PhyslibAlpha.ClassicalMechanics.MomentMap.GalileanMassCocycle

For any orthogonal matrix $R \in \text{O}(3)$ and any vector $v \in \mathbb{R}^3$ in three-dimensional Euclidean space, the group action of $R$ on $v$, denoted $R \cdot v$, is given by the matrix-vector product $R v$.

For any real scalar $k \in \mathbb{R}$ and any vector $v$ in the 3D Euclidean space $\mathbb{R}^3$, the coordinate representation of the scalar product $k \cdot v$ is equal to $k$ times the coordinate representation of $v$. This is expressed as: $$\operatorname{ofLp}(k \cdot v) = k \cdot \operatorname{ofLp}(v)$$ where $\operatorname{ofLp}$ is the map identifying the Euclidean space with its underlying coordinate space.

For an orthogonal matrix $R$, the product of its transpose $R^T$ and itself is equal to the identity matrix $I$: $$R^T R = I$$

For an orthogonal matrix $R$, the product of $R$ and its transpose $R^\top$ is the identity matrix $I$, i.e., $R R^\top = I$.

For any orthogonal matrix $R$, its inverse $R^{-1}$ is equal to its transpose $R^\top$.

For an orthogonal matrix $R$, the square of its determinant is $1$, that is, $\det(R) \cdot \det(R) = 1$.

For any orthogonal matrix $R$, the determinant of its inverse is equal to the determinant of $R$, denoted as $\det(R^{-1}) = \det(R)$.

For any orthogonal matrix $R$ and any two vectors $u, v \in \mathbb{R}^3$, the dot product is invariant under the transformation by $R$: $$(Ru) \cdot (Rv) = u \cdot v$$ where $Ru$ and $Rv$ denote matrix-vector multiplication and $\cdot$ denotes the standard dot product.

For any vectors $x, y \in \mathbb{R}^3$ and an orthogonal matrix $R$, the dot product satisfies the following identity involving the transpose $R^\top$: $$x \cdot (R^\top y) = (Rx) \cdot y$$

For an orthogonal matrix $R$, its adjugate matrix $\operatorname{adj}(R)$ is equal to the product of its determinant and its transpose, expressed as: $$\operatorname{adj}(R) = (\det R) R^T$$

For any orthogonal matrix $R$ and vectors $u, v \in \mathbb{R}^3$, the matrix-vector product of $R$ and the cross product $u \times v$ is equal to the determinant of $R$ multiplied by the cross product of the transformed vectors $R u$ and $R v$: $$R(u \times v) = \det(R) (R u \times R v)$$ If $R$ is a rotation (i.e., $\det(R) = 1$), then $R$ preserves the cross product: $R(u \times v) = R u \times R v$.

Let $R$ be an orthogonal matrix in $O(3)$. For any two vectors $u, v \in \mathbb{R}^3$, the transpose matrix $R^T$ applied to their cross product satisfies the identity $$R^T (u \times v) = \det(R) (R^T u \times R^T v),$$ where $\times$ denotes the standard cross product in $\mathbb{R}^3$ and $\det(R)$ is the determinant of the matrix $R$.

For any vectors $u, v, w \in \mathbb{R}^3$ and an orthogonal matrix $R$, the following identity holds: $$(u \times v) \cdot (\det(R) R^T w) = (R u \times R v) \cdot w$$ where $\times$ denotes the cross product in $\mathbb{R}^3$, $\cdot$ denotes the standard dot product, $\det(R)$ is the determinant of $R$, and $R^T$ is the transpose of $R$. This identity relates the pairing of an axial vector (the cross product) with a vector under orthogonal transformations.

This definition provides the action of the Galilean group on the evolution space of $N$ material points. For an element $a = (R, b, c, e)$ of the Galilean group—where $R$ is an orthogonal matrix representing rotation, $b$ is a velocity boost vector, $c$ is a spatial translation vector, and $e$ is a time translation—and a point $y$ in the evolution space characterized by time $t$ and the positions $r_j$ and velocities $v_j$ of $N$ particles, the action $a \cdot y = (t^*, r_j^*, v_j^*)$ is defined as: - $t^* = t + e$ - $r_j^* = R r_j + t b + c$ - $v_j^* = R v_j + b$ for each $j \in \{1, \dots, N\}$.

Let $a$ be an element of the Galilean group in 3 dimensions and $y$ be a point in the evolution space of $N$ material points. If $t$ is the time component of $y$ and $e$ is the time translation component of $a$, then the time component of the transformed point $a \cdot y$ is $t + e$.

Let $a = (R, \mathbf{b}, \mathbf{c}, e)$ be an element of the Galilean group $\mathcal{G}(3)$, where $R$ is an orthogonal matrix, $\mathbf{b} \in \mathbb{R}^3$ is the velocity boost, $\mathbf{c} \in \mathbb{R}^3$ is the spatial translation, and $e \in \mathbb{R}$ is the time translation. Let $y$ be a point in the evolution space of $N$ material points, characterized by a time coordinate $t$ and particle positions $\mathbf{r}_k$ for $k \in \{1, \dots, N\}$. The position of the $j$-th particle after the action of the group element $a$, denoted $(a \cdot y).r_j$, is given by $$(a \cdot y).r_j = R \mathbf{r}_j + t \mathbf{b} + \mathbf{c}$$

For an element $a$ of the Galilean group with rotation matrix $R$ and velocity boost $\mathbf{b}$, and for any point $y$ in the evolution space where the $j$-th particle has velocity $\mathbf{v}_j$, the velocity of the $j$-th particle in the transformed state $a \cdot y$ is given by $$(a \cdot y).\mathbf{v}_j = R \mathbf{v}_j + \mathbf{b}$$

The Galilean group $\text{GalileanGroup}(3)$ acts on the evolution space of $N$ material points. For a group element $a = (R, \mathbf{b}, \mathbf{c}, e)$—where $R$ is an orthogonal matrix (rotation), $\mathbf{b}$ is a velocity boost, $\mathbf{c}$ is a spatial translation, and $e$ is a time translation—and a point $y$ in the evolution space with time $t$, particle positions $\mathbf{r}_j$, and velocities $\mathbf{v}_j$, the action $a \cdot y$ is defined by: - $t \mapsto t + e$ - $\mathbf{r}_j \mapsto R \mathbf{r}_j + t \mathbf{b} + \mathbf{c}$ - $\mathbf{v}_j \mapsto R \mathbf{v}_j + \mathbf{b}$ This mapping satisfies the axioms of a group action: the identity element of the group acts as the identity transformation ($1 \cdot y = y$), and the action is compatible with the group composition ($(a_1 a_2) \cdot y = a_1 \cdot (a_2 \cdot y)$).

For a Galilean group element $a$ with rotation $R$ and velocity boost $\mathbf{b}$, and a tangent vector $dy = (dt, d\mathbf{r}_j, d\mathbf{v}_j)_{j=1}^N$ in the evolution space, this function returns the transformed tangent vector $(dt, R d\mathbf{r}_j + \mathbf{b} dt, R d\mathbf{v}_j)_{j=1}^N$. This corresponds to the differential of the affine action of the Galilean group on the evolution space.

For an element $a$ of the Galilean group and a point $y$ and tangent vector $dy$ in the evolution space of $N$ material points, the Galilean action maps the line $s \mapsto y + s \, dy$ to the line $s \mapsto (a \cdot y) + s \, (T_a \, dy)$, where $T_a$ is the tangent action (the linear part of the Galilean transformation). That is, for any $s \in \mathbb{R}$: $$a \cdot (y + s \, dy) = (a \cdot y) + s \, (T_a \, dy)$$

Let $a$ be an element of the Galilean group $\text{GalileanGroup}(3)$ and $y$ be a point in the evolution space of $N$ material points. For any index $j \in \{1, \dots, N\}$, let $t$ be the time and $\mathbf{r}_j$ be the position of the $j$-th particle in the state $y$. The action of $a$ on the spacetime point $(t, \mathbf{r}_j)$ is equal to the pair $(t', \mathbf{r}_j')$ representing the time and position of the $j$-th particle in the transformed state $a \cdot y$ in the evolution space.

Let $N$ be a natural number representing the number of material points with masses $m_j \in \mathbb{R}$ for $j \in \{1, \dots, N\}$. Let $\sigma$ denote the Lagrange form on the evolution space of these $N$ free material points. For any element $a$ of the Galilean group $\text{GalileanGroup}(3)$, any point $y$ in the evolution space, and any two tangent vectors $dy$ and $\delta y$ at $y$, the action of the Galilean group preserves the Lagrange form: $$\sigma(a \cdot y)(a \cdot dy, a \cdot \delta y) = \sigma(y)(dy, \delta y)$$ where $a \cdot y$ is the action of the group on the evolution space and $a \cdot dy$ (and $a \cdot \delta y$) is the induced tangent action (the differential of the group action) on the tangent vectors.

Given an element $a = (R, \mathbf{b}, \mathbf{c}, e)$ of the Galilean group (consisting of a rotation $R \in O(3)$, a velocity boost $\mathbf{b}$, a spatial translation $\mathbf{c}$, and a time translation $e$) and an element $Z = (\omega, \beta, \gamma, \varepsilon)$ of the Galilean Lie algebra, the adjoint action $a \cdot Z = (\omega^*, \beta^*, \gamma^*, \varepsilon^*)$ is defined by: - $\omega^* = (\det R) R \omega$ - $\beta^* = R \beta - \omega^* \times \mathbf{b}$ - $\gamma^* = R \gamma + \varepsilon \mathbf{b} - e R \beta - \omega^* \times (\mathbf{c} - e \mathbf{b})$ - $\varepsilon^* = \varepsilon$ In this transformation, the rotation vector $\omega$ transforms as an axial vector (weighted by $\det R$), and the action corresponds to the conjugation $a Z a^{-1}$ in the matrix representation of the group.

For an element $a$ of the Galilean group with rotation matrix $R$ and an element $Z$ of the Galilean Lie algebra with rotation vector component $\omega$, the rotation vector component of the adjoint action $a \cdot Z$ is given by $(\det R) R \omega$. This shows that the rotation vector $\omega$ transforms as an axial vector under the Galilean group.

Consider an element $a = (R, \mathbf{b}, \mathbf{c}, e)$ of the Galilean group, where $R \in O(3)$ is a rotation, $\mathbf{b}$ is a velocity boost, $\mathbf{c}$ is a spatial translation, and $e$ is a time translation. Let $Z = (\boldsymbol{\omega}, \boldsymbol{\beta}, \boldsymbol{\gamma}, \varepsilon)$ be an element of the Galilean Lie algebra. The velocity component $\boldsymbol{\beta}^*$ of the adjoint action $a \cdot Z$ is given by $$(a \cdot Z)_{\boldsymbol{\beta}} = R \boldsymbol{\beta} - ((\det R) R \boldsymbol{\omega}) \times \mathbf{b},$$ where $\times$ denotes the cross product in $\mathbb{R}^3$.

Let $a = (R, \mathbf{b}, \mathbf{c}, e)$ be an element of the Galilean group, where $R \in O(3)$ is a rotation, $\mathbf{b} \in \mathbb{R}^3$ is a velocity boost, $\mathbf{c} \in \mathbb{R}^3$ is a spatial translation, and $e \in \mathbb{R}$ is a time translation. Let $Z = (\omega, \beta, \gamma, \varepsilon)$ be an element of the Galilean Lie algebra, where $\omega, \beta, \gamma \in \mathbb{R}^3$ represent rotation, boost, and spatial translation components respectively, and $\varepsilon \in \mathbb{R}$ is the time translation component. The spatial translation component $(a \cdot Z)_\gamma$ of the adjoint action of $a$ on $Z$ is given by: $$(a \cdot Z)_\gamma = R \gamma + \varepsilon \mathbf{b} - e R \beta - ((\det R) R \omega) \times (\mathbf{c} - e \mathbf{b})$$ where $\times$ denotes the cross product in $\mathbb{R}^3$.

For any element $a$ of the 3D Galilean group and any element $Z = (\omega, \beta, \gamma, \varepsilon)$ of the Galilean Lie algebra, where $\varepsilon$ is the time translation component, the adjoint action of $a$ on $Z$ leaves the time translation component invariant: $(a \cdot Z)_\varepsilon = \varepsilon$.

The 3D Galilean group $\text{GalileanGroup}(3)$ acts on its Lie algebra through the adjoint action. This structure defines a group action (multiplicative action), which means that the identity element $1$ of the group acts as the identity transformation ($1 \cdot Z = Z$) and the action is compatible with the group composition law: $(a_1 a_2) \cdot Z = a_1 \cdot (a_2 \cdot Z)$ for all group elements $a_1, a_2$ and all Lie algebra elements $Z$.

Let $a = (R, \mathbf{b}, \mathbf{c}, e)$ be an element of the 3D Galilean group, where $R \in O(3)$ is a rotation matrix, $\mathbf{b} \in \mathbb{R}^3$ is a velocity boost, $\mathbf{c} \in \mathbb{R}^3$ is a spatial translation, and $e \in \mathbb{R}$ is a time translation. Let $Z = (\omega, \beta, \gamma, \varepsilon)$ be an element of the Galilean Lie algebra, where $\omega$ is the rotation vector, $\beta$ is the velocity component, $\gamma$ is the spatial translation component, and $\varepsilon$ is the time translation component. The adjoint action of the inverse element $a^{-1}$ on $Z$, denoted by $a^{-1} \cdot Z$, is given by the following tuple in the Lie algebra: $$a^{-1} \cdot Z = \left( (\det R) R^\top \omega, \, R^\top (\beta + \omega \times \mathbf{b}), \, R^\top (\gamma - \varepsilon \mathbf{b} + e \beta + \omega \times \mathbf{c}), \, \varepsilon \right)$$ where $R^\top$ denotes the transpose of $R$ and $\times$ denotes the vector cross product in $\mathbb{R}^3$.

For any element $a$ of the Galilean group, any element $Z$ of its Lie algebra, and any point $y$ in the evolution space of $N$ material points, the vector field associated with the adjoint action $a \cdot Z$ evaluated at the transformed point $a \cdot y$ is equal to the tangent action of $a$ applied to the vector field of $Z$ at $y$. That is, $$(a \cdot Z)_V(a \cdot y) = D(a \cdot)(y)(Z_V(y)),$$ where $Z_V$ denotes the vector field on the evolution space corresponding to the algebra element $Z$, and $D(a \cdot)(y)$ denotes the differential of the group action at the point $y$.

For a system of $N \geq 1$ particles, let $Z$ and $Z'$ be elements of the Galilean Lie algebra, and let $Z_V$ denote the vector field on the evolution space associated with an algebra element $Z$. If $Z_V(y) = Z'_V(y)$ for all points $y$ in the evolution space, then $Z = Z'$. In other words, the map $Z \mapsto Z_V$ from the Galilean Lie algebra to the space of vector fields on the evolution space is injective.

This equivalence identifies a Galilean torsor $\mu$ with its component representation in the product space $\mathbb{R}^3 \times \mathbb{R}^3 \times \mathbb{R}^3 \times \mathbb{R}$. The map sends $\mu$ to the quadruple $(\mathbf{l}, \mathbf{g}, \mathbf{p}, E)$, where $\mathbf{l}$ is the angular momentum, $\mathbf{g}$ is the static momentum, $\mathbf{p}$ is the linear momentum, and $E$ is the energy. This identification is used to transport the vector space structure from the product space to the space of Galilean torsors.

The space of Galilean torsors forms an additive abelian group. The addition and negation operations are defined component-wise via the isomorphism $\text{GalileanTorsor} \cong \mathbb{R}^3 \times \mathbb{R}^3 \times \mathbb{R}^3 \times \mathbb{R}$, which identifies a torsor with its physical components: angular momentum $\mathbf{l}$, static momentum $\mathbf{g}$, linear momentum $\mathbf{p}$, and energy $E$.

The space of Galilean torsors $\text{GalileanTorsor}$ is a vector space over $\mathbb{R}$. The vector space operations (addition and scalar multiplication) are defined component-wise through the isomorphism $\text{GalileanTorsor} \cong \mathbb{R}^3 \times \mathbb{R}^3 \times \mathbb{R}^3 \times \mathbb{R}$, which identifies a torsor with its physical components: angular momentum $\mathbf{l} \in \mathbb{R}^3$, static momentum $\mathbf{g} \in \mathbb{R}^3$, linear momentum $\mathbf{p} \in \mathbb{R}^3$, and energy $E \in \mathbb{R}$.

For any two Galilean torsors $\mu, \nu$ and any element $Z$ of the Galilean Lie algebra, the pairing of their sum with $Z$ is equal to the sum of their individual pairings with $Z$: $$(\mu + \nu)(Z) = \mu(Z) + \nu(Z)$$ Here, the pairing $\mu(Z)$ represents the duality between the torsor (representing physical quantities like angular momentum and energy) and the Lie algebra (representing infinitesimal transformations).

For any two Galilean torsors $\mu, \nu$ and any element $Z$ of the Galilean Lie algebra, the pairing of their difference with $Z$ satisfies $(\mu - \nu)(Z) = \mu(Z) - \nu(Z)$.

For any element $Z$ of the Galilean Lie algebra, the pairing of the zero Galilean torsor with $Z$ is zero. The pairing $\mu(Z)$ represents the evaluation of a torsor $\mu = (\mathbf{l}, \mathbf{g}, \mathbf{p}, E)$ on an algebra element $Z = (\omega, \beta, \gamma, \varepsilon)$, defined as $\mu(Z) = \mathbf{l} \cdot \omega - \mathbf{g} \cdot \beta + \mathbf{p} \cdot \gamma - E \varepsilon$.

For any scalar $k \in \mathbb{R}$, any Galilean torsor $\mu$, and any element $Z$ of the Galilean Lie algebra, the pairing of the scaled torsor $k \cdot \mu$ with $Z$ is equal to $k$ times the pairing of $\mu$ with $Z$, that is, $(k \cdot \mu)(Z) = k \cdot \mu(Z)$.

Let $\mu$ and $\nu$ be Galilean torsors. If for every element $Z$ of the Galilean Lie algebra, the pairing $\mu(Z) = \nu(Z)$ holds, then $\mu = \nu$. This pairing $\mu(Z)$ represents the evaluation of the torsor on an element of the Lie algebra.

This definition establishes the coadjoint action of the Galilean group on the space of Galilean torsors. For an element of the Galilean group $a = (R, \mathbf{b}, \mathbf{c}, e)$, where $R \in O(3)$ is a rotation (or reflection), $\mathbf{b}$ is a velocity vector, $\mathbf{c}$ is a space translation, and $e$ is a time translation, and for a Galilean torsor $\mu = (\mathbf{l}, \mathbf{g}, \mathbf{p}, E)$ representing angular momentum, static moment, linear momentum, and energy respectively, the action $a \cdot \mu$ is defined as: $$a \cdot \mu = \left( (\det R) R \mathbf{l} - \mathbf{b} \times R \mathbf{g} + \mathbf{c} \times R \mathbf{p}, \ R \mathbf{g} - e R \mathbf{p}, \ R \mathbf{p}, \ E + \langle R \mathbf{p}, \mathbf{b} \rangle \right)$$ where $\times$ denotes the vector cross product in $\mathbb{R}^3$ and $\langle \cdot, \cdot \rangle$ denotes the standard Euclidean inner product. This formula provides the closed-form expression for the coadjoint action such that $(a \cdot \mu)(Z) = \mu(a^{-1} \cdot Z)$ for any element $Z$ of the Galilean Lie algebra.

For any element $a$ of the 3D Galilean group, any Galilean torsor $\mu$, and any element $Z$ of the Galilean Lie algebra, the pairing of the transformed torsor $a \cdot \mu$ with $Z$ satisfies the identity: $$(a \cdot \mu)(Z) = \mu(a^{-1} \cdot Z)$$ This identity (Souriau 11.15) characterizes the coadjoint action of the Galilean group on its torsors.

The coadjoint action of the 3D Galilean group $G = \text{GalileanGroup}(3)$ on the space of Galilean torsors is a multiplicative group action. For any elements $a_1, a_2 \in G$ and any Galilean torsor $\mu$, the action satisfies the identity property $1 \cdot \mu = \mu$ and the associativity property $(a_1 a_2) \cdot \mu = a_1 \cdot (a_2 \cdot \mu)$. This action is defined by the duality relation $(a \cdot \mu)(Z) = \mu(a^{-1} \cdot Z)$ for any element $Z$ of the Galilean Lie algebra.

The coadjoint action of the 3D Galilean group $\text{GalileanGroup}(3)$ on the space of Galilean torsors is a distributive multiplicative action. For any Galilean transformation $a \in \text{GalileanGroup}(3)$ and any two Galilean torsors $\mu$ and $\nu$, the action satisfies: $$a \cdot (\mu + \nu) = a \cdot \mu + a \cdot \nu$$ and $$a \cdot 0 = 0$$ where addition and the zero element are defined by the additive group structure of the Galilean torsors. This property confirms that the coadjoint representation is additive on the space of torsors.

For any element $a$ of the 3D Galilean group, any real scalar $k \in \mathbb{R}$, and any Galilean torsor $\mu$, the coadjoint action of $a$ on the torsor commutes with scalar multiplication: $$a \cdot (k \mu) = k (a \cdot \mu)$$ This demonstrates that the coadjoint representation of the Galilean group on the space of torsors is linear with respect to real scalars.

Given an element $a = (R, \mathbf{b}, \mathbf{c}, e)$ of the 3D Galilean group `GalileanGroup 3`, where $R$ is an orthogonal rotation, $\mathbf{b} \in \mathbb{R}^3$ is a velocity boost, $\mathbf{c} \in \mathbb{R}^3$ is a spatial translation, and $e \in \mathbb{R}$ is a time translation, this function defines Souriau's mass 1-cocycle $\theta_0(a)$ as a `GalileanTorsor`. The cocycle is defined by the following components: $$\theta_0(a) = \left( \mathbf{c} \times \mathbf{b}, \mathbf{c} - e\mathbf{b}, \mathbf{b}, \frac{1}{2} \|\mathbf{b}\|^2 \right)$$ where $\times$ denotes the cross product in $\mathbb{R}^3$ and $\|\mathbf{b}\|^2$ is the squared Euclidean norm of the velocity vector $\mathbf{b}$.

Souriau's mass cocycle $\theta_0$ evaluated at the identity element $1$ of the 3D Galilean group is equal to the zero Galilean torsor: $$\theta_0(1) = 0$$ where $0$ represents the additive identity in the space of Galilean torsors.

Let $\theta_0: \text{GalileanGroup}(3) \to \text{GalileanTorsor}$ be Souriau's mass cocycle, which maps an element of the 3D Galilean group $a = (R, \mathbf{b}, \mathbf{c}, e)$ to a Galilean torsor defined by: $$\theta_0(a) = \left( \mathbf{c} \times \mathbf{b}, \mathbf{c} - e\mathbf{b}, \mathbf{b}, \frac{1}{2} \|\mathbf{b}\|^2 \right)$$ Then $\theta_0$ satisfies the 1-cocycle identity with respect to the coadjoint action of the Galilean group on the space of torsors. That is, for all $a, a' \in \text{GalileanGroup}(3)$, it holds that: $$\theta_0(a a') = a \cdot \theta_0(a') + \theta_0(a)$$ where $a \cdot \mu$ denotes the coadjoint action of the group element $a$ on the torsor $\mu$.

Let $\theta_0: \text{GalileanGroup}(3) \to \text{GalileanTorsor}$ be Souriau's mass 1-cocycle of the 3D Galilean group (where rotations are in $O(3)$). For any real number $M$, the map $a \mapsto M \theta_0(a)$ is a 1-coboundary with respect to the coadjoint action on the space of Galilean torsors if and only if $M = 0$. Specifically, there exists a fixed Galilean torsor $\mu_0$ such that $M \theta_0(a) = a \cdot \mu_0 - \mu_0$ for all $a \in \text{GalileanGroup}(3)$ if and only if $M = 0$.

Souriau's mass 1-cocycle $\theta_0$ of the 3D Galilean group (where rotations are in $O(3)$) is not a 1-coboundary with respect to the coadjoint action on Galilean torsors. Specifically, there exists no fixed Galilean torsor $\mu_0$ such that for all elements $a$ of the Galilean group, $\theta_0(a) = a \cdot \mu_0 - \mu_0$.

The proper Galilean group is the subgroup of the 3D Galilean group $\text{GalileanGroup}(3)$ consisting of elements $(R, v, c, e)$ such that the determinant of the rotation matrix $R$ is equal to $1$. In other words, it is the subgroup where the spatial transformations are restricted to proper rotations, meaning $R \in SO(3)$ rather than the full orthogonal group $O(3)$.

The restriction of Souriau's mass cocycle $\theta_0$ to the proper Galilean group in 3 dimensions (consisting of transformations with $\det R = 1$) is a 1-cocycle with respect to the coadjoint action on the space of Galilean torsors. Specifically, for any two elements $a_1, a_2$ of the proper Galilean group, the following identity holds: $$\theta_0(a_1 a_2) = \theta_0(a_1) + a_1 \cdot \theta_0(a_2)$$ where $\theta_0(a) = \left( \mathbf{c} \times \mathbf{b}, \mathbf{c} - e\mathbf{b}, \mathbf{b}, \frac{1}{2} \|\mathbf{b}\|^2 \right)$ for a transformation $a = (R, \mathbf{b}, \mathbf{c}, e)$, and $a_1 \cdot \theta_0(a_2)$ denotes the coadjoint action of $a_1$ on the torsor $\theta_0(a_2)$.

Let $\mathcal{G}_{\text{proper}}$ be the proper Galilean group (where rotations have determinant 1) and let $\theta_0: \mathcal{G}_{\text{proper}} \to \text{GalileanTorsor}$ be Souriau's mass cocycle. For any total mass $M \in \mathbb{R}$, the map $a \mapsto M \theta_0(a)$ is a 1-coboundary with respect to the coadjoint action of the group on the space of torsors if and only if $M = 0$. That is, there exists a fixed Galilean torsor $\mu_0$ such that for all $a \in \mathcal{G}_{\text{proper}}$, $M \theta_0(a) = a \cdot \mu_0 - \mu_0$, if and only if $M = 0$.

Let $\mathcal{G}_+$ denote the proper Galilean group in 3D, which consists of transformations $a = (R, \mathbf{b}, \mathbf{c}, e)$ where $R \in SO(3)$ is a proper rotation, $\mathbf{b} \in \mathbb{R}^3$ is a velocity boost, $\mathbf{c} \in \mathbb{R}^3$ is a spatial translation, and $e \in \mathbb{R}$ is a time translation. Souriau's mass 1-cocycle $\theta_0 : \mathcal{G}_+ \to \text{GalileanTorsor}$ is defined as: $$\theta_0(a) = \left( \mathbf{c} \times \mathbf{b}, \mathbf{c} - e\mathbf{b}, \mathbf{b}, \frac{1}{2} \|\mathbf{b}\|^2 \right)$$ This cocycle $\theta_0$ is not a 1-coboundary. That is, there does not exist a fixed Galilean torsor $\mu_0$ such that for all $a \in \mathcal{G}_+$, $\theta_0(a) = a \cdot \mu_0 - \mu_0$ holds under the coadjoint action of the group on the space of torsors.

Consider a system of $N$ material points with masses $m_j$ and total mass $M = \sum_{j=1}^N m_j$. Let $y$ be a point in the evolution space representing the state of the system (time, positions, and velocities). For any element $a$ of the 3D Galilean group $\mathcal{G}$, let $a \cdot y$ be the action of the group on the evolution space and $a \cdot \mu$ be the coadjoint action of the group on the space of Galilean torsors. The moment $\mu_m(y)$ of the system (comprising angular momentum, static moment, linear momentum, and energy) satisfies the following transformation law: $$\mu_m(a \cdot y) - a \cdot \mu_m(y) = M \theta_0(a)$$ where $\theta_0(a)$ is Souriau's mass cocycle. Specifically, for an element $a = (R, \mathbf{b}, \mathbf{c}, e)$ where $R \in O(3)$ is a rotation, $\mathbf{b}$ is a velocity boost, $\mathbf{c}$ is a spatial translation, and $e$ is a time translation, the cocycle is: $$\theta_0(a) = \left( \mathbf{c} \times \mathbf{b}, \mathbf{c} - e\mathbf{b}, \mathbf{b}, \frac{1}{2} \|\mathbf{b}\|^2 \right)$$