15.4 Examples of Hamiltonian group actions

15.4 Examples of Hamiltonian group actions

Concept links · terms present in this machine draft; source roles are unverified: vector space · dual space · Lie algebra · group action

Some examples of Hamiltonian group actions are the following:

• For $d=3,$ an element $a$ of the translation group $G={\mathbf{R}}^3$ acts on the phase space $M={\mathbf{R}}^6$ by translation

$$m\in M\to \mathbf{a}\cdot m\in M$$

such that the coordinates satisfy

$$\mathbf{q}(\mathbf{a}\cdot m)=\mathbf{q}+\mathbf{a},\mathbf{p}(\mathbf{a}\cdot m)=\mathbf{p}(m)$$

Taking a to be the corresponding element in the Lie algebra of $G={\mathbf{R}}^3$ 2 the vector field on $M$ corresponding to this action (by 15.10) is

$$X_{\mathbf{a}}=a_1\frac{\partial }{\partial q_1}+a_2\frac{\partial }{\partial q_2}+a_3\frac{\partial }{\partial q_3}$$

and the moment map is given by

$${\mu }_{\mathbf{a}}(m)=\mathbf{a}\cdot \mathbf{p}(m)\text{\hspace{1em}(15.12)}$$

This can be interpreted as a function $a$ · $p$ on $M$ for each element $a$ of the Lie algebra, or as an element $\mathbf{p}(m)$ of the dual of the Lie algebra ${\mathbf{R}}^3$ for each point $m$ $\in \mathcal{M}$

• For another example, consider the action of the group $G=SO(3)$ of rotations on phase space $M={\mathbf{R}}^6$ given by performing the same $SO(3)$ rotation on position and momentum vectors. This gives a map from so(3) to vector fields on ${\mathbf{R}}^6$ , taking for example

$$l_1\in \mathfrak{so}(3)\to X_{l_1}=-q_3\frac{\partial }{\partial q_2}+q_2\frac{\partial }{\partial q_3}-p_3\frac{\partial }{\partial p_2}+p_2\frac{\partial }{\partial p_3}$$

(this is the vector field for an infinitesimal counter-clockwise rotation in the $q_2-q_3$ and $p_2-p_3$ planes, in the opposite direction to the case of the vector field $X_{\frac{1}{2}(q^2+p^2)}$ in the qp plane of section 15.2). The moment map here gives the usual expression for the 1-component of the angular momentum

$${\mu }_{l_1}=q_2{p}_3-q_3{p}_2$$

since one can check from equation 15.2 that $X_{l_1}=X_{{\mu }_{l_1}}$ . On basis elements of so(3) one has

$${\mu }_{l_j}(m)=(\mathbf{q}(m)\times \mathbf{p}(m){)}_j$$

Formulated as a map from M to $\mathfrak{so}(3{)}^{\ast }$ , the moment map is

$$\mu (m)(\mathbf{l})=(\mathbf{q}(m)\times \mathbf{p}(m))\cdot \mathbf{l}$$

where $\mathbf{l}\in \mathfrak{so}(3)$

• While most of the material of this chapter also applies to the case of a general symplectic manifold $M$, the case of M a vector space has the feature that G can be taken to be a group of linear transformations of M, and the moment map will give quadratic polynomials. The previous example is a special case of this and more general linear transformations will be studied in great detail in later chapters. In this linear case it turns out that it is generally best to work not with $M$ but with its dual space $M$.

Previous section: 15.3 Group actions on M and the moment map · Next section: 15.5 The dual of a Lie algebra and symplectic geometry · 中文版本

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来源版本：2025-10-20

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