Definite quadratic form - Wikipedia

Definite quadratic form - Wikipedia: "the quadratic form is called positive definite or negative definite."



'via Blog this'

Riemannian manifold - Wikipedia

Riemannian manifold - Wikipedia: " real, smooth manifold M equipped with an inner product {\displaystyle g_{p}} on the tangent space {\displaystyle T_{p}M} at each point {\displaystyle p} that varies smoothly from point to point "



'via Blog this'

Lie Groups (why do mirrors reverse left and right, but not up and down?)

Lie Groups: "Digression: the well-known puzzle, ``why do mirrors reverse left and right, but not up and down?'' is resolved mathematically by pointing out that a mirror perpendicular to the y-axis performs the reflection:

"



'via Blog this'

Group action - Wikipedia

Group action - Wikipedia: "stabilizer subgroup of G with respect to x (also called the isotropy group "



'via Blog this'

Differential of a function - Wikipedia

Differential of a function - Wikipedia: "The differential dy is defined by

{\displaystyle dy=f'(x)\,dx,}
where {\displaystyle f'(x)} is the derivative of f with respect to x, and dx is an additional real variable (so that dy is a function of x and dx). "



'via Blog this'

What first-order linearity really is

‘to first order everything is linear’. This is tautological, of course, but it really says
that most functions are differentiable and we can approximate by the first derivative.



'via Blog this'

Differential (infinitesimal) - Wikipedia

Differential (infinitesimal) - Wikipedia: "the differential dy of y is related to dx by the formula

{\displaystyle \mathrm {d} y={\frac {\mathrm {d} y}{\mathrm {d} x}}\,\mathrm {d} x,}
where dy/dx denotes the derivative of y with respect to x."



'via Blog this'

One-form - Wikipedia

One-form - Wikipedia: "Many real-world concepts can be described as one-forms:

Indexing into a vector"



'via Blog this'

One-form - Wikipedia

One-form - Wikipedia: "Many real-world concepts can be described as one-forms:

"



'via Blog this'

Linear form - Wikipedia

Linear form - Wikipedia: "Any linear functional L is either trivial (equal to 0 everywhere) or surjective onto the scalar field."



'via Blog this'

Linear form - Wikipedia

Linear form - Wikipedia: "A state of a quantum mechanical system can be identified with a linear functional."



'via Blog this'

Quadrature (mathematics) - Wikipedia

Quadrature (mathematics) - Wikipedia: "In mathematics, quadrature is a historical term which means determining area."



'via Blog this'

One-form - Wikipedia

One-form - Wikipedia: "The usage of one-form in this context usually distinguishes the one-forms from higher-degree multilinear functionals on the space"



'via Blog this'

List of mathematical jargon - Wikipedia

List of mathematical jargon - Wikipedia: "canonical"



'via Blog this'

Function space - Wikipedia

Function space - Wikipedia: " if X is also vector space over F, the set of linear maps X → V form a vector space over F with pointwise operations (often denoted Hom(X,V)). One such space is the dual space of V: the set of linear functionals V → F with addition and scalar multiplication defined pointwise."



'via Blog this'

Function space - Wikipedia

Function space - Wikipedia: "the set of functions from any set X into a vector space have a natural vector space structure given by pointwise addition and scalar multiplication."



'via Blog this'

Pointwise - Wikipedia

Pointwise - Wikipedia: "By taking some algebraic structure {\displaystyle A} in the place of {\displaystyle R} , we can turn the set of all functions {\displaystyle X} to the carrier set of {\displaystyle A} into an algebraic structure of the same type in an analogous way."



'via Blog this'

Hamiltonian

It turns out that giving the position and momentum of a particle specifies a point
in a space called the ‘phase space’ or ‘state space’ of the particle. Mathematically this is often a
‘cotangent bundle’ — an important concept from the theory of manifolds. A cotangent bundle is
an example of a ‘symplectic manifold’, which in turn is an example of a more general thing called a
‘Poisson manifold’.



'via Blog this'

Classical mechanics observables a commutative and Lie algebra

In fact for all classical mechanics problems, the algebra of observables C∞(X) is always both a
commutative algebra and a Lie algebra, but even better, they fit together to form a Poisson algebra.



'via Blog this'

Observables

In our example of a particle in R
n, we said an observable is a smooth function
F: R
n × R
n → R
by
(q, p) 7→ F(q, p)
from the phase space to R, sending each point of phase space (or state of our system) to the value
of the observables.



'via Blog this'

configuration vs phase spaces

Hamiltonian mechanics focuses not on the configuration space but on the phase space or
state space, where a point specifies the position and momentum of the system



'via Blog this'

Lorentz group

Schuller lecture 13 defines Lorentz group in a general way as a vector space equipped with a pseudo inner product (that is little bit less positive definite)

Configuration space (physics) - Wikipedia

Configuration space (physics) - Wikipedia: "In classical mechanics, the parameters that define the configuration of a system are called generalized coordinates, and the vector space defined by these coordinates is called the configuration space of the physical system. It is often the case that these parameters satisfy mathematical constraints, such that the set of actual configurations of the system is a manifold in the space of generalized coordinates. This manifold is called the configuration manifold of the system.

"



'via Blog this'

Configuration space (mathematics) - Wikipedia

Configuration space (mathematics) - Wikipedia: "these are used to describe the state of a whole system as a single point in a high-dimensional space."



'via Blog this'

schuller | KUPDF

schuller | KUPDF:



'via Blog this'

Picturing the Correspondences

Picturing the Correspondences: "Since the action preserves distances, one can also consider the action of the group just on the set of vectors of norm 1, i.e., on the sphere.)"



'via Blog this'

Center (group theory) - Wikipedia

Center (group theory) - Wikipedia: "The center of an abelian group, G, is all of G.
"



'via Blog this'

Symplectic group - Wikipedia

Symplectic group - Wikipedia: "Since all symplectic matrices have determinant 1, the symplectic group is a subgroup of the special linear group SL(2n, F).

Notational warning: What is here called Sp(2n, F) is often referred to as Sp(n, F).

More abstractly, the symplectic group can be defined as the set of linear transformations of a 2n-dimensional vector space over F that preserve a non-degenerate, skew-symmetric, bilinear form, see classical group for this definition. Such a vector space is called a symplectic vector space. The symplectic group of an abstract symplectic vector space V is also denoted Sp(V)."



'via Blog this'

Unitary group - Wikipedia

Unitary group - Wikipedia: "Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group.

In the simple case n = 1, the group U(1) corresponds to the circle group, consisting of all complex numbers with absolute value 1 under multiplication. All the unitary groups contain copies of this group."



'via Blog this'

Lie Groups

Lie Groups: " and . These are, colloquially, the groups of rotations in 2-space and 3-space. O(n)  is the group of reflections and rotations."



'via Blog this'

Homogeneous space - Wikipedia

Homogeneous space - Wikipedia: "homogeneous space can be thought of as a coset space without a choice of origin."



'via Blog this'

Group action - Wikipedia

Group action - Wikipedia: "Free (or semiregular or fixed point free) if, given g, h in G, the existence of an x in X with g⋅x = h⋅x implies g = h. Equivalently: if g is a group element and there exists an x in X with g⋅x = x (that is, if g has at least one fixed point), then g is the identity. "



"fix point free" that is, the action transforms free of fixed points

'via Blog this'

Group action - Encyclopedia of Mathematics

Group action - Encyclopedia of Mathematics: "If the homomorphism ρρ is injective the action is faithful: GG may be regarded as a subgroup of SXSX"



'via Blog this'

Transitive Group Action -- from Wolfram MathWorld

Transitive Group Action -- from Wolfram MathWorld: "The space , which has a transitive group action, is called a homogeneous space when the group is a Lie group."



(w) Regular (or simply transitive or sharply transitive) if it is both transitive and free; this is equivalent to saying that for every two x, y in X there exists precisely one g in Gsuch that g⋅x = y. In this case, X is called a principal homogeneous space for G or a G-torsor. 

'via Blog this'

Group action - Wikipedia

Group action - Wikipedia: "The group action is transitive if and only if it has exactly one orbit, i.e., if there exists x in X with G⋅x = X. This is the case if and only if G⋅x = X for all x in X.

"



'via Blog this'

Orthogonal Group -- from Wolfram MathWorld

Orthogonal Group -- from Wolfram MathWorld: "In fact, the orthogonal group is a smooth -dimensional submanifold"
Because the orthogonal group is a group and a manifold, it is a Lie group. 

... preserve the quadratic form , and so they also preserve circles , which are the group orbits.

'via Blog this'

Group Orbit -- from Wolfram MathWorld

Group Orbit -- from Wolfram MathWorld: "For example, consider the action by the circle group on the sphere by rotations along its axis. Then the north pole is an orbit, as is the south pole. The equator is a one-dimensional orbit, as is a general orbit, corresponding to a line of latitude.

"



'via Blog this'

torsors

torsors: " This combination of translations and dilations arises because R is not just a group, but a ring. "



'via Blog this'

torsors

torsors: "You can always pretend a torsor is a group. But, it involves an arbitrary choice!"

:

Any group G is a G-torsor, and every other G-torsor is isomorphic to G - but not canonically!



'via Blog this'

Group representation - Wikipedia

Group representation - Wikipedia: "formally, a "representation" means a homomorphism from the group to the automorphism group of an object"



'via Blog this'