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angle between planes The angle between angular variables Let M be a manifold and
1
two planes is given by the angle between the two S the unit circle. A smooth map ω : M → S 1
normal vectors to these planes. is called an angular variable on M.
n
angle between vectors Let u ∈ R and v ∈ angular velocity If a particle is moving in
n
n
R be two vectors in R . The angle θ between u a plane, its angular velocity about a point in the
and v is given by plane is the rate of change per unit time of the
angle between a fixed line and the line joining
u · v
cos θ = , the moving particle to the fixed point.
u v
n anion A monoatomic or polyatomic species
where the dot product u · v = i=1 u v and the
i i
2 n 2 having one or more elementary charges of the
norm is u = u ,if u = (u , ..., u ) and
i=1 i 1 n
v = (v , ..., v ). electron.
1 n
angle of depression The angle between the annihilation operator For the harmonic
2 2
1
2
horizontal plane and the line from the observer’s oscillator with Hamiltonian H = (p + ω q )
2
eye to some object lower than the line of her eyes. the annihilation operator a is given by a =
1
∗
√ (p −iωq). The creation operator a is given
2ω
angle of elevation The angle between the ∗ 1 ω ∗
by a = √ (p + miωq). Then H = (aa +
2
horizontal plane and the line from the observer’s 2ω √
∗
a a) and we have aψ(N) = Nψ(N − 1) and
eye to some object above her eyes. √
∗
a ψ(N) = N + 1ψ(N + 1).
angle of incidence The angle that a line (as
annihilator Let X be a vector space, X its
∗
a ray of light) falling on a surface or interface
dual vector space, and Y a subspace of X. The
makes with the normal vector drawn at the point
⊥ ⊥
of incidence to that surface. annihilator M of M is defined as M ={f ∈
∗
X | f(x) = 0,for all x ∈ M}.
angle of reflection The angle between a
reflected ray and the normal vector drawn at the annulus The region of a plane bounded by
point of incidence to a reflecting surface. two concentric circles in the plane. Let R> r,
the annulus A determined by the two circles of
angle of refraction The angle between a radius R and r, respectively, (centered at 0) is
refracted ray and the normal vector drawn at the given by
point of incidence to the interface at which the
2
refraction occurs. A ={ x = (x, y) ∈ R | r< x <R}
2 2
angular Measured by angle. where x = x + y .
angularacceleration Therateofchangeper anomalies In quantum field theories anom-
unit time of angular velocity; i.e., if the angular alies are quantum effects of conservation laws;
velocity is represented by a vector ω along the i.e., if one has a conservation law at the classical
axis of rotation, then the angular acceleration α level which is not true at the quantum level, this
d ω
is given by α = . is expressed by an anomaly, e.g., scale invariance
dt
is violated when quantized, which gives rise to a
angular momentum, L L L (or moment of momen- scale factor, the anomaly.
tum of a particle about a point) A vector
quantity equal to the vector product of the Anosov system A diffeomorphism on a
position vector of the particle and its momentum, manifold which has a hyperbolic structure every-
d
L = r × p where r(t) = r(t) is the velocity where is called an Anosov system.
dt
vector and p = m · r is the momentum. For spe-
cial angular momenta of particles in atomic and ansatz An “assumed form” for a solution; a
molecular physics different symbols are used. simplified assumption.
© 2003 by CRC Press LLC