Free particle in R³

A single particle freely moving in ${\displaystyle \mathbb {R} ^{3}}$ has 3 degrees of freedom. The configuration space is ${\displaystyle M=\mathbb {R} ^{3},}$ and ${\displaystyle P(M)=C^{\infty }([t_{0},t_{1}],M).}$ For every path ${\displaystyle \gamma \in P(M)}$ and a variation ${\displaystyle \Gamma (t,\epsilon )}$ of ${\displaystyle \gamma ,}$ there exists a unique ${\displaystyle \sigma \in T_{0}\mathbb {R} ^{3}}$ such that ${\displaystyle \Gamma (t,\epsilon )=\gamma (t)+\sigma (t)\epsilon +o(\epsilon ),}$ as ${\displaystyle \epsilon \to 0.}$ By the definition,

${\displaystyle \delta \gamma (t)=\left({\frac {d}{d\epsilon }}{\Bigl (}\gamma (t)+\sigma (t)\epsilon +o(\epsilon ){\Bigr )}\right){\Biggl |}_{\epsilon =0}}$

which leads to

${\displaystyle \delta \gamma (t)=\sigma (t)\in T_{\gamma (t)}\mathbb {R} ^{3}.}$

Free particles on a surface

${\displaystyle N}$ particles moving freely on a two-dimensional surface ${\displaystyle S\subset \mathbb {R} ^{3}}$ have ${\displaystyle 2N}$ degree of freedom. The configuration space here is

${\displaystyle M=\{(\mathbf {r} _{1},\ldots ,\mathbf {r} _{N})\in \mathbb {R} ^{3\,N}\mid \mathbf {r} _{i}\in \mathbb {R} ^{3};\ \mathbf {r} _{i}\neq \mathbf {r} _{j}\ {\text{if}}\ i\neq j\},}$

where ${\displaystyle \mathbf {r} _{i}\in \mathbb {R} ^{3}}$ is the radius vector of the ${\displaystyle i^{\text{th}}}$ particle. It follows that

${\displaystyle T_{(\mathbf {r} _{1},\ldots ,\mathbf {r} _{N})}M=T_{\mathbf {r} _{1}}S\oplus \ldots \oplus T_{\mathbf {r} _{N}}S,}$

and every path ${\displaystyle \gamma \in P(M)}$ may be described using the radius vectors ${\displaystyle \mathbf {r} _{i}}$ of each individual particle, i.e.

${\displaystyle \gamma (t)=(\mathbf {r} _{1}(t),\ldots ,\mathbf {r} _{N}(t)).}$

This implies that, for every ${\displaystyle \delta \gamma (t)\in T_{(\mathbf {r} _{1}(t),\ldots ,\mathbf {r} _{N}(t))}M,}$

${\displaystyle \delta \gamma (t)=\delta \mathbf {r} _{1}(t)\oplus \ldots \oplus \delta \mathbf {r} _{N}(t),}$

where ${\displaystyle \delta \mathbf {r} _{i}(t)\in T_{\mathbf {r} _{i}(t)}S.}$ Some authors express this as

${\displaystyle \delta \gamma =(\delta \mathbf {r} _{1},\ldots ,\delta \mathbf {r} _{N}).}$

Rigid body rotating around fixed point

A rigid body rotating around a fixed point with no additional constraints has 3 degrees of freedom. The configuration space here is ${\displaystyle M=SO(3),}$ the special orthogonal group of dimension 3 (otherwise known as 3D rotation group), and ${\displaystyle P(M)=C^{\infty }([t_{0},t_{1}],M).}$ We use the standard notation ${\displaystyle {\mathfrak {so}}(3)}$ to refer to the three-dimensional linear space of all skew-symmetric three-dimensional matrices. The exponential map ${\displaystyle \exp$ :{\mathfrak {so}}(3)\to SO(3)} guarantees the existence of ${\displaystyle \epsilon _{0}>0}$ such that, for every path ${\displaystyle \gamma \in P(M),}$ its variation ${\displaystyle \Gamma (t,\epsilon ),}$ and ${\displaystyle t\in [t_{0},t_{1}],}$ there is a unique path ${\displaystyle \Theta ^{t}\in C^{\infty }([-\epsilon _{0},\epsilon _{0}],{\mathfrak {so}}(3))}$ such that ${\displaystyle \Theta ^{t}(0)=0}$ and, for every ${\displaystyle \epsilon \in [-\epsilon _{0},\epsilon _{0}],}$ ${\displaystyle \Gamma (t,\epsilon )=\gamma (t)\exp(\Theta ^{t}(\epsilon )).}$ By the definition,

${\displaystyle \delta \gamma (t)=\left({\frac {d}{d\epsilon }}{\Bigl (}\gamma (t)\exp(\Theta ^{t}(\epsilon )){\Bigr )}\right){\Biggl |}_{\epsilon =0}=\gamma (t){\frac {d\Theta ^{t}(\epsilon )}{d\epsilon }}{\Biggl |}_{\epsilon =0}.}$

Since, for some function ${\displaystyle \sigma$ :[t_{0},t_{1}]\to {\mathfrak {so}}(3),} ${\displaystyle \Theta ^{t}(\epsilon )=\epsilon \sigma (t)+o(\epsilon )}$, as ${\displaystyle \epsilon \to 0}$,

${\displaystyle \delta \gamma (t)=\gamma (t)\sigma (t)\in T_{\gamma (t)}SO(3).}$