User talk:173.206.33.141

September 2020
Hello, I'm DVdm. I noticed that you added or changed content in an article, Elastic collision, but you didn't provide a reliable source. It's been removed and archived in the page history for now, but if you'd like to include a citation and re-add it, please do so. You can have a look at the tutorial on citing sources. If you think I made a mistake, you can leave me a message on my talk page. Thank you. - DVdm (talk) 09:20, 2 September 2020 (UTC)

Hi DVdm,

Re: Full revert of an edit (derivation of a physics formula) in Elastic collision article due to lack of citation

Thank you for the notice of lack of citation. The derivation is elementary, so it does not require a source & its validity can be verified easily. Nevertheless, citation has been added.

However, my edit should have remained but been tagged for citation. It is not appreciable that the edit was reverted in its entirety, which is disproportionate and is demotivating in participating further in Wikipedia. But, I really do appreciate that you reached out directly on my talk page, which I wish established editors would engage in more.

FYI, I may be adding to that article further re: Body assumptions; Programming; Ideal & non-ideal classical mechanics; Statistical mechanics. Will keep in mind citations and will appreciate discussion before full reverts.

Thank you,

173.206.33.141 (talk) 19:17, 2 September 2020 (UTC)


 * Regarding your remark that "the derivation is elementary, so it does not require a source", please note that all challenged content requires a citation—see wp:BURDEN. Yes, there is an exception for simple routine calculations (—see wp:CALC—), but your content is way betond that. Please keep that in mind in the future.
 * As I have no access to the source that you provided when you undid my revert, I have put an informal request for comment on the physics project page Wikipedia talk:WikiProject Physics. Feel free to comment there. By the way, there is no need to also notify me on my talk page, as I have yours on my watch list. - DVdm (talk) 19:44, 2 September 2020 (UTC)

Sandbox, pls ignore
Sandbox. Pls ignore. - 173.206.33.141 (talk) 22:47, 5 September 2020 (UTC)

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Finite invariance condition
Definition: Finite invariance condition

A function $$F(x_i)$$ is invariant under a continuous symmetry transformation or Lie group $$\overline{x}_j = T_j(x_i, \varepsilon)$$ if and only if, for all $$\varepsilon$$,


 * $$ F( \overline{x}_j ) = F( T_j (x_i, \varepsilon) ) = F(x_i) .$$

This is called the finite invariance condition. The Lie group is defined such that for $$\varepsilon = 0$$, the Lie group is just identity


 * $$ [ \overline{x}_j ]_{\varepsilon = 0} = [ T_j (x_i, \varepsilon) ]_{\varepsilon = 0} = x_j .$$

Expanding $$T_j(x_i, \varepsilon)$$ in a Taylor series about $$\varepsilon = 0$$ gives


 * $$ \overline{x}_j = x_i + \varepsilon \left[ \frac{\partial T_j}{\partial \varepsilon} \right]_{\varepsilon = 0} + O(\varepsilon^2) .$$

The infinitesimals of the group are defined as


 * $$ \xi_j(x_i) \equiv \left[ \frac{\partial T_j}{\partial \varepsilon} \right]_{\varepsilon = 0}, \quad j = 1, 2, 3, ..., n . $$

The vector $$\xi_j$$ is also called the vector field of the group.

Due to the general non-linear dependence of $$T_j(x_i, \varepsilon)$$ on the parameter $$\varepsilon$$, the finite invariance condition is difficult to apply in practice. Lie's insight was that the finite invariance condition can be linearized and thus substantially simplified. Expanding $$F( \overline{x}_j ) = F( T_j (x_i, \varepsilon) )$$ in a Taylor series about $$\varepsilon = 0$$ gives


 * $$ F( \overline{x}_j ) = F( T_j (x_i, \varepsilon) ) = F(x_i) + \varepsilon \left[ \frac{\partial F}{\partial \varepsilon} \right]_{\varepsilon = 0} + \frac{\varepsilon^2}{2!} \left[ \frac{\partial^2 F}{\partial \varepsilon^2} \right]_{\varepsilon = 0} + \frac{\varepsilon^3}{3!} \left[ \frac{\partial^3 F}{\partial \varepsilon^3} \right]_{\varepsilon = 0} + \quad ...$$

Applying the chain rule


 * $$ \left[ \frac{\partial F}{\partial \varepsilon} \right]_{\varepsilon = 0} = \frac{\partial F}{\partial T_j} \left[ \frac{\partial T_j}{\partial \varepsilon} \right]_{\varepsilon = 0} = \xi_j \frac{\partial F}{\partial T_j} $$

to the Taylor expansion gives the Lie series expansion of the function $$F(x_i)$$:



F( \overline{x}_j ) = F( T_j (x_i, \varepsilon) ) = F(x_j) + \varepsilon \left( \xi_j \frac{\partial F}{\partial x_j} \right) + \frac{\varepsilon^2}{2!} \left( \xi_j \frac{\partial}{\partial x_j} \left( \xi_k \frac{\partial F}{\partial x_k} \right) \right) + \frac{\varepsilon^3}{3!} \left( \xi_j \frac{\partial}{\partial x_j} \left( \xi_k \frac{\partial}{\partial x_k} \left( \xi_l \frac{\partial F}{\partial x_l} \right) \right) \right) + \quad ... $$

The finite invariance condition $$F( \overline{x}_j ) = F(x_i)$$ is satisfied if and only if the following infinitesimal invariance condition is satisfied:


 * $$ \xi_j \frac{\partial F}{\partial x_j} = 0 .$$

So the finite invariance condition is linearized, which substantially simplifies problems.

The operator


 * $$ X_j \equiv \xi_j \frac{\partial}{\partial x_j} $$

is called the group operator or infinitesimal operator, and $$X_j F$$ is called the Lie derivative of $$F$$.

The Lie series can be expanded in terms of the group operator as an exponential


 * $$ F( \overline{x}_j ) = F(x_j) + \varepsilon (X_j F) + \frac{\varepsilon^2}{2!} (X_j (X_j F)) + \frac{\varepsilon^3}{3!} (X_j (X_j (X_j F))) + \quad ... = e^{\varepsilon X_j} F(x_i) $$

Continuous groups
A symmetry is a transformation of an object which leaves that object unchanged or "invariant". The object can be a shape, a function, or also a differential equation. is usually a function, and the symmetry is a transformation of the variables of the function.

Consider, for example, the rotation of a circle of radius $$r$$. The object in this case is a circle and the transformation is rotation. Intuitively, the form of the circle is unchanged by rotation. Hence, the transformation of rotation is a symmetry of a circle, so the circle has rotational symmetry. The form of a circle can be mathematically characterized by the function $$F(x,y) = x^2 + y^2 - r^2 = 0$$. A point $$(x, y)$$ on a circle can be determined by $$(x, y) = (r \operatorname{cos} \theta, r \operatorname{sin} \theta)$$ where $$\theta$$ is a reference angle.

The transformation is a rotation, which transforms the point $$(x, y)$$ into another point $$(\overline{x}, \overline{y})$$, by a change $$\varepsilon$$ in the angle


 * $$ \overline{x} = g(x, \varepsilon) = x(\theta + \varepsilon) = r \operatorname{cos} (\theta + \varepsilon) = r \operatorname{cos} \theta \operatorname{cos} \varepsilon - r \operatorname{sin} \theta \operatorname{sin} \varepsilon = x \operatorname{cos} \varepsilon - y \operatorname{sin} \varepsilon $$
 * $$ \overline{y} = h(y, \varepsilon) = y(\theta + \varepsilon) = r \operatorname{sin} (\theta + \varepsilon) = r \operatorname{sin} \theta \operatorname{cos} \varepsilon + r \operatorname{cos} \theta \operatorname{sin} \varepsilon = y \operatorname{cos} \varepsilon + x \operatorname{sin} \varepsilon .$$

The characteristic form of a circle remains unchanged under the transformation of a rotation, since


 * $$ \begin{align}

F(\overline{x}, \overline{y}) &= \overline{x}^2 + \overline{y}^2 - r^2 \\ &= (x \operatorname{cos} \varepsilon - y \operatorname{sin} \varepsilon)^2 + (y \operatorname{cos} \varepsilon + x \operatorname{sin} \varepsilon)^2 - r^2 \\ &= x^2 \operatorname{cos}^2 \varepsilon - 2xy \operatorname{cos} \varepsilon \operatorname{sin} \varepsilon + y^2 \operatorname{sin}^2 \varepsilon + y^2 \operatorname{cos}^2 \varepsilon + 2xy \operatorname{cos} \varepsilon \operatorname{sin} \varepsilon + x^2 \operatorname{sin}^2 \varepsilon - r^2 \\ &= x^2 (\operatorname{sin}^2 \varepsilon + \operatorname{cos}^2 \varepsilon) + y^2 (\operatorname{sin}^2 \varepsilon + \operatorname{cos}^2 \varepsilon) - r^2 \\ &= x^2 + y^2 - r^2 \\ &= F(x, y) \\ \end{align} $$

Hence, the transformation of rotation is a symmetry of the object of a circle.

More generally, let $$ x_i \equiv (x_1, x_2, x_3, ..., x_n) $$ be a point in the $$n$$-dimensional Euclidean space $$ \mathbb{R}^n $$. In two dimensions, the point $$ x_i $$ is commonly denoted $$ (x, y) $$, and in three dimensions, it is commonly denoted $$ (x, y, z) $$.

The point $$ x_i $$ can be continuously transformed to another point $$ \overline{x}_i $$ with respect to a continuous parameter $$ \epsilon $$ by


 * $$ \overline{x}_i = f_i (x_j, \epsilon), \quad i, j = 1, 2, 3, ..., n, $$

where the transformations $$ f_i $$ are smooth functions of the variables $$ x_j $$, and are analytic functions of the parameter $$ \epsilon $$ (in other words, the $$ f_i $$ have a convergent Taylor series in $$ \epsilon $$).

For example, Consider the points (x, y) and (x, y), on the circumference of a circle of radiusr(Figure 1.4). We can write these in terms of the radius and the angles 𝜃 (a reference angle) and 𝜃 + 𝜀, (after rotation), that is, These then become x = r cos 𝜃, x = r cos(𝜃 + 𝜀), (1.1a) y = rsin 𝜃, y = rsin(𝜃 + 𝜀), (1.1b) or, after eliminating 𝜃 x = x cos 𝜀 − y sin 𝜀 (1.2a) y = y cos 𝜀 + x sin 𝜀.

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