaces" doesn't really address what I asked
https://imechanica.org/comment/10501#comment-10501
<a id="comment-10501"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/10493#comment-10493">my understanding</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Xu,
</p>
<p>
Thanks for your reply.
</p>
<p>
I am not sure about the right mathematical jargon/language. But yes, I can see that as a functional, Dirac's delta would have to take as its input a set of functions and produce as its outputs a set of corresponding numbers. Now, referring to the <a href="http://mathworld.wolfram.com/DeltaFunction.html" target="_blank">MathWorld web page</a>, I gather that the input set consists of those test functions, and so, the output numbers are going to be nothing but the values of those test functions at x = 0. This, essentially, is what you say, and there is no question of disagreement about it.
</p>
<p>
But all of it still does not address the basic issue that I had raised.
</p>
<p>
Another way to state that basic issue is to scroll down the MathWorld page a little bit and make a reference to eqs. (2), (3) etc. in that page. The eq. (2), in particular, gives the property of Dirac's delta involving a certain "a".
</p>
<p>
Now my question is: In reference to eq. (2), how do we capture the fact that the vertical line for Dirac's delta is going to be erected at x = a but not at x = 0 or at any other point? How do we communicate this basic fact?
</p>
<p>
The short and sweet way to communicate it would be to say that Dirac's delta is <em>defined</em> at x = a but not at x = 0. Now, what harm is there with <em>that</em>? That is the basic question I have. As I indicated above, personally, I can see no harm at all because the fact that Dirac's delta is not an ordinary function is, already, a part of the context, and so, it need not come in the way of saying that the delta is defined at x = a.
</p>
<p>
And, if mathematicians, when they talk to each other, do not say that Dirac's delta is <em>defined</em> at x = a, then how do they tell each other the distinction of "a" from all other points? Or is it that they always remain flying high up in the abstraction of linear functionals/function spaces and never come down to that domain in which x = a is defined? That too is a secondary question I have as an engineer.
</p>
<p>
Thanks in advance for clarifying these specific matters.
</p>
<p>
</p>
<p>
- - - - - <br />
Even as you read this, I remain jobless (as I have, for years)
</p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Mon, 13 Apr 2009 07:31:57 +0000Ajit R. Jadhavcomment 10501 at https://imechanica.orgδ(x) as a functional
https://imechanica.org/comment/10500#comment-10500
<a id="comment-10500"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/10498#comment-10498">Dirac Delta association with the domain of x:</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Venkatesan,
</p>
<p>
In mechanics, δ(x) is usually used to REPRESENT a unit concentrated load applied at x=0. it is not the mathematical EXPRESSION of a "force function" as you mentioned.It is a functional defined on the test function space. It is not a "function" in the ordinary sense.
</p>
<p>
When the external load is not smooth or regular enough (for example, concentrated load), the classical solution of the correesponding PDE does not exist. We can only talk about the generalized solution of the PDE. In this case, the weak form of the considered PDE is often used. In term of mechanics, this is just the "virtual work" form of the PDE.
</p>
<p>
The right hand side of this "virtual work" form, is the virtual work done by the applied
</p>
<p>
foce on the virtual displacement (test function). Then for the given external load, for every virtual displacement (test function), there is a real number (work) associated with it. From
</p>
<p>
this sense, we can take the "external load" as a functional. δ(x) is just one of such functional, which gives 1*u(0) for the test function u(x) defined in some suitable function
</p>
<p>
spaces.
</p>
<p>
δ(x) is often described as a "function" such that δ(x)= inifinity at x=0 and δ(x)=0 elsewhere. From my unstanding, this is just to give us some "feelings" of this mathematical
</p>
<p>
object, not its "expression". Some of the properties of δ(x) cannot be understood if we viewed it as a ordinary function.
</p>
<p>
Of course, δ(x) also has some relationship with the ordinary functions. Since it can be
</p>
<p>
viewed as a fucntional which is the weak* limit of some functionals generated by ordinary functions. These functions are often pointwise converged to a "function" such that
</p>
<p>
"=" infinity at x=0; =0 otherwise.
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Mon, 13 Apr 2009 06:39:20 +0000Xu Guocomment 10500 at https://imechanica.orgDirac Delta association with the domain of x:
https://imechanica.org/comment/10498#comment-10498
<a id="comment-10498"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/10493#comment-10493">my understanding</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Xu,
</p>
<p>
I thought the δ(x) function is defined in the domain of x. For example, the δ(x) is used in the definition of moving loads problem, where the moving load P acts on the bridge structures. So to capture the different points of contact due to moving loads, the delta function is defined as δ(x-ct), and the force function accordingly as δ(x-ct) times P, where c is the speed of the moving load, and t is the time, so ct defines the point of action of load P.
</p>
<p>
Vgn
</p>
<p>
Graduate Student
</p>
<p>
University of Oklahoma
</p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Sun, 12 Apr 2009 21:18:55 +0000Gopinath Venkatesancomment 10498 at https://imechanica.orgRich, Good that you find
https://imechanica.org/comment/10495#comment-10495
<a id="comment-10495"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/10494#comment-10494">Ajit,
OK, You make an</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Rich,
</p>
<p>
Good that you find it all interesting.
</p>
<p>
But I am not sure if even mathematics allows you what you say it does.
</p>
<p>
Do you have any evidence or any support issuing forth from the science of mathematics (let alone of mechanics) for saying that you can begin with a volume of zero measure and still be able to integrate around its faces (each, obviously, of a zero measure, too), rather than begin with a finite volume and then approach the zero size for it via an appropriate limiting process?
</p>
<p>
I think not. Newton himself was at pains in emphasizing precisely this point (even though calculus was too new to be communicated very effectively even by he himself). And, no functional analysits or measure theorist has been able to override the basic considerations that were known to Newton himself.
</p>
<p>
Simply put, it's best to remember it all this way, with twin-points:
</p>
<p>
(1) All continuum theoretical definitions ultimately make reference to a differential element. [BTW, it was precisely this point which I had pointed out right in <a href="http://www.imechanica.org/node/3570#comment-8357" target="_blank"> my very first comment w.r.t. Falk Koenemann's theory</a>---right last year. It's been satisfying to note that many / all of those points have later been noted by other iMechanicians, this year.] This includes definitions of "scalars" like pressure or temperature too. (Not even a temperature is a point phenomenon---it, too, requires a differential element for its definition.)
</p>
<p>
(2) As Newton himself had emphasized (and all mathematicians since then have), a differential element has infinitesimal size, i.e. a size that is, emphatically, not zero even though it can be made as small as you would like it to be (simply because it's being used in a limiting process). Both epsilon and delta in the famous epsilon-delta way of putting it are <em>non</em>-zero in size.
</p>
<p>
Conclusion: <em>Mathematically</em> as well as mechanically what you say cannot be done. If you have other evidence, I would like to know of it.
</p>
<p>
-----
</p>
<p>
If the reader would permit me a bit of a relevant aside: The Objectivist philosopher <a href="http://www.hblist.com" target="_blank">Harry Binswanger</a> has argued, perhaps based on certain ancient Greeks' ideas, that even something as basic and simple as motion itself cannot be described as something that happens <em>at</em> a point; it can only be described in reference to the ever smaller measurements of length (dl) and duration (dt) neither of which can at all be made zero, <em>in principle</em>. It was the later rationalist tradition which popularized the idea of something happening at a point of space even if the point itself was, as everyone knows, left undefined in Euclid's original texts.
</p>
<p>
In other words, zero cannot be the basis of even a mathematical definition of motion. (I keep it in mind with a very meaningful pun (self-made) about it: "Zero is at the base of nothing" or, stronger: "Nothing can have zero as its basis.") Zero is strictly a derivative concept, meaningful only for closure in the context of certain higher-level operations.
</p>
<p>
The idea is straightforward to understand if you consider space as a continuum (a concept wherein materiality is not retained, only spatial attributes are).
</p>
<p>
[BTW, as to Binswanger's position, I could only glean some bits about it by going over the free course pamphlets etc. available at the Ayn Rand Book Store and other similar sources. I have been having no money for years (including the time I was in the USA) to be able to buy his products. So, for clarifications on his positions, contact him directly.]
</p>
<p>
- - - - - <br />
Even as you read this, I remain jobless (as I have, for years)
</p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Sun, 12 Apr 2009 08:30:57 +0000Ajit R. Jadhavcomment 10495 at https://imechanica.orgAjit,
OK, You make an
https://imechanica.org/comment/10494#comment-10494
<a id="comment-10494"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/10483#comment-10483">You can't talk of a force then</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Ajit,
</p>
<p>
OK, You make an interesting point. When I speak of a zero volume, and then integrate, this is allowable mathematically, although maybe not meaningful from a mechanical perspective.
</p>
<p>
rich
</p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Sun, 12 Apr 2009 04:31:04 +0000Rich Lehoucqcomment 10494 at https://imechanica.orgmy understanding
https://imechanica.org/comment/10493#comment-10493
<a id="comment-10493"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/10485#comment-10485">Re: Clarification on Dirac's delta...</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Ajit,
</p>
<p>
The domainin in which \delta function defined is a function space, not the domain
</p>
<p>
where f(x) (the test function that \delta function operates on) is defined. The "point" associated with \delta function is f(x) not x.
</p>
<p>
From this meaning, we can say the value of \delta function at "point" f(x) is
</p>
<p>
f(0). There is no need (or even meaningless) to talk about the value of a functional at x, especially for \delta function, which is a functional that cannot be generated by some ordinary functions.
</p>
<p>
As for the "harm" of defining the value of \delta function at x, I think that it is mainly from the mathematical concept considerations as metioned above.
</p>
<p>
Hope this helps.
</p>
<p>
</p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Sun, 12 Apr 2009 03:13:08 +0000Xu Guocomment 10493 at https://imechanica.orgRe: Clarification on Dirac's delta...
https://imechanica.org/comment/10485#comment-10485
<a id="comment-10485"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/10477#comment-10477">Clear things up a bit</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Rich,
</p>
<p>
(1) That's a nice clarification... (Actually, about the first part, I got what you must have meant soon after my posting... Happens...)
</p>
<p>
(2) Once again, I would like a mathematician to clear up this matter for me:
</p>
<p>
Is Dirac's delta really <em>undefined</em> at the only point where it has support?
</p>
<p>
After all, the requirement that the range value must be unique applies only to a function, not to a distribution---am I right? If I am wrong, then why be comfortable calling Dirac's delta a distribution either? On the other hand, if I am right, then what's the harm in taking Dirac's delta as being defined at the support point? Isn't the whole point in having an idea like Dirac's delta only about distinguishing that one point---the point of support? That's my point.
</p>
<p>
(Someone who is strong in mathematics even if not a professional mathematician himself, is also welcome to clarify.)
</p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Sat, 11 Apr 2009 06:00:40 +0000Ajit R. Jadhavcomment 10485 at https://imechanica.org"Closure"?
https://imechanica.org/comment/10484#comment-10484
<a id="comment-10484"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/10463#comment-10463">Averaged Equations are one thing; closure quite another</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Closure? With respect to what operation(s), precisely?
</p>
<p>
</p>
<p>
[An aside: Since I have used precise mathematical terms, this should be enough to convey my ideas well about this topic, right? Or, is it that I am actually wrong? <em>Can </em>I be---if I said what I did in this aside?]
</p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Sat, 11 Apr 2009 05:38:44 +0000Ajit R. Jadhavcomment 10484 at https://imechanica.orgYou can't talk of a force then
https://imechanica.org/comment/10483#comment-10483
<a id="comment-10483"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/10478#comment-10478">Ajit,
I agree with</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Rich,
</p>
<p>
No, you didn't agree with me, really speaking.
</p>
<p>
I still maintain, force is fundamentally an undefined term within continuum mechanics; only (some form of a) force-density is. In contrast, you think that it would be possible to speak meaningfully of a force when the volume is zero. I maintain that you can't. That's the difference in our positions.
</p>
<p>
Note, the limit of a function at some point is not the same as the value of that function at that point. The two concepts differ. Sometimes, only the limit may exist at a given point; the function itself may not even be defined at that point. Such, precisely, happens to be the case with the idea of a "point-force" within a continuum---it is not defined.
</p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Sat, 11 Apr 2009 05:22:57 +0000Ajit R. Jadhavcomment 10483 at https://imechanica.orgPradeep,
I think the
https://imechanica.org/comment/10479#comment-10479
<a id="comment-10479"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/10465#comment-10465">I agree....Noll's paper is</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Pradeep,
</p>
<p>
I think the paper will appear sometime this summer. I'll post to imechanica.
</p>
<p>
Due credit should be given to Eliot Fried and Roger Fosdick, who heard that I had translated for my own research, and then made the effort to get the translation published.
</p>
<p>
rich
</p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Sat, 11 Apr 2009 01:27:40 +0000Rich Lehoucqcomment 10479 at https://imechanica.orgAjit,
I agree with
https://imechanica.org/comment/10478#comment-10478
<a id="comment-10478"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/10462#comment-10462">Force density vs. force</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Ajit,
</p>
<p>
I agree with you.
</p>
<p>
My understanding when we talk about force in continuum mechanics, we've integrated a force density about some volume. If the volume is of zero measure, then the force is zero at the point but certainly the force density may not be.
</p>
<p>
rich
</p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Sat, 11 Apr 2009 01:22:00 +0000Rich Lehoucqcomment 10478 at https://imechanica.orgClear things up a bit
https://imechanica.org/comment/10477#comment-10477
<a id="comment-10477"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/10460#comment-10460">Re: Dirac's delta function...</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Ajit,
</p>
<p>
Let me clear up my rather loose statements.
</p>
<p>
(1)
</p>
<p>
The well-known identity \int ( f(x) \delta(x) ) dx = f(0) assumes that f is continuous at 0.
</p>
<p>
(2)
</p>
<p>
We have \int ( f(x) \phi_n(x) ) dx has limit f(0) when the \phi_n(x) are peaked about zero and \int ( f(x) \phi_n(x) ) dx = 1 for all n and a converging sequence of functions \phi_n(x) continuous at zero. This is a helpful way to understand what is going on. Think of normalized Gaussians.
</p>
<p>
Arash provided a concise but precise explanation of delta functions in the sense of distributions. \delta(x) is undefined at zero but it's integral over any volume containing zero in its interior is defined. The term <em>delta function</em> is stricly formal and only makes mathematical sense as a linear functional. And so it doesn't matter whether \delta(0) is undefined.
</p>
<p>
rich
</p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Sat, 11 Apr 2009 01:15:42 +0000Rich Lehoucqcomment 10477 at https://imechanica.orgAmit, thanks....I was not
https://imechanica.org/comment/10466#comment-10466
<a id="comment-10466"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/10463#comment-10463">Averaged Equations are one thing; closure quite another</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>Amit, thanks....I was not aware of Babic's paper. I will take a look</p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Fri, 10 Apr 2009 11:28:19 +0000Pradeep Sharmacomment 10466 at https://imechanica.orgI agree....Noll's paper is
https://imechanica.org/comment/10465#comment-10465
<a id="comment-10465"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/10447#comment-10447">Irving and Kirkwood</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>I agree....Noll's paper is great....I did not realize that this paper has been translated (by you!) and put on arxiv--thanks for pointing this out. I will re-read it. We have used Noll's results as well. When is the translated paper expected to appear in J. El? I will take a look at the peridynamics paper....</p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Fri, 10 Apr 2009 11:20:46 +0000Pradeep Sharmacomment 10465 at https://imechanica.orgRich,
I particularly
https://imechanica.org/comment/10464#comment-10464
<a id="comment-10464"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/10448#comment-10448">symmetry group</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Rich,
</p>
<p>
I particularly enjoyed the following book as it is fairly systematic (e.g. uses group theory) and the treatment is elementary (not usually the case): <a href="http://www.amazon.com/Crystal-Properties-via-Group-Theory/dp/052141945X">Crystal Properties Via Group Theory</a>.
</p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Fri, 10 Apr 2009 11:14:18 +0000Pradeep Sharmacomment 10464 at https://imechanica.orgAveraged Equations are one thing; closure quite another
https://imechanica.org/comment/10463#comment-10463
<a id="comment-10463"></a>
<p><em>In reply to <a href="https://imechanica.org/node/5214">a point and a particle</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Pradeep,
</p>
<p>
A not so well-known paper by a not well-known author using space-time averaging which I have found useful is:
</p>
<p>
Babic, M. (1997) Average balance equations for Granular Materials, Intl. Journal of Engineering Sciences, v. 35, p. 523-548.
</p>
<p>
This actually deals with, if i might say, the MD problem with collisions.
</p>
<p>
As a somewhat relevant, I hope, aside - Ultimately, the main question, in my opinion, is that of closure - how do you write the terms appearing in these average balances, defined in terms of microscopic entities, in tems of the averaged quantities or a small augmentation to the set of these averaged quantities. This *is* the non-trivial question and at the heart of prediction of memory effects, macroscopic dissipation related to microscopic conservative physics, macroscopic stick-slip as a limit of mixing up microscopically fast motions with dead stops, etc....
</p>
<p>
A thought-provoking book addressing small parts of this very big question of closure is:
</p>
<p>
From Hyperbolic Systems to Kinetic Theory A Personalized Quest<br />
Series:<br /><a href="http://www.springer.com/series/7172">Lecture Notes of the Unione Matematica Italiana</a><br />
, Vol. 6<br /><br /><strong>Tartar</strong>, Luc<br /></p>
<p>2008, XXVIII, 282 p., Softcover</p>
<p>ISBN: 978-3-540-77561-4</p>
<p> </p>
<p>The technical parts (that I understand ! - I figure it will take me two lifetimes of dedicated learning to really understand Tartar) are pure luminous gold - you will also get an interesting and personal view on science and scientists by one of the most serious thinkers of our time and the last century, in my opinion.</p>
<p> </p>
<p>- Amit </p>
<p>
</p>
<p>
</p>
<p>
</p>
<p>
</p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Fri, 10 Apr 2009 10:47:32 +0000Amit Acharyacomment 10463 at https://imechanica.orgRe: Rui's zeroth point
https://imechanica.org/comment/10461#comment-10461
<a id="comment-10461"></a>
<p><em>In reply to <a href="https://imechanica.org/node/5214">a point and a particle</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Dear Rui (and others),
</p>
<p>
In your point no. (0) above, you say:
</p>
<p>
"... The state at a point of a continuum such as temperature and pressure is a statistical average of many particles in a representative volume."</p>
<p>
I think that the inclusion of the word "statistical" here makes it all a bit too narrow, theoretically speaking.
</p>
<p>
The word "statistical" suggests randomness... Now, do we have to assume that state properties such as temperatures and pressures must be suffering random fluctuations inside every continuum? Why can't these fluctuations be just nonrandom or systematic fluctuations? After all, continuum is just an abstraction, right? (Here, concerning the basic nature of continuum, I was reminded of Zhigang's excellent post on the topic of why we would have to invent the continuum if it were not to exist already...) Since the continuum is basically an abstraction, you can always hypothesize a nonrandom fluctuation for it, right? An easy example: Electronic wavefunction is, by QM, random; but its representation in MD simulations is completely deterministic.
</p>
<p>
Thinking further, I am not even sure if the state at a point within a continuum has to be an <em>average</em> (whether statistical or deterministic). In fact, doesn't this suggest circularity in a sense? Consider this: The state at a point P is determined by the state at other points (an average of whose values is to be taken at the point P), and the state at those other points is determined by the state here at point P... That, clearly, is a circularity, with the concept "value of a state" being basically undefined all along...
</p>
<p>
This circularity is easily broken by recognizing that the continuum description basically (even axiomatically) includes a state definition at each point.
</p>
<p>
Averaging would be useful for having a steadier or lower-differential order description of more complex phenomena such as thermal fluctuations. Yet, the continuum model itself need not be bound by the process of taking an average---that's the basic point.
</p>
<p>
-----
</p>
<p>
Apart from it all, Rui, your effort is very much laudable and appreciated. I mean, it's a very good list of points.
</p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Fri, 10 Apr 2009 10:05:36 +0000Ajit R. Jadhavcomment 10461 at https://imechanica.orgForce density vs. force
https://imechanica.org/comment/10462#comment-10462
<a id="comment-10462"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/10422#comment-10422">points and particles</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Dear Rich,
</p>
<p>
Inside a continuum, you can only define a density of a force---whether it's a body force (the volume density) or a surface/line traction force (the area/lineal density).
</p>
<p>
Defining a density is the only way theoretical available to bring the particle mechanical concept of force into correspondence with the concepts defined within the continuum mechanical framework (and vice versa: to take those continuum densities outside of CM and into the realm of the particle mechanics).
</p>
<p>
As such, the notion of a "force inside a continuum" is in principle undefined. (Actually, it would be an example of an arbitrary concept.)
</p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Fri, 10 Apr 2009 10:01:41 +0000Ajit R. Jadhavcomment 10462 at https://imechanica.orgRe: Dirac's delta function...
https://imechanica.org/comment/10460#comment-10460
<a id="comment-10460"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/10445#comment-10445">dirac functions</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Dear Rich,
</p>
<p>
(1) I did not quite understand what you mean by this:
</p>
<p>
"... (assuming that the integrand not including the Dirac is continuous at the point where the Dirac is undefined)..."</p>
<p>
What other integrand did you have in mind? One of the series of those (finite) test functions whose limit the delta represents? Or something else?
</p>
<p>
(2) Also, in your next paragraph, you say that:
</p>
<p>
"...Dirac as the limit of really nice functions peaked about the point where the Dirac is undefined."</p>
<p>
I get a feel that, if we were to refer to the first diagram shown on <a href="http://en.wikipedia.org/wiki/Dirac_delta_function" target="_blank"> the Wiki page for Dirac's delta "function"</a>, you would take Dirac's delta to be <em>undefined</em> at x = 0.
</p>
<p>
Well, if it is not defined at x = 0, and if it is anyway known to vanish everywhere else in the domain (i.e. for all other values of x), then how do we describe "it" at all? ...
</p>
<p>
... I think it's OK to consider Dirac's delta as defined at x= 0. It's OK to use the word "defined" here, because the delta is not a function anyway. (And, just in case I am wrong, I would very much like to know what word I should use in the place of "defined.")
</p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Fri, 10 Apr 2009 08:57:41 +0000Ajit R. Jadhavcomment 10460 at https://imechanica.orgRe: symmetry group
https://imechanica.org/comment/10452#comment-10452
<a id="comment-10452"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/10448#comment-10448">symmetry group</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Rich,
</p>
<p>
Another text that I have found beneficial in this regard is Robert Newnham's "Properties of Crystals: Anisotropy|Symmetry|Structure".
</p>
<p>
Jason
</p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Thu, 09 Apr 2009 20:28:13 +0000Jason Mayeurcomment 10452 at https://imechanica.orgDistribution Theory
https://imechanica.org/comment/10450#comment-10450
<a id="comment-10450"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/10430#comment-10430">Force as a dirac function</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>Dear Mikael:</p>
<p>The rigorous way of looking at Dirac "function" is what S. Sobolev and L. Schwratz (among others) built in the thirties and forties). In the rigorous theory, one does not think of point values of "generalized functions". What you would do is the following: </p>
<p>First a set of test functions are defined. These are infinitely smooth functions that have compact supports, i.e. are zero outside compact sets (a closed and bounded interval in the case of R^n). Now a distribution (generalized function) is dual of this set. In other words, given a distribution, it associates a real number to any given test function; distributions are linear, continuous functionals on the space of test functions.</p>
<p>In the case of Dirac delta, when it acts on any test function, it gives you the value of the test function at the origin (or the support point). One can define derivatives of distributions, and many other operators. </p>
<p>I'm not answering your question regarding discrete systems but if there is a rigorous theory it should be based on these ideas.</p>
<p>Regards,<br />
Arash</p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Thu, 09 Apr 2009 19:47:11 +0000arash_yavaricomment 10450 at https://imechanica.orgRe: symmetry group
https://imechanica.org/comment/10449#comment-10449
<a id="comment-10449"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/10448#comment-10448">symmetry group</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>Dear Rich:</p>
<p>There are many books that discuss applications of group theory in physics (my favorite is "Group Theory in Physics" by Wu-Ki Tung) but for applications in crystals the best book I've seen is:</p>
<p>Continuum Models for Phase Transitions and Twinning in Crystals by M. Pitteri and G. Zanzotto</p>
<p>Regards,<br />
Arash</p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Thu, 09 Apr 2009 19:36:33 +0000arash_yavaricomment 10449 at https://imechanica.orgsymmetry group
https://imechanica.org/comment/10448#comment-10448
<a id="comment-10448"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/10404#comment-10404">infinite vs finite</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>Arash,</p>
<p>Do you have a source, or more, where the relationships between lattices and symmetry groups are introduced/derived? This is something I'd like to learn more about but lack a good source (let alone the time to do a careful search).</p>
<p>Thanks in advance,</p>
<p>rich</p>
</div></div></div><ul class="links inline"><li class="comment_forbidden first last"><span><a href="/user/login?destination=node/5214%23comment-form">Log in</a> or <a href="/user/register?destination=node/5214%23comment-form">register</a> to post comments</span></li>
</ul>Thu, 09 Apr 2009 19:16:39 +0000Rich Lehoucqcomment 10448 at https://imechanica.org