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$\def \cG {\cal G}$ $\def \RR {\mathbf R}$

The moment map is one of the most important constructions in symplectic mechanics. There are at least two main reasons for that:

1. The moment map $\mu : M \to \cG^*$ on a homogeneous symplectic manifold $(M,\omega)$ under the action of a Lie group $G$ is a covering onto its image [Sou70]. That classifies what is called now elementary systems, or elementary particles when the group $G$ is the Galilee group (Galilean mechanics) or Poincaré group (for relativity). This is called the theorem KKS for Kirillov-Kostant-Souriau.

2. In the case of a presymplectic system $(M,\omega)$, that is: $$ \dim\big(\ker (\omega) \big) = \mathrm{cst.} $$ Theorem (Noether-Souriau) The moment map $\mu$ is constant on the characteristics of $\omega$.

This theorem is the generalisation in the context of symplectic dynamic of the classical Noether theorem since in the case of $\omega = d\lambda$ and $\lambda$ is given by a Lagrangian, it coincides with the Noether theorem.

The characteristics of $\omega$ are the integral sub-manifolds of the distribution of tangent subspaces $$ x \mapsto \ker(\omega). $$ In symplectic mechanics the characteristics of a presymplectic form are regarded as the solutions of the dynamical system characterised by $\omega$ on the space of initial conditions $M$. Hence, the values of $\mu$ are the conserved quantities associated with the action of $G$.

In the case of the Galilee group, represented by the matricies: $$ g = \begin{pmatrix} A & b & c\\ 0 & 1 & e \\ 0 & 0 & 1 \end{pmatrix} \quad \text{with} \quad \begin{array}{l} A \in \mathrm{SO}(3) \\ b,c \in \RR^3 \\ e \in \RR. \end{array} $$ Acting on $\RR^3 \times \RR^3 \times \RR$, the space of initial conditions of a free particle: $$ \begin{pmatrix} A & b & c\\ 0 & 1 & e \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} r\\ t \\ 1 \end{pmatrix} = \begin{pmatrix} Ar + bt + c\\ t +e \\ 1 \end{pmatrix} $$ The moment map is given by: $$ \mu(r,v,t) = \left\{ \begin{array}{ll} l = m r \times v & \text{the kinetic momentum relative to $A \in \mathrm{SO}(3)$}. \\ p = mv & \text{the momentum relative to the boost $b \in \RR^3$. } \\ g = r - vt & \text{the center of mass relative to the displacement $c \in \RR^3$.}\\ E = {mv^2 \over 2} & \text{the kinetic energy relative to the time translation $e \in \RR$.} \end{array} \right. $$ That gives the 10 components of the moment map in Galilean mechanics for the exemple of a free particle. Of course, this extends to any free Galilean mechanical system described by a presymplectic manifold invariant by the group of Galilee. Actually this is the definition of a free Galilean dynamical system in symplectic mechanics.

Note. The moment map has been generalized in diffeology [PIZ10], and then the classification theorem above has been extended in to any symplectic manifold [DIZ22]:

Theorem. Every connected symplectic manifold $(M,\omega)$ is a coadjoint orbit of a central extension by the torus of periods $T_\omega$ of its group $\mathrm{Ham}(M,\omega)$ of Hamiltonian diffeomorphisms.

References:

[Sou70] Jean-Marie Souriau. Structure des Systèmes Dynamiques, Dunod 1970.

[PIZ10] Patrick Iglesias-Zemmour. The moment map in diffeology (2010). Memoirs of the American Mathematical Society. Volume 207.

[DIZ22] Paul Donato & Patrick Iglesias-Zemmour. Every symplectic manifold is a (linear) coadjoint orbit. Canadian Mathematical Bulletin, Volume 65, Issue 2, June 2022, pp. 345 - 360.

$\def \cG {\cal G}$ $\def \RR {\mathbf R}$

The moment map is one of the most important constructions in symplectic mechanics. There are at least two main reasons for that:

1. The moment map $\mu : M \to \cG^*$ on a homogeneous symplectic manifold $(M,\omega)$ under the action of a Lie group $G$ is a covering onto its image [Sou70]. That classifies what is called now elementary systems, or elementary particles when the group $G$ is the Galileo group (Galilean mechanics) or Poincaré group (for relativity). This is called the theorem KKS for Kirillov-Kostant-Souriau.

2. In the case of a presymplectic system $(M,\omega)$, that is: $$ \dim\big(\ker (\omega) \big) = \mathrm{cst.} $$ Theorem (Noether-Souriau) The moment map $\mu$ is constant on the characteristics of $\omega$.

This theorem is the generalisation in the context of symplectic dynamic of the classical Noether theorem since in the case of $\omega = d\lambda$ and $\lambda$ is given by a Lagrangian, it coincides with the Noether theorem.

The characteristics of $\omega$ are the integral sub-manifolds of the distribution of tangent subspaces $$ x \mapsto \ker(\omega). $$ In symplectic mechanics the characteristics of a presymplectic form are regarded as the solutions of the dynamical system characterised by $\omega$ on the space of initial conditions $M$. Hence, the values of $\mu$ are the conserved quantities associated with the action of $G$.

In the case of the Galileo group, represented by the matrices: $$ g = \begin{pmatrix} A & b & c\\ 0 & 1 & e \\ 0 & 0 & 1 \end{pmatrix} \quad \text{with} \quad \begin{array}{l} A \in \mathrm{SO}(3) \\ b,c \in \RR^3 \\ e \in \RR. \end{array} $$ Acting on $\RR^3 \times \RR^3 \times \RR$, the space of initial conditions of a free particle: $$ \begin{pmatrix} A & b & c\\ 0 & 1 & e \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} r\\ t \\ 1 \end{pmatrix} = \begin{pmatrix} Ar + bt + c\\ t +e \\ 1 \end{pmatrix} $$ The moment map is given by: $$ \mu(r,v,t) = \left\{ \begin{array}{ll} l = m r \times v & \text{the kinetic momentum relative to $A \in \mathrm{SO}(3)$}. \\ p = mv & \text{the momentum relative to the boost $b \in \RR^3$. } \\ g = r - vt & \text{the center of mass relative to the displacement $c \in \RR^3$.}\\ E = {mv^2 \over 2} & \text{the kinetic energy relative to the time translation $e \in \RR$.} \end{array} \right. $$ That gives the 10 components of the moment map in Galilean mechanics for the example of a free particle. Of course, this extends to any free Galilean mechanical system described by a presymplectic manifold invariant by the Galileo group. Actually this is the definition of a free Galilean dynamical system in symplectic mechanics.

Note. The moment map has been generalized in diffeology [PIZ10], and then the classification theorem above has been extended in to any symplectic manifold [DIZ22]:

Theorem. Every connected symplectic manifold $(M,\omega)$ is a coadjoint orbit of a central extension by the torus of periods $T_\omega$ of its group $\mathrm{Ham}(M,\omega)$ of Hamiltonian diffeomorphisms.

References:

[Sou70] Jean-Marie Souriau. Structure des Systèmes Dynamiques, Dunod 1970.

[PIZ10] Patrick Iglesias-Zemmour. The moment map in diffeology (2010). Memoirs of the American Mathematical Society. Volume 207.

[DIZ22] Paul Donato & Patrick Iglesias-Zemmour. Every symplectic manifold is a (linear) coadjoint orbit. Canadian Mathematical Bulletin, Volume 65, Issue 2, June 2022, pp. 345 - 360.

$\def \cG {\cal G}$ $\def \RR {\mathbf R}$

The moment map is one of the most important constructions in symplectic mechanics. There are at least two main reasons for that:

1. The moment map $\mu : M \to \cG^*$ on a homogeneous symplectic manifold $(M,\omega)$ under the action of a Lie group $G$ is a covering onto its image [Sou70]. That classifies what is called now elementary systems, or elementary particles when the group $G$ is the Galilee group (Galilean mechanics) or Poincaré group (for relativity). This is called the theorem KKS for Kirillov-Kostant-Souriau.

2. In the case of a presymplectic system $(M,\omega)$, that is: $$ \dim\big(\ker (\omega) \big) = \mathrm{cst.} $$ Theorem (Noether-Souriau) The moment map $\mu$ is constant on the characteristics of $\omega$.

This theorem is the generalisation in the context of symplectic dynamic of the classical Noether theorem since in the case of $\omega = d\lambda$ and $\lambda$ is given by a Lagrangian, it coincides with the Noether theorem.

The characteristics of $\omega$ are the integral sub-manifolds of the distribution of tangent subspaces $$ x \mapsto \ker(\omega). $$ In symplectic mechanics the characteristics of a presymplectic form are regarded as the solutions of the dynamical system characterised by $\omega$ on the space of initial conditions $M$. Hence, the values of $\mu$ are the conserved quantities associated with the action of $G$.

In the case of the Galilee group, represented by the matricies: $$ g = \begin{pmatrix} A & b & c\\ 0 & 1 & e \\ 0 & 0 & 1 \end{pmatrix} \quad \text{with} \quad \begin{array}{l} A \in \mathrm{SO}(3) \\ b,c \in \RR^3 \\ e \in \RR. \end{array} $$ Acting on $\RR^3 \times \RR^3 \times \RR$, the space of initial conditions of a free particle: $$ \begin{pmatrix} A & b & c\\ 0 & 1 & e \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} r\\ t \\ 1 \end{pmatrix} = \begin{pmatrix} Ar + bt + c\\ t +e \\ 1 \end{pmatrix} $$ The moment map is given by: $$ \mu(r,v,t) = \left\{ \begin{array}{ll} l = m r \times v & \text{the kinetic momentum relative to $A \in \mathrm{SO}(3)$}. \\ p = mv & \text{the momentum relative to the boost $b \in \RR^3$. } \\ g = r - vt & \text{the center of mass relative to the displacement $c \in \RR^3$.}\\ E = {mv^2 \over 2} & \text{the kinetic energy relative to the time translation $e \in \RR$.} \end{array} \right. $$ That gives the 10 components of the moment map in Galilean mechanics for the exemple of a free particle. Of course, this extends to any free Galilean mechanical system described by a presymplectic manifold invariant by the group of Galilee. Actually this is the definition of a free Galilean dynamical system in symplectic mechanics.

Note. The moment map has been generalized [PIZ10] in diffeology, and then the classification theorem above has been extended [DIZ22] in to any symplectic manifold:

Theorem. Every connected symplectic manifold $(M,\omega)$ is a coadjoint orbit of a central extension by the torus of periods $T_\omega$ of its group $\mathrm{Ham}(M,\omega)$ of Hamiltonian diffeomorphisms.

References:

[Sou70] Jean-Marie Souriau. Structure des Systèmes Dynamiques, Dunod 1970.

[PIZ10] Patrick Iglesias-Zemmour. The moment map in diffeology (2010). Memoirs of the American Mathematical Society. Volume 207.

[DIZ22] Paul Donato & Patrick Iglesias-Zemmour. Every symplectic manifold is a (linear) coadjoint orbit. Canadian Mathematical Bulletin, Volume 65, Issue 2, June 2022, pp. 345 - 360.

$\def \cG {\cal G}$ $\def \RR {\mathbf R}$

The moment map is one of the most important constructions in symplectic mechanics. There are at least two main reasons for that:

1. The moment map $\mu : M \to \cG^*$ on a homogeneous symplectic manifold $(M,\omega)$ under the action of a Lie group $G$ is a covering onto its image [Sou70]. That classifies what is called now elementary systems, or elementary particles when the group $G$ is the Galilee group (Galilean mechanics) or Poincaré group (for relativity). This is called the theorem KKS for Kirillov-Kostant-Souriau.

2. In the case of a presymplectic system $(M,\omega)$, that is: $$ \dim\big(\ker (\omega) \big) = \mathrm{cst.} $$ Theorem (Noether-Souriau) The moment map $\mu$ is constant on the characteristics of $\omega$.

This theorem is the generalisation in the context of symplectic dynamic of the classical Noether theorem since in the case of $\omega = d\lambda$ and $\lambda$ is given by a Lagrangian, it coincides with the Noether theorem.

The characteristics of $\omega$ are the integral sub-manifolds of the distribution of tangent subspaces $$ x \mapsto \ker(\omega). $$ In symplectic mechanics the characteristics of a presymplectic form are regarded as the solutions of the dynamical system characterised by $\omega$ on the space of initial conditions $M$. Hence, the values of $\mu$ are the conserved quantities associated with the action of $G$.

In the case of the Galilee group, represented by the matricies: $$ g = \begin{pmatrix} A & b & c\\ 0 & 1 & e \\ 0 & 0 & 1 \end{pmatrix} \quad \text{with} \quad \begin{array}{l} A \in \mathrm{SO}(3) \\ b,c \in \RR^3 \\ e \in \RR. \end{array} $$ Acting on $\RR^3 \times \RR^3 \times \RR$, the space of initial conditions of a free particle: $$ \begin{pmatrix} A & b & c\\ 0 & 1 & e \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} r\\ t \\ 1 \end{pmatrix} = \begin{pmatrix} Ar + bt + c\\ t +e \\ 1 \end{pmatrix} $$ The moment map is given by: $$ \mu(r,v,t) = \left\{ \begin{array}{ll} l = m r \times v & \text{the kinetic momentum relative to $A \in \mathrm{SO}(3)$}. \\ p = mv & \text{the momentum relative to the boost $b \in \RR^3$. } \\ g = r - vt & \text{the center of mass relative to the displacement $c \in \RR^3$.}\\ E = {mv^2 \over 2} & \text{the kinetic energy relative to the time translation $e \in \RR$.} \end{array} \right. $$ That gives the 10 components of the moment map in Galilean mechanics for the exemple of a free particle. Of course, this extends to any free Galilean mechanical system described by a presymplectic manifold invariant by the group of Galilee. Actually this is the definition of a free Galilean dynamical system in symplectic mechanics.

Note. The moment map has been generalized in diffeology [PIZ10], and then the classification theorem above has been extended in to any symplectic manifold [DIZ22]:

Theorem. Every connected symplectic manifold $(M,\omega)$ is a coadjoint orbit of a central extension by the torus of periods $T_\omega$ of its group $\mathrm{Ham}(M,\omega)$ of Hamiltonian diffeomorphisms.

References:

[Sou70] Jean-Marie Souriau. Structure des Systèmes Dynamiques, Dunod 1970.

[PIZ10] Patrick Iglesias-Zemmour. The moment map in diffeology (2010). Memoirs of the American Mathematical Society. Volume 207.

[DIZ22] Paul Donato & Patrick Iglesias-Zemmour. Every symplectic manifold is a (linear) coadjoint orbit. Canadian Mathematical Bulletin, Volume 65, Issue 2, June 2022, pp. 345 - 360.

$\def \cG {\cal G}$ $\def \RR {\mathbf R}$

The moment map is one of the most important constructions in symplectic mechanics. There are at least two main reasons for that:

1. The moment map $\mu : M \to \cG^*$ on a homogeneous symplectic manifold $(M,\omega)$ under the action of a Lie group $G$ is a covering onto its image [Sou70]. That classifies what is called now elementary systems, or elementary particles when the group $G$ is the Galilee group (Galilean mechanics) or Poincaré group (for relativity). This is called the theorem KKS for Kirillov-Kostant-Souriau.

2. In the case of a presymplectic system $(M,\omega)$, that is: $$ \dim\big(\ker (\omega) \big) = \mathrm{cst.} $$ Theorem (Noether-Souriau) The moment map $\mu$ is constant on the characteristics of $\omega$.

This theorem is the generalisation in the context of symplectic dynamic of the classical Noether theorem since in the case of $\omega = d\lambda$ and $\lambda$ is given by a Lagrangian, it coincides with the Noether theorem.

The characteristics of $\omega$ are the integral sub-manifolds of the distribution of tangent subspaces $$ x \mapsto \ker(\omega). $$ In symplectic mechanics the characteristics of a presymplectic form are regarded as the solutions of the dynamical system characterised by $\omega$ on the space of initial conditions $M$. Hence, the values of $\mu$ are the conserved quantities associated with the action of $G$.

In the case of the Galilee group, represented by the matricies: $$ g = \begin{pmatrix} A & b & c\\ 0 & 1 & e \\ 0 & 0 & 1 \end{pmatrix} \quad \text{with} \quad \begin{array}{l} A \in \mathrm{SO}(3) \\ b,c \in \RR^3 \\ e \in \RR. \end{array} $$ Acting on $\RR^3 \times \RR^3 \times \RR$, the space of initial conditions of a free particle: $$ \begin{pmatrix} A & b & c\\ 0 & 1 & e \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} r\\ t \\ 1 \end{pmatrix} = \begin{pmatrix} Ar + bt + c\\ t +e \\ 1 \end{pmatrix} $$ The moment map is given by: $$ \mu(r,v,t) = \left\{ \begin{array}{ll} l = m r \times v & \text{the kinetic momentum relative to $A \in \mathrm{SO}(3)$}. \\ p = mv & \text{the momentum relative to the boost $b \in \RR^3$. } \\ g = r - vt & \text{the center of mass relative to the displacement $c \in \RR^3$.}\\ E = {mv^2 \over 2} & \text{the kinetic energy relative to the time translation $e \in \RR$.} \end{array} \right. $$ That gives the 10 components of the moment map in Galilean mechanics for the exemple of a free particle. Of course, this extends to any free Galilean mechanical system described by a presymplectic manifold invariant by the group of Galilee. Actually this is the definition of a free Galilean dynamical system in symplectic mechanics.

Note. The moment map has been generalized [PIZ10] in diffeology, and then the classification theorem above has been extended [IZD21] in to any symplectic manifold:

Theorem. Every connected symplectic manifold $(M,\omega)$ is a coadjoint orbit of a central extension by the torus of periods $T_\omega$ of its group of Hamiltonian diffeomorphisms.

References:

[Sou70] Jean-Marie Souriau. Structure des Systèmes Dynamiques, Dunod 1970.

[PIZ10] Patrick Iglesias-Zemmour. The moment map in diffeology (2010). Memoirs of the American Mathematical Society. Volume 207.

[DIZ22] Paul Donato & Patrick Iglesias-Zemmour. Every symplectic manifold is a (linear) coadjoint orbit. Canadian Mathematical Bulletin, Volume 65, Issue 2, June 2022, pp. 345 - 360.

$\def \cG {\cal G}$ $\def \RR {\mathbf R}$

The moment map is one of the most important constructions in symplectic mechanics. There are at least two main reasons for that:

1. The moment map $\mu : M \to \cG^*$ on a homogeneous symplectic manifold $(M,\omega)$ under the action of a Lie group $G$ is a covering onto its image [Sou70]. That classifies what is called now elementary systems, or elementary particles when the group $G$ is the Galilee group (Galilean mechanics) or Poincaré group (for relativity). This is called the theorem KKS for Kirillov-Kostant-Souriau.

2. In the case of a presymplectic system $(M,\omega)$, that is: $$ \dim\big(\ker (\omega) \big) = \mathrm{cst.} $$ Theorem (Noether-Souriau) The moment map $\mu$ is constant on the characteristics of $\omega$.

This theorem is the generalisation in the context of symplectic dynamic of the classical Noether theorem since in the case of $\omega = d\lambda$ and $\lambda$ is given by a Lagrangian, it coincides with the Noether theorem.

The characteristics of $\omega$ are the integral sub-manifolds of the distribution of tangent subspaces $$ x \mapsto \ker(\omega). $$ In symplectic mechanics the characteristics of a presymplectic form are regarded as the solutions of the dynamical system characterised by $\omega$ on the space of initial conditions $M$. Hence, the values of $\mu$ are the conserved quantities associated with the action of $G$.

In the case of the Galilee group, represented by the matricies: $$ g = \begin{pmatrix} A & b & c\\ 0 & 1 & e \\ 0 & 0 & 1 \end{pmatrix} \quad \text{with} \quad \begin{array}{l} A \in \mathrm{SO}(3) \\ b,c \in \RR^3 \\ e \in \RR. \end{array} $$ Acting on $\RR^3 \times \RR^3 \times \RR$, the space of initial conditions of a free particle: $$ \begin{pmatrix} A & b & c\\ 0 & 1 & e \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} r\\ t \\ 1 \end{pmatrix} = \begin{pmatrix} Ar + bt + c\\ t +e \\ 1 \end{pmatrix} $$ The moment map is given by: $$ \mu(r,v,t) = \left\{ \begin{array}{ll} l = m r \times v & \text{the kinetic momentum relative to $A \in \mathrm{SO}(3)$}. \\ p = mv & \text{the momentum relative to the boost $b \in \RR^3$. } \\ g = r - vt & \text{the center of mass relative to the displacement $c \in \RR^3$.}\\ E = {mv^2 \over 2} & \text{the kinetic energy relative to the time translation $e \in \RR$.} \end{array} \right. $$ That gives the 10 components of the moment map in Galilean mechanics for the exemple of a free particle. Of course, this extends to any free Galilean mechanical system described by a presymplectic manifold invariant by the group of Galilee. Actually this is the definition of a free Galilean dynamical system in symplectic mechanics.

Note. The moment map has been generalized [PIZ10] in diffeology, and then the classification theorem above has been extended [DIZ22] in to any symplectic manifold:

Theorem. Every connected symplectic manifold $(M,\omega)$ is a coadjoint orbit of a central extension by the torus of periods $T_\omega$ of its group $\mathrm{Ham}(M,\omega)$ of Hamiltonian diffeomorphisms.

References:

[Sou70] Jean-Marie Souriau. Structure des Systèmes Dynamiques, Dunod 1970.

[PIZ10] Patrick Iglesias-Zemmour. The moment map in diffeology (2010). Memoirs of the American Mathematical Society. Volume 207.

[DIZ22] Paul Donato & Patrick Iglesias-Zemmour. Every symplectic manifold is a (linear) coadjoint orbit. Canadian Mathematical Bulletin, Volume 65, Issue 2, June 2022, pp. 345 - 360.