aarjaneiro/planetmath huggingface dataset:
affine parameter

affine parameter

Given a

geodesic curve

, an

affine parameterization

for that curve is a parameterization by a

parameter

t

such that the

parametric equations

for the curve satisfy the

geodesic equation

.

Put another way, if one picks a parameterization of a geodesic curve by an arbitrary parameter

s

and sets

u

μ

=

d

⁢

x

μ

/

d

⁢

s

, then we have

u

μ

⁢

∇

μ

⁡

u

ν

=

f

⁢

(

s

)

⁢

u

ν

for some function

f

. In general, the

right hand side

of this equation does not equal zero — it is only zero in the special case where

t

is an

affine parameter

.

The reason for the name “affine parameter” is that, if

t

1

and

t

2

are affine parameters for the same geodesic curve, then they are related by an affine transform, i.e. there exist constants

a

and

b

such that

t

1

=

a

⁢

t

2

+

b

Conversely, if

t

is an affine parameter, then

a

⁢

t

+

b

is also an affine parameter.

From this it follows that an affine parameter

t

is uniquely determined if we specify its value at two points on the

geodesic

or if we specify both its value and the value of

d

⁢

x

μ

/

d

⁢

t

at a single point of the geodesic.

Title

affine parameter

Canonical

name

AffineParameter

Date of creation

2013-03-22 14:35:47

Last modified on

2013-03-22 14:35:47

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

9

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 53C22

Defines

affinely-parameterized

Generated on Fri Feb 9 22:28:05 2018 by

LaTeXML
geodesic

geodesic

Let

M

be a

differentiable manifold

(at least two times

differentiable

) with

affine connection

∇

. The solution to the equation

∇

γ

˙

⁡

γ

˙

=

0

defined in the

interval

[

0

,

a

]

, is called a

geodesic

or a

geodesic curve

.
It can be shown that if

∇

is a

Levi-Civita connection

and

a

is ‘small enough’, then the curve

γ

is the shortest possible curve between the points

γ

⁢

(

0

)

and

γ

⁢

(

a

)

, and is often referred to as a

minimizing geodesic

between these points.

Conversely, any curve which minimizes the between two arbitrary points in a

manifold

, is a geodesic.

examples of geodesics includes

straight lines

in

Euclidean space

(

ℝ

n

) and

great circles

on spheres (such as the equator of earth). The latter of which is not minimizing if the geodesic from the point

p

is extended beyond its

antipodal point

. This example also points out to us that between any two points there may be more than one geodesic. In fact, between a point and its antipodal point on the sphere, there are an infinite number of geodesics. Given a

p

, it is also a

property

for a point

q

(known as a

focal point

of

p

) where different geodesics issuing from

p

intersects, to be the point where any given geodesic from

p

ceases to be minimizing.

Coordinates

In

coordinates

the equation is given by the system

d

2

⁢

x

k

d

⁢

t

2

+

∑

i

,

j

Γ

i

⁢

j

k

⁢

d

⁢

x

i

d

⁢

t

⁢

d

⁢

x

j

d

⁢

t

=

0

1

≤

k

≤

n

where

Γ

i

⁢

j

k

is the

Christoffel symbols

(see entry about

connection

),

t

is the

parameter

of the curve and

{

x

1

,

…

,

x

n

}

are coordinates on

M

.

The formula follows since if

γ

˙

=

∑

i

d

⁢

x

i

d

⁢

t

⁢

∂

x

i

, where

{

∂

x

1

,

…

,

∂

x

n

}

are the corresponding coordinate vectors, we have

∇

γ

˙

⁡

γ

˙

=

∇

∑

i

d

⁢

x

i

d

⁢

t

⁢

∂

x

i

⁢

∑

j

d

⁢

x

j

d

⁢

t

⁢

∂

x

j

=

∑

k

γ

˙

⁢

(

d

⁢

x

k

d

⁢

t

)

⁢

∂

x

k

+

∑

i

,

j

d

⁢

x

j

d

⁢

t

⁢

d

⁢

x

i

d

⁢

t

⁢

∇

∂

x

i

⁡

∂

x

j

=

∑

k

(

d

2

⁢

x

k

d

⁢

t

2

+

∑

i

,

j

d

⁢

x

i

d

⁢

t

⁢

d

⁢

x

j

d

⁢

t

⁢

Γ

i

⁢

j

k

)

⁢

∂

x

k

.

Metric spaces

A geodesic in a metric space

(

X

,

d

)

is simply a

continuous

f

:

[

0

,

a

]

→

X

such that the
length (

http://planetmath.org/LengthOfCurveInAMetricSpace

) of

f

is

a

.
Of course, the may be infinite. A geodesic metric space is a metric space
where the distance between two points may be realized by a geodesic.

Title

geodesic

Canonical

name

Geodesic

Date of creation

2013-03-22 14:06:37

Last modified on

2013-03-22 14:06:37

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

22

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 53C22

Related topic

connection

Related topic

Connection

Defines

focal point

Defines

minimizing geodesic

Defines

geodesic curve

Generated on Fri Feb 9 22:28:07 2018 by

LaTeXML
geodesic completeness

geodesic completeness

A

Riemannian metric

on a

manifold

M

is said to be

geodesically

complete

iff its geodesic flow is a complete flow,
i.e. iff for every point

p

∈

M

and every

tangent vector

v

∈

T

p

⁢

M

at

p

the solution to the

geodesic equation

∇

γ

˙

⁡

γ

˙

=

0

with

initial condition

γ

⁢

(

0

)

=

p

,

γ

˙

⁢

(

0

)

=

v

is defined for all time.
The Hopf-Rinow theorem
asserts that a Riemannian metric is complete if and only if the corresponding metric on

M

defined by

d

⁢

(

p

,

q

)

:=

inf

⁡

{

L

⁢

(

c

)

,

c

:

[

0

,

1

]

→

M

,

c

⁢

(

0

)

=

p

,

c

⁢

(

1

)

=

q

}

is a complete metric (i.e.

Cauchy sequences

converge). Here

L

⁢

(

c

)

denote the length of the smooth curve

c

, i.e.

L

⁢

(

c

)

:=

∫

0

1

∥

c

˙

⁢

(

t

)

∥

c

⁢

(

t

)

⁢

𝑑

t

For a proof of the Hopf-Rinow theorem see Milnor’s monograph

Morse Theory

Princeton Annals of Math Studies

51

page 62.

Title

geodesic completeness

Canonical name

GeodesicCompleteness

Date of creation

2013-06-03 13:04:01

Last modified on

2013-06-03 13:04:01

Owner

jacou (1000048)

Last modified by

unlord (1)

Numerical id

14

Author

jacou (1)

Entry type

Definition

Classification

msc 53C22

Generated on Fri Feb 9 22:28:09 2018 by

LaTeXML
geodesic triangle

geodesic triangle

Let

M

be a

differentiable manifold

(at least two times

differentiable

) and

A

,

B

,

C

∈

M

(not necessarily distinct). Let

x

1

,

x

2

,

x

3

∈

[

0

,

∞

)

. Let

γ

1

:

[

0

,

x

1

]

→

M

,

γ

2

:

[

0

,

x

2

]

→

M

, and

γ

3

:

[

0

,

x

3

]

→

M

be

geodesics

such that all of the following hold:

•

γ

1

⁢

(

0

)

=

A

;

•

γ

1

⁢

(

x

1

)

=

B

;

•

γ

2

⁢

(

0

)

=

A

;

•

γ

2

⁢

(

x

2

)

=

C

;

•

γ

3

⁢

(

0

)

=

B

;

•

γ

3

⁢

(

x

3

)

=

C

.

Then the figure determined by

γ

1

,

γ

2

, and

γ

3

is a

geodesic triangle

.

Note that a geodesic triangle need not be a

triangle

. For example, in

ℝ

2

, if

A

=

(

0

,

0

)

,

B

=

(

1

,

2

)

, and

C

=

(

3

,

6

)

, then the geodesic triangle determined by

A

,

B

, and

C

is

{

(

x

,

2

⁢

x

)

:

x

∈

[

0

,

3

]

}

, which is not a triangle.

geodesic metric space (

http://planetmath.org/GeodesicMetricSpace

)

Title

geodesic triangle

Canonical

name

GeodesicTriangle

Date of creation

2013-03-22 17:11:31

Last modified on

2013-03-22 17:11:31

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

7

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 53C22

Generated on Fri Feb 9 22:28:12 2018 by

LaTeXML
spray space

spray space

Take a fibred

manifold

π

:

B

→

X

. Choose a

vector field

S

over

B

that satisfies

D

⁢

π

∘

S

⁢

(

y

)

=

y

for the

Jacobian

map

D

⁢

π

:

T

⁢

B

→

B

over all

coordinate vectors

y

=

(

y

1

,

…

,

y

n

)

∈

B

. A

spray

field

G

over

B

is a globally defined
smooth vector field associated to the first

jet bundle

J

B

1

⁢

X

of

X

that is given in local coordinates

x

=

(

x

1

,

…

,

x

n

)

∈

B

as

𝐆

=

y

i

⁢

∂

∂

⁡

x

i

-

G

i

⁢

∂

∂

⁡

y

i

.

The

spray coefficients

G

i

⁢

(

y

)

are second degree

homogeneous functions

which correspond up to nonlinear

connections

on

M

. Thus by

D

⁢

π

the

integral curves

of

𝐆

must be of second order, and so given the constraints of the spray coefficients, satisfy

c

¨

i

⁢

i

=

2

⁢

G

i

⁢

(

c

˙

)

. Subsequently, the pair

(

X

,

𝐆

)

is called a

spray space

.

Example 1: Choose a system of second order quasilinear

ordinary differential
equations

that satisfy

c

¨

i

⁢

i

+

2

⁢

G

i

⁢

(

c

˙

)

=

0

for a family
of

parameterized curves

c

, and let the system induce its
corresponding

spray

. Then when

c

is also a Finsler

geodesic

in

B

with constant speed so that the

covariant
derivative

gives

D

V

⁢

V

=

0

along a vector field

V

, the
corresponding autoparallels of the spray coefficients completely
characterize a

path space

for

B

.

Title

spray space

Canonical name

SpraySpace

Date of creation

2013-05-03 16:21:46

Last modified on

2013-05-03 16:21:46

Owner

Orphanage (1000048)

Last modified by

jacou (1000048)

Numerical id

17

Author

Orphanage (1000048)

Entry type

Definition

Classification

msc 53C60

Synonym

Spray

Synonym

geodesic spray

Synonym

finsler spray

Defines

spray spaces

Generated on Fri Feb 9 22:28:17 2018 by

LaTeXML
almost complex structure

almost complex structure

Let

V

be a

vector space

over

ℝ

. Recall that a

complex structure

on

V

is a

linear operator

J

on

V

such that

J

2

=

-

I

, where

J

2

=

J

∘

J

, and

I

is the

identity operator

on

V

. A prototypical example of a complex structure is given by the map

J

:

V

→

V

defined by

J

⁢

(

v

,

w

)

=

(

-

w

,

v

)

where

V

=

ℝ

n

⊕

ℝ

n

.

An

almost complex structure

on a

manifold

M

is a

differentiable map

J

:

T

⁢

M

→

T

⁢

M

on the tangent bundle

T

⁢

M

of

M

, such that

•

J

preserves each fiber, that is, the following diagram is

commutative

:

\xymatrix

⁢

T

⁢

M

⁢

\ar

⁢

[

r

]

J

⁢

\ar

⁢

[

d

]

π

⁢

&

⁢

T

⁢

M

⁢

\ar

⁢

[

d

]

π

⁢

M

⁢

\ar

⁢

[

r

]

i

⁢

&

⁢

M

or

π

∘

J

=

π

, where

π

is the standard

projection

onto

M

, and

i

is the identity map on

M

;

•

J

is linear on each fiber, and whose square is minus the

identity

. This means that, for each fiber

F

x

:=

π

-

1

⁢

(

x

)

⊆

T

⁢

M

, the

restriction

J

x

:=

J

∣

F

x

is a complex structure on

F

x

.

Remark

. If

M

is a

complex manifold

, then

multiplication

by

i

on each tangent space gives an almost complex structure.

Title

almost complex structure

Canonical name

AlmostComplexStructure

Date of creation

2013-03-22 13:15:34

Last modified on

2013-03-22 13:15:34

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

7

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 53D05

Related topic

KahlerManifold

Related topic

HyperkahlerManifold

Related topic

MathbbCIsAKahlerManifold

Generated on Fri Feb 9 22:28:19 2018 by

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characteristic polynomial of a symplectic matrix is a reciprocal polynomial

characteristic polynomial of a symplectic matrix is a reciprocal polynomial

Theorem 1

.

The

characteristic polynomial

of a symplectic matrix is a

reciprocal polynomial

.

Proof.

Let

A

be the symplectic matrix, and let

p

⁢

(

λ

)

=

det

⁡

(

A

-

λ

⁢

I

)

be
its characteristic polynomial. We wish to prove that

p

⁢

(

λ

)

=

±

λ

n

⁢

p

⁢

(

1

/

λ

)

.

By

definition

,

A

⁢

J

⁢

A

T

=

J

where

J

is the matrix

J

=

(

0

I

-

I

0

)

.

Since

A

and

J

are symplectic matrices, their

determinants

are

1

, and

p

⁢

(

λ

)

=

det

⁡

(

A

⁢

J

-

λ

⁢

J

)

=

det

⁡

(

A

⁢

J

-

λ

⁢

A

⁢

J

⁢

A

T

)

=

det

⁡

(

-

λ

⁢

A

)

⁢

det

⁡

(

J

)

⁢

det

⁡

(

-

1

λ

⁢

J

+

J

⁢

A

T

)

=

±

λ

n

⁢

det

⁡

(

A

-

1

λ

⁢

I

)

.

as claimed.
∎

Title

characteristic polynomial of a symplectic matrix is a reciprocal polynomial

Canonical

name

CharacteristicPolynomialOfASymplecticMatrixIsAReciprocalPolynomial

Date of creation

2013-03-22 15:33:18

Last modified on

2013-03-22 15:33:18

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

7

Author

matte (1858)

Entry type

Theorem

Classification

msc 53D05

Related topic

ReciprocalPolynomial

Related topic

CharacteristicPolynomialOfAOrthogonalMatrixIsAReciprocalPolynomial

Generated on Fri Feb 9 22:28:22 2018 by

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coadjoint orbit

coadjoint orbit

Let

G

be a Lie group, and

𝔤

its

Lie algebra

. Then

G

has a natural action on

𝔤

*

called the coadjoint action, since it is dual to the

adjoint action

of

G

on

𝔤

. The orbits of this action are

submanifolds

of

𝔤

*

which carry a natural

symplectic structure

, and are in a certain sense, the minimal

symplectic manifolds

on which

G

acts. The orbit through a point

λ

∈

𝔤

*

is typically denoted

𝒪

λ

.

The

tangent space

T

λ

⁢

𝒪

λ

is naturally idenified by the action with

𝔤

/

𝔯

λ

, where

𝔯

λ

is the Lie algebra of the

stabilizer

of

λ

. The

symplectic form

on

𝒪

λ

is given by

ω

λ

⁢

(

X

,

Y

)

=

λ

⁢

(

[

X

,

Y

]

)

. This is obviously anti-symmetric and non-degenerate since

λ

⁢

(

[

X

,

Y

]

)

=

0

for all

Y

∈

𝔤

if and only if

X

∈

𝔯

λ

. This also shows that the form is well-defined.

There is a close association between coadoint orbits and the

representation theory

of

G

, with

irreducible representations

being realized as the space of sections of

line bundles

on

coadjoint orbits

. For example, if

𝔤

is compact, coadjoint orbits are partial

flag manifolds

, and this follows from the

Borel-Bott-Weil theorem

.

Title

coadjoint orbit

Canonical name

CoadjointOrbit

Date of creation

2013-03-22 13:59:11

Last modified on

2013-03-22 13:59:11

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

5

Author

bwebste (988)

Entry type

Definition

Classification

msc 53D05

Generated on Fri Feb 9 22:28:24 2018 by

LaTeXML
concepts in symplectic geometry

concepts in symplectic geometry

Linear theory

1.

symplectic vector space

,

symplectic matrix

2.

symplectic complement

Symplectic manifolds

1.

symplectic manifold

(examples (

http://planetmath.org/ExamplesOfSymplecticManifolds

))

2.

Lagrangian submanifold

,

isotropic submanifold

3.

symplectic vector field

4.

symplectomorphism, canonical transformation

5.

Hamilton equations

6.

Darboux’s Theorem (

http://planetmath.org/DarbouxsTheoremSymplecticGeometry

)

7.

Poisson bracket

8.

Moser’s theorem

9.

Gray stability theorem

10.

momentum map

Title

concepts in symplectic geometry

Canonical name

ConceptsInSymplecticGeometry

Date of creation

2013-03-22 14:46:37

Last modified on

2013-03-22 14:46:37

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

8

Author

matte (1858)

Entry type

Topic

Classification

msc 53D05

Generated on Fri Feb 9 22:28:26 2018 by

LaTeXML
Darboux’s theorem (symplectic geometry)

Darboux’s theorem (symplectic geometry)

If

(

M

,

ω

)

is a

2

⁢

n

-dimensional

symplectic manifold

, and

m

∈

M

, then there exists a

neighborhood

U

of

m

with a

coordinate chart

x

=

(

x

1

,

…

,

x

2

⁢

n

)

:

U

→

ℝ

2

⁢

n

,

such that

ω

=

∑

i

=

1

n

d

⁢

x

i

∧

d

⁢

x

n

+

i

.

These are called canonical or

Darboux coordinates

. On

U

,

ω

is the pullback by

X

of the standard

symplectic form

on

ℝ

2

⁢

n

, so

x

is a symplectomorphism. Darboux’s theorem
implies that there are no local invariants in

symplectic geometry

, unlike in

Riemannian geometry

,
where there is

curvature

.

Title

Darboux’s theorem (symplectic geometry)

Canonical name

DarbouxsTheoremsymplecticGeometry

Date of creation

2013-03-22 13:15:31

Last modified on

2013-03-22 13:15:31

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

6

Author

bwebste (988)

Entry type

Theorem

Classification

msc 53D05

Synonym

Darboux coordinates

Generated on Fri Feb 9 22:28:28 2018 by

LaTeXML
examples of Lagrangian submanifolds

examples of Lagrangian submanifolds

1.

Suppose

M

is a

manifold

, and

f

:

M

→

ℝ

is a

smooth function

.
Then

{

(

d

⁢

f

)

⁢

(

x

)

∈

T

x

∗

⁢

(

M

)

:

x

∈

M

}

is a

Lagrangian submanifold

of the

cotangent bundle

T

∗

⁢

M

when
this is equipped with the canonical

symplectic structure

.

Title

examples of Lagrangian submanifolds

Canonical name

ExamplesOfLagrangianSubmanifolds

Date of creation

2013-03-22 15:18:02

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2013-03-22 15:18:02

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matte (1858)

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6

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matte (1858)

Entry type

Example

Classification

msc 53D05

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examples of symplectic manifolds

examples of symplectic manifolds

Examples of symplectic manifolds

:
The most basic example of a

symplectic manifold

is

ℝ

2

⁢

n

. If we choose

coordinate

functions

x

1

,

…

,

x

n

,

y

1

,

…

⁢

y

n

, then

ω

=

∑

m

=

1

n

d

⁢

x

m

∧

d

⁢

y

m

is a

symplectic form

, and one can easily check that it is closed.

Any

orientable

2

-

manifold

is symplectic. Any

volume form

is a symplectic form.

If

M

is any manifold, then the

cotangent bundle

T

*

⁢

M

is symplectic.
If

x

1

,

…

,

x

n

are coordinates on a coordinate patch

U

on

M

, and

ξ

1

,

…

,

ξ

n

are the functions

T

*

⁢

(

U

)

→

ℝ

ξ

i

⁢

(

m

,

η

)

=

η

⁢

(

∂

∂

⁡

x

i

)

⁢

(

m

)

at any point

(

m

,

η

)

∈

T

*

⁢

(

M

)

, then

ω

=

∑

i

=

1

n

d

⁢

x

i

∧

d

⁢

ξ

i

.

(Equivalently, using the notation

α

from the entry Poincare 1-form, we can define

ω

=

-

d

⁢

α

.)

One can check that this behaves well under coordinate

transformations

, and thus defines a form on the whole manifold. One can easily check that this is closed and non-degenerate.

All orbits in the coadjoint action of a Lie group on the dual of it Lie algebra are symplectic. In particular, this includes complex

Grassmannians

and

complex projective spaces

.

Examples of non-symplectic manifolds: Obviously, all odd-dimensional manifolds are non-symplectic.

More subtlely, if

M

is

compact

,

2

⁢

n

dimensional and

M

is a closed 2-form, consider the form

ω

n

. If this form is exact, then

ω

n

must be 0 somewhere, and so

ω

is somewhere degenerate. Since the wedge of a closed and an

exact form

is exact, no power

ω

m

of

ω

can be exact. In particular,

H

2

⁢

m

⁢

(

M

)

≠

0

for all

0

≤

m

≠

n

, for any compact symplectic manifold.

Thus, for example,

S

n

for

n

>

2

is not symplectic. Also, this means that any symplectic manifold must be orientable.

Finally, it is not generally the case that

connected sums

of compact symplectic manifolds are again symplectic: Every symplectic manifold admits an almost

complex structure

(a symplectic form and a

Riemannian metric

on a manifold are sufficient to define an almost complex structure which is compatible with the symplectic form in a nice way). In the case of a connected sum of two symplectic manifolds, there does not necessarily exist such an almost complex structure, and hence connected sums cannot be (

generically

) symplectic.

Title

examples of symplectic manifolds

Canonical name

ExamplesOfSymplecticManifolds

Date of creation

2013-03-22 13:12:31

Last modified on

2013-03-22 13:12:31

Owner

mathcam (2727)

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mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Example

Classification

msc 53D05

Generated on Fri Feb 9 22:28:36 2018 by

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Hamilton equations

Hamilton equations

The

Hamilton equations

are a formulation of the equations of motion in classical mechanics.

Local formulation

Suppose

U

⊆

ℝ

n

is an open set, suppose

I

is an interval
(representing time), and

H

:

U

×

ℝ

n

×

I

→

ℝ

is a

smooth function

. Then the

equations

q

˙

j

=

∂

⁡

H

∂

⁡

p

j

⁢

(

q

⁢

(

t

)

,

p

⁢

(

t

)

,

t

)

,

(1)

p

˙

j

=

-

∂

⁡

H

∂

⁡

q

j

⁢

(

q

⁢

(

t

)

,

p

⁢

(

t

)

,

t

)

,

(2)

are the

Hamilton equations

for the curve

(

q

,

p

)

=

(

q

1

,

…

,

q

n

,

p

1

,

…

,

p

n

)

:

I

→

U

×

ℝ

n

.

Such a solution is called a

bicharacteristic

, and

H

is
called a

Hamiltonian

function

. Here we use classical notation;
the

q

i

’s represent the location of the particles,
the

p

i

’s represent the momenta of the particles.

Global formulation

Suppose

P

is a

symplectic manifold

with

symplectic form

ω

and that

H

:

P

→

ℝ

is a smooth function. Then

X

H

, the

Hamiltonian
vector field

corresponding to

H

is determined by

d

⁢

H

=

ω

⁢

(

X

H

,

⋅

)

.

The most common case is when

P

is the cotangent bundle of a

manifold

Q

equipped with the canonical symplectic form

ω

=

-

d

⁢

α

,
where

α

is the Poincaré

1

-form (

http://planetmath.org/Poincare1Form

). (Note that other authors may have different sign convention.) Then Hamilton’s equations are the equations for the flow of the vector field

X

H

. Given a system of coordinates

x

1

,

…

⁢

x

2

⁢

n

on the manifold

P

, they can be written as follows:

x

˙

i

=

(

X

H

)

i

⁢

(

x

1

,

…

⁢

x

2

⁢

n

,

t

)

The

relation

with the former definition is that in canonical
local coordinates

(

q

i

,

p

j

)

for

T

∗

⁢

Q

, the flow of

X

H

is determined by equations (

1

)-(

2

).

Also, the following terminology is frequently encountered — the manifold

P

is known as the phase space, the manifold

Q

is known as the

configuration space

, and the

product

P

×

ℝ

is known as

state space

.

Title

Hamilton equations

Canonical name

HamiltonEquations

Date of creation

2013-03-22 14:45:58

Last modified on

2013-03-22 14:45:58

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 53D05

Classification

msc 70H05

Related topic

Quantization

Generated on Fri Feb 9 22:28:40 2018 by

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Hamiltonian vector field

Hamiltonian vector field

Let

(

M

,

ω

)

be a

symplectic manifold

, and

ω

~

:

T

⁢

M

→

T

*

⁢

M

be the isomorphism from the

tangent
bundle

to the

cotangent bundle

X

↦

ω

⁢

(

⋅

,

X

)

and let

f

:

M

→

ℝ

is a

smooth function

.
Then

H

f

=

ω

~

-

1

⁢

(

d

⁢

f

)

is the

Hamiltonian vector field

of

f

. The vector field

H

f

is symplectic (

http://planetmath.org/SymplecticVectorField

),
and a symplectic vector field

X

is

http://planetmath.org/node/6410

Hamiltonian

if and only if the 1-form

ω

~

⁢

(

X

)

=

ω

⁢

(

⋅

,

X

)

is exact.

If

T

*

⁢

Q

is the cotangent bundle of a

manifold

Q

, which is naturally identified with the phase
space of one particle on

Q

, and

f

is the Hamiltonian, then the flow of the Hamiltonian
vector field

H

f

is the time flow of the physical system.

Title

Hamiltonian vector field

Canonical

name

HamiltonianVectorField

Date of creation

2013-03-22 13:14:07

Last modified on

2013-03-22 13:14:07

Owner

rspuzio (6075)

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rspuzio (6075)

Numerical id

7

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 53D05

Generated on Fri Feb 9 22:28:42 2018 by

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isotropic submanifold

isotropic submanifold

If

(

M

,

ω

)

is a

symplectic manifold

, then a

submanifold

L

⊂

M

is

isotropic

if the

symplectic form

vanishes on the

tangent space

of

L

, that is,

ω

⁢

(

v

1

,

v

2

)

=

0

for all

v

1

,

v

2

∈

T

ℓ

⁢

L

for all

ℓ

∈

L

.

Title

isotropic submanifold

Canonical name

IsotropicSubmanifold

Date of creation

2013-03-22 13:12:26

Last modified on

2013-03-22 13:12:26

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

4

Author

bwebste (988)

Entry type

Definition

Classification

msc 53D05

Related topic

LagrangianSubmanifold

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