Ø (disambiguation)

In mathematics, specifically in symplectic geometry, the momentum map (or moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The moment map generalizes the classical notions of linear and angular momentum. It is an essential ingredient in various constructions of symplectic manifolds, including symplectic (Marsden–Weinstein) quotients, discussed below, and symplectic cuts and sums.

Formal definition

Let M be a manifold with symplectic form ω. Suppose that a Lie group G acts on M via symplectomorphisms (that is, the action of each g in G preserves ω). Let ${\mathfrak {g}}$ be the Lie algebra of G, ${\mathfrak {g}}^{*}$ its dual, and

$\langle ,\rangle :{\mathfrak {g}}^{*}\times {\mathfrak {g}}\to \mathbf {R}$

the pairing between the two. Any ξ in ${\mathfrak {g}}$ induces a vector field ρ(ξ) on M describing the infinitesimal action of ξ. To be precise, at a point x in M the vector $\rho (\xi )_{x}$ is

$\left.{\frac {d}{dt}}\right|_{t=0}\exp(t\xi )\cdot x,$

where $\exp :{\mathfrak {g}}\to G$ is the exponential map and $\cdot$ denotes the G-action on M. Let $\iota _{\rho (\xi )}\omega \,$ denote the contraction of this vector field with ω. Because G acts by symplectomorphisms, it follows that $\iota _{\rho (\xi )}\omega \,$ is closed for all ξ in ${\mathfrak {g}}$.

A moment map for the G-action on (M, ω) is a map $\mu :M\to {\mathfrak {g}}^{*}$ such that

$d(\langle \mu ,\xi \rangle )=\iota _{\rho (\xi )}\omega$

for all ξ in ${\mathfrak {g}}$. Here $\langle \mu ,\xi \rangle$ is the function from M to R defined by $\langle \mu ,\xi \rangle (x)=\langle \mu (x),\xi \rangle$. The moment map is uniquely defined up to an additive constant of integration.

A moment map is often also required to be G-equivariant, where G acts on ${\mathfrak {g}}^{*}$ via the coadjoint action. If the group is compact or semisimple, then the constant of integration can always be chosen to make the moment map coadjoint equivariant; however in general the coadjoint action must be modified to make the map equivariant (this is the case for example for the Euclidean group). The modification is by a 1-cocycle on the group with values in ${\mathfrak {g}}^{*}$, as first described by Souriau (1970).

Hamiltonian group actions

The definition of the moment map requires $\iota _{\rho (\xi )}\omega$ to be exact. In practice it is useful to make an even stronger assumption. The G-action is said to be Hamiltonian if and only if the following conditions hold. First, for every ξ in ${\mathfrak {g}}$ the one-form $\iota _{\rho (\xi )}\omega$ is exact, meaning that it equals $dH_{\xi }$ for some smooth function

$H_{\xi }:M\to \mathbf {R} .$

If this holds, then one may choose the $H_{\xi }$ to make the map $\xi \mapsto H_{\xi }$ linear. The second requirement for the G-action to be Hamiltonian is that the map $\xi \mapsto H_{\xi }$ be a Lie algebra homomorphism from ${\mathfrak {g}}$ to the algebra of smooth functions on M under the Poisson bracket.

If the action of G on (M, ω) is Hamiltonian in this sense, then a moment map is a map $\mu :M\to {\mathfrak {g}}^{*}$ such that writing $H_{\xi }=\langle \mu ,\xi \rangle$ defines a Lie algebra homomorphism $\xi \mapsto H_{\xi }$ satisfying $\rho (\xi )=X_{H_{\xi }}$. Here $X_{H_{\xi }}$ is the vector field of the Hamiltonian $H_{\xi }$, defined by

$\iota _{X_{H_{\xi }}}\omega =dH_{\xi }.$

Examples

In the case of a Hamiltonian action of the circle G = U(1), the Lie algebra dual ${\mathfrak {g}}^{*}$ is naturally identified with R, and the moment map is simply the Hamiltonian function that generates the circle action.

Another classical case occurs when M is the cotangent bundle of R³ and G is the Euclidean group generated by rotations and translations. That is, G is a six-dimensional group, the semidirect product of SO(3) and R³. The six components of the moment map are then the three angular momenta and the three linear momenta.

Symplectic quotients

Suppose that the action of a compact Lie group G on the symplectic manifold (M, ω) is Hamiltonian, as defined above, with moment map $\mu :M\to {\mathfrak {g}}^{*}$. From the Hamiltonian condition it follows that $\mu ^{-1}(0)$ is invariant under G.

Assume now that 0 is a regular value of μ and that G acts freely and properly on $\mu ^{-1}(0)$. Thus $\mu ^{-1}(0)$ and its quotient $\mu ^{-1}(0)/G$ are both manifolds. The quotient inherits a symplectic form from M; that is, there is a unique symplectic form on the quotient whose pullback to $\mu ^{-1}(0)$ equals the restriction of ω to $\mu ^{-1}(0)$. Thus the quotient is a symplectic manifold, called the Marsden–Weinstein quotient, symplectic quotient or symplectic reduction of M by G and is denoted $M/\!\!/G$. Its dimension equals the dimension of M minus twice the dimension of G.

See also

• Poisson-Lie group
• Toric manifold
• Geometric Mechanics

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

• J.-M. Souriau, Structure des systèmes dynamiques, Maîtrises de mathématiques, Dunod, Paris, 1970. ISSN 0750-2435.
• S. K. Donaldson and P. B. Kronheimer, The Geometry of Four-Manifolds, Oxford Science Publications, 1990. ISBN 0-19-850269-9.
• Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, Oxford Science Publications, 1998. ISBN 0-19-850451-9.
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534