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Classical mechanics

Rongzhou — Thu, 24 Sep 2026 15:13:07 GMT

Gravity: any two mass attracts

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	The potential energy as a result of gravitational attraction is known as [[gravitational potential energy]].		The potential energy as a result of gravitational attraction is known as [[gravitational potential energy]].

			<blockquote>
			See also: [[Two body problem]]
			</blockquote>

Two body problem

Rongzhou — Thu, 24 Sep 2026 15:04:13 GMT

Classic form: planetary/satellite orbit (point mass approximation)

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	Where $\mu=G(M+m)$.		Where $\mu=G(M+m)$.

			The orbit described by the equation is usually elliptical:

			[[File:Two-body-problem.ggb]]

	=== Derivation ===		=== Derivation ===

File:Two-body-problem.ggb

Rongzhou — Thu, 24 Sep 2026 15:01:52 GMT

Rongzhou uploaded File:Two-body-problem.ggb

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== Licensing ==
[[Q320|CC BY-SA 4.0]]
== Attribution ==
Author: Rong Martin-Siebler ZHOU

Item:Q1786

Rongzhou — Thu, 24 Sep 2026 15:01:52 GMT

Created the image item for Two-body-problem

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		File:Two-body-problem.ggb

Item:Q1786

Rongzhou — Thu, 24 Sep 2026 15:01:52 GMT

Created the image item for Two-body-problem

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Two body problem

Rongzhou — Thu, 24 Sep 2026 14:41:17 GMT

Created page with "In astronomy, two body problems normally refer to the study of the gravitational interaction of two celestial objects. == Classic form: planetary/satellite orbit (point mass approximation) == Approximate a star/planet as point mass $M$, and a planet/satellite orbiting around it as point mass $m$. The orbit of the planet in the reference frame of the star satisfies the equation: {{#content:Q1785}} Where $\mu=G(M+m)$. === Derivation === Let $$ \mathbf{r}=\math..."

New page

In [[astronomy]], two body problems normally refer to the study of the gravitational interaction of two celestial objects.

== Classic form: planetary/satellite orbit (point mass approximation) ==

Approximate a star/planet as point mass $M$, and a planet/satellite orbiting around it as point mass $m$.

The orbit of the planet in the reference frame of the star satisfies the equation:

{{#content:Q1785}}

Where $\mu=G(M+m)$.

=== Derivation ===

Let

$$
\mathbf{r}=\mathbf{r}_m-\mathbf{r}_M
$$

be the vector from $M$ to $m$.

Let $r=|\mathbf{r}|$.

Gravity pulls $m$ toward $M$:

$$
m\ddot{\mathbf{r}}_m=-GMm\frac{\mathbf{r}}{r^3} \text{ (1)}
$$

Gravity pulls $M$ toward $m$:

$$
M\ddot{\mathbf{r}}_M=+GMm\frac{\mathbf{r}}{r^3} \text{ (2)}
$$

Therefore:

$$
\begin{cases}
\ddot{\mathbf{r}}_m=-GM\frac{\mathbf{r}}{r^3} \text{ (1)/m} \\
\ddot{\mathbf{r}}_M=+Gm\frac{\mathbf{r}}{r^3} \text{ (2)/M}
\end{cases}
$$

The relative acceleration of $m$ according to $M$, $\ddot{\mathbf{r}}$, is

$$
\ddot{\mathbf{r}}
=
\ddot{\mathbf{r}}_m-\ddot{\mathbf{r}}_M
$$

Substitute:

$$
\ddot{\mathbf{r}}
=
-GM\frac{\mathbf{r}}{r^3}
-
Gm\frac{\mathbf{r}}{r^3}
$$

$$
\boxed{
\ddot{\mathbf{r}}
=
-G(M+m)\frac{\mathbf{r}}{r^3}
}
$$

Usually, we note

$$
\boxed{
\mu=G(M+m)
}
$$

Item:Q1785

Rongzhou — Thu, 24 Sep 2026 14:35:31 GMT

Added content item: Planetary orbit

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Item:Q1785

Rongzhou — Thu, 24 Sep 2026 14:35:31 GMT

Added content item: Planetary orbit

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Gravitational potential energy

Rongzhou — Thu, 24 Sep 2026 13:42:55 GMT

Created page with "Gravitational potential energy, commonly noted $U$, refers to the energy incorporated in a system as a result of the gravitational attraction between its elements. == Zero reference point == Generally, we define the zero reference point: {{#content:Q1779}} Since there is no gravitational interaction when objects are at an infinite distance from each other. Then, the gravitational potential energy of a system measures the amount of work done by gravitational forces t..."

New page

Gravitational potential energy, commonly noted $U$, refers to the energy incorporated in a system as a result of the gravitational attraction between its elements.

== Zero reference point ==

Generally, we define the zero reference point:

{{#content:Q1779}}

Since there is no gravitational interaction when objects are at an infinite distance from each other.

Then, the gravitational potential energy of a system measures the amount of work done by gravitational forces to move all masses of the system away from each other until the distance between them is infinite.

== Basic case: 2 point masses ==

For two point masses $m$, $M$, we can calculate the gravitational potential energy of the system mostly easily by constructing a cartesian coordinate system with one of the point masses, $M$, at the origin.

Then, if $m$ has position vector $ \mathbf r_1$:

$$\mathbf F_g(m)=-G\frac{Mm}{r_1^2}\hat{\mathbf r}_1$$

The potential energy of the system is:

$$U=-\int_{\infty}^{r_1}\left(-G\frac{Mm}{r^2}\right) \hat{\mathbf r} \cdot dr=-G\frac{Mm}{r}$$

<blockquote>
$$\hat{\mathbf r} \cdot dr = dr$$
</blockquote>

== Many point masses ==

Recall the gravitational potential energy between 2 point masses is

{{#content:Q1780}}

Applying the formula to all pairs of point masses in the system:

{{#content:Q1781}}

We can similarly derive a formula for the gravitational potential energy of a point mass external to the system:

{{#content:Q1782}}

== Continuous mass distribution ==

Any generic mass system can be modelled by a continuous mass distribution, with $\rho=0$ where there is no mass.

If the density at point with position vector $\mathbf{r}$ is $\rho(\mathbf{r})$:

{{#content:Q1783}}

<blockquote>
$dV=dr^3$: infinitesimal volume
</blockquote>

== Gravitational potential ==

If a test mass $m$ has gravitational potential energy $U(\mathbf{r})$ at a point, then the gravitational potential at that point is

$$\Phi(\mathbf{r})=\frac{U(\mathbf{r})}{m}$$

For another test mass $m_1$, the gravitational potential energy at the same point is then simply:

{{#content:Q1784}}

Item:Q1784

Rongzhou — Thu, 24 Sep 2026 13:40:52 GMT

Added content item: Gravitational potential energy and gravitational potential

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	~~Rong Martin-Siebler ZHOU~~
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		Property / attributed to: Rong Martin-Siebler ZHOU / rank
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Item:Q1784

Rongzhou — Thu, 24 Sep 2026 13:40:52 GMT

Added content item: Gravitational potential energy and gravitational potential

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Classical mechanics

Rongzhou — Thu, 24 Sep 2026 13:34:58 GMT

Gravity: any two mass attracts

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	! Symbol !! Meaning		! Symbol !! Meaning
	\|-		\|-
	\| $\mathbf r_A$ \|\| position of ~~a tiny~~ piece of body $A$.		\| $\mathbf r_A$ \|\| position vector of an infinitesimal piece of body $A$.
	\|-		\|-
	\| $\mathbf r_B$ \|\| position of ~~a tiny~~ piece of body $B$.		\| $\mathbf r_B$ \|\| position of an infinitesimal piece of body $B$.
	\|-		\|-
	\| $\rho_A(\mathbf r_A), \rho_B(\mathbf r_B)$ \|\| density at those points.		\| $\rho_A(\mathbf r_A), \rho_B(\mathbf r_B)$ \|\| density at those points.
	\|-		\|-
	\| $dV_A, dV_B$ \|\| ~~tiny~~ volumes.		\| $dV_A=dr_A^3, dV_B=dr_B^3$ \|\| infinitesimal volumes.
	\|-		\|-
	\| $dm_A=\rho_A dV_A$, $dm_B=\rho_B dV_B$ \|\| ~~tiny~~ masses.		\| $dm_A=\rho_A dV_A$, $dm_B=\rho_B dV_B$ \|\| infinitesimal masses.
	\|-		\|-
	\| $\lvert \mathbf r_B-\mathbf r_A\rvert$ \|\| distance between the two ~~tiny~~ pieces.		\| $\lvert \mathbf r_B-\mathbf r_A\rvert$ \|\| distance between the two infinitesimal pieces.
	\|-		\|-
	\| $\dfrac{\mathbf r_B-\mathbf r_A}{\lvert \mathbf r_B-\mathbf r_A\rvert^3}$ \|\| unit direction from $A$-piece to $B$-piece, divided by distance squared.		\| $\dfrac{\mathbf r_B-\mathbf r_A}{\lvert \mathbf r_B-\mathbf r_A\rvert^3}$ \|\| unit direction from $A$-piece to $B$-piece, divided by distance squared.
	\|}		\|}

			The potential energy as a result of gravitational attraction is known as [[gravitational potential energy]].

Item:Q1783

Rongzhou — Thu, 24 Sep 2026 13:22:00 GMT

Added content item: Gravitational potential energy of continuous mass distribution

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	~~U=-\\frac12 G\\int\\int\\frac{\\rho(\\mathbf{r})\\rho(\\mathbf{r}_1)}{\|\\mathbf{r}-\\mathbf{r}_1\|}d^3r\\,d^3r_1~~
Property / LaTeX source: U=-\\frac12 G\\int\\int\\frac{\\rho(\\mathbf{r})\\rho(\\mathbf{r}_1)}{\|\\mathbf{r}-\\mathbf{r}_1\|}d^3r\\,d^3r_1 / rank
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			U=-\\frac12 G\\int\\int\\frac{\\rho(\\mathbf{r})\\rho(\\mathbf{r}_1)}{\|\\mathbf{r}-\\mathbf{r}_1\|}d^3r\\,d^3r_1
		Property / LaTeX source: U=-\\frac12 G\\int\\int\\frac{\\rho(\\mathbf{r})\\rho(\\mathbf{r}_1)}{\|\\mathbf{r}-\\mathbf{r}_1\|}d^3r\\,d^3r_1 / rank
			Normal rank
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			Rong Martin-Siebler ZHOU
		Property / attributed to: Rong Martin-Siebler ZHOU / rank
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Item:Q1783

Rongzhou — Thu, 24 Sep 2026 13:22:00 GMT

Added content item: Gravitational potential energy of continuous mass distribution

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Item:Q1782

Rongzhou — Thu, 24 Sep 2026 13:12:53 GMT

Added content item: Gravitational potential energy between a point mass and a system of many point masses

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	~~U(\\mathbf{r})=-Gm\\sum_i \\frac{M_i}{\\mathbf{r}-\\mathbf{r}_i}~~
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Item:Q1782

Rongzhou — Thu, 24 Sep 2026 13:12:53 GMT

Added content item: Gravitational potential energy between a point mass and a system of many point masses

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Item:Q1781

Rongzhou — Thu, 24 Sep 2026 12:59:17 GMT

Added content item: Gravitational potential energy between many point masses

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	~~U=-G\\sum_{i<j}\\frac{m_i m_j}{\\mathbf{r_i}-\\mathbf{r_j}}~~
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Item:Q1781

Rongzhou — Thu, 24 Sep 2026 12:59:17 GMT

Added content item: Gravitational potential energy between many point masses

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Item:Q1780

Rongzhou — Thu, 24 Sep 2026 12:54:09 GMT

Added content item: Gravitational potential energy between two point masses

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Property / instance of
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	~~U=-G\\frac{Mm}{r}~~
Property / LaTeX source: U=-G\\frac{Mm}{r} / rank
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Item:Q1780

Rongzhou — Thu, 24 Sep 2026 12:54:09 GMT

Added content item: Gravitational potential energy between two point masses

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Item:Q1779

Rongzhou — Thu, 24 Sep 2026 12:37:27 GMT

Added content item: Zero reference point for gravitational potential energy

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	~~U(r=\\infty)=0~~
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Item:Q1779

Rongzhou — Thu, 24 Sep 2026 12:37:27 GMT

Added content item: Zero reference point for gravitational potential energy

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	For large Wikimedia files the upload itself also happens through your browser, so the wiki server never downloads the file from Wikimedia.		For large Wikimedia files the upload itself also happens through your browser, so the wiki server never downloads the file from Wikimedia.
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			{\| class="wikitable" style="width:100%"
			\|-
			! You type !! What it does
			\|-
			\| <syntaxhighlight lang="text" inline>[[File:Parabola.ggb]]</syntaxhighlight>
			\| The interactive worksheet at the default size (800 × 600).
			\|-
			\| <syntaxhighlight lang="text" inline>[[File:Parabola.ggb\|600px]]</syntaxhighlight>
			\| Fixes the width to 600 pixels; the height keeps the 4:3 proportions.
			\|-
			\| <syntaxhighlight lang="text" inline>[[File:Parabola.ggb\|thumb\|Caption text]]</syntaxhighlight>
			\| A framed applet with a caption.
			\|-
			\| <syntaxhighlight lang="text" inline>[[:File:Parabola.ggb]]</syntaxhighlight>
			\| A text link to the file page — the leading colon prevents embedding.
			\|}

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Classical mechanics

Rongzhou — Wed, 23 Sep 2026 12:48:26 GMT

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	{{#content:Q1778}}		{{#content:Q1778}}

			{\| class="wikitable"
			! Symbol !! Meaning
			\|-
			\| $\mathbf r_A$ \|\| position of a tiny piece of body $A$.
			\|-
			\| $\mathbf r_B$ \|\| position of a tiny piece of body $B$.
			\|-
			\| $\rho_A(\mathbf r_A), \rho_B(\mathbf r_B)$ \|\| density at those points.
			\|-
			\| $dV_A, dV_B$ \|\| tiny volumes.
			\|-
			\| $dm_A=\rho_A dV_A$, $dm_B=\rho_B dV_B$ \|\| tiny masses.
			\|-
			\| $\lvert \mathbf r_B-\mathbf r_A\rvert$ \|\| distance between the two tiny pieces.
			\|-
			\| $\dfrac{\mathbf r_B-\mathbf r_A}{\lvert \mathbf r_B-\mathbf r_A\rvert^3}$ \|\| unit direction from $A$-piece to $B$-piece, divided by distance squared.
			\|}

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Classical mechanics

Rongzhou — Wed, 23 Sep 2026 06:47:23 GMT

Created page with "Classical mechanics is one framework to the study of mechanics. == Validity == == Vector description == Concise description of mechanical systems can be provided with vectors. === Reference frame === A reference frame is defined by an origin $O$, and the three unit vectors for each dimension of space: $\overrightarrow{x}$, $\overrightarrow{y}$, and $\overrightarrow{z}$. It can be noted as: $$R(O;\overrightarrow{x},\overrightarrow{y},\overrightarrow{z})$$..."

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Classical mechanics is one framework to the study of [[mechanics]].

== Validity ==

== Vector description ==

Concise description of mechanical systems can be provided with [[vector]]s.

=== Reference frame ===

A reference frame is defined by an origin $O$, and the three unit vectors for each dimension of space: $\overrightarrow{x}$, $\overrightarrow{y}$, and $\overrightarrow{z}$.

It can be noted as:

$$R(O;\overrightarrow{x},\overrightarrow{y},\overrightarrow{z})$$

Conventionally, we choose unit vectors perpendicular to each other ($\overrightarrow{x}\bot\overrightarrow{y}\bot\overrightarrow{z}$), so each spatial direction is independent of another.

=== Objects in the reference frame ===

==== Motion: Position, velocity, accelaration ====

For an object (S) centered at point $P$, its '''position''' relative to reference frame $R$ can be described by its position vector $\overrightarrow{OP}=(\overrightarrow{x_P},\overrightarrow{y_P},\overrightarrow{z_P})$.

Then velocity, defined as the instantaneous change in position with respect to time, is then:

$$\overrightarrow{v_S}=\frac{d \overrightarrow{OP}}{dt}=(\overrightarrow{v_x},\overrightarrow{v_y},\overrightarrow{v_z})$$

Finally, acceleration, defined as the instantaneous change in velocity with respect to time, is then:

$$\overrightarrow{a_S}=\frac{d \overrightarrow{v_S}}{dt}=(\overrightarrow{a_x},\overrightarrow{a_y},\overrightarrow{a_z})$$

==== Mechanical contact: Force and moments ====

Any ''force'' $\overrightarrow{F}$ on object $(S)$ can be decomposed into component forces in each of the three independent spatial direction:

$$\overrightarrow{F}= \overrightarrow{F_x}+\overrightarrow{F_y}+\overrightarrow{F_z}$$

Therefore, it can be described as a vector:

$$\overrightarrow{F}= (F_x,F_y,F_z)$$

where $$F_x=\left\| \overrightarrow{F_x} \right\|$$ and so on.

As a result, if the force $\overrightarrow{F}$ is applied to point $A$ the ''moment'' $M(\overrightarrow{F}\to S)_B$ caused by such force at point $B$ is:

$$\overrightarrow{M}(\overrightarrow{F}\to S)_B=\overrightarrow{BA}\wedge \overrightarrow{F}$$

It can also be decomposed into components along three independent spatial directions:

$$\overrightarrow{M}=(M_x,M_y,M_z)$$

== Key principles ==

=== Newton's 1st law ===

Newton's 1st law describes the phenomenon '''inertia''': when the overall net external force is zero, the velocity of an object is maintained.<ref>{{#cite:Q1761}}</ref>

In vector notation, it can be written as:

{{#content:Q1766}}

The state that $\sum \overrightarrow{F}_{ext}=0$ is known '''mechanical equilibrium'''.

=== Newton's 2nd law ===

Newton's 2nd law describes the relationship between force, mass, and acceleration: the acceleration is proportional to force and the inverse of mass.<ref>{{#cite:Q1761}}</ref>

In vector notation, it can be written as:

{{#content:Q1767}}

=== Newton's 3rd law ===

Newton's 3rd law describes the reciprocity of force: when object $(S1)$ applies a force on object $(S2)$, $(S2)$ must apply a force opposite to $(S1)$ and equal in magnitude.<ref>{{#cite:Q1761}}</ref>

In vector notation, it can be written as:

{{#content:Q1768}}

=== Conservation of momentum ===

For any closed system, the momentum, defined as the product of mass and velocity (m\overrightarrow{v}), is conserved before and after any mechanical interaction within the system (including collision).<ref>{{#cite:Q1761}}</ref>

In vector notation, for a closed system of $n$ objects, the principle can be expressed as:

{{#content:Q1769}}

where $C$ is a constant.

Any change to momentum must be a direct result of an external force $\overrightarrow{F_{ext}}$ acting on a member of the system for a duration $t$. This change in momentum, known as ''impulse'', satisfies the following relationship:

{{#content:Q1770}}

=== Any motion carries kinetic energy ===

The kinetic energy of an object of mass $m$ and velocity $\overrightarrow{v}$ is:

{{#content:Q1771}}

Any change in kinetic energy, is the result of an external force acting over a given distance:<ref>{{#cite:Q1761}}</ref>

{{#content:Q1773}}

=== Conservation of energy ===

In a closed system, energy is conserved. It can be transformed from one form to another, but not created or destroyed:<ref>{{#cite:Q1761}}</ref>

{{#content:Q1772}}

=== Gravity: any two mass attracts ===

Two point masses at points $A$ and $B$ exerts a gravitational force on each other:<ref>{{#cite:Q1761}}</ref>

{{#content:Q1777}}

According to [[shell theorem]], spherically symmetric shapes can be approximated by point masses.

For real-world objects, whose shape may be irregular, if the distance between the object is much larger than the size of a given object ($\left\| \overrightarrow{AB} \right\|\gg R_A$), that object may be approximated as a point mass.

Otherwise, for arbitrary real-world objects:

{{#content:Q1778}}