Laplace Vector - Runge - Lenza

In this article, vectors are in bold and their absolute values in italics, for example,

|\mathbf {A} |=A

${\ displaystyle | \ mathbf {A} | = A}$ $| {\ mathbf {A}} | = A$ .

In classical mechanics, the Laplace – Runge – Lentz vector is a vector that is mainly used to describe the shape and orientation of the orbit in which one celestial body revolves around another (for example, the orbit in which the planet rotates around a star). In the case of two bodies, the interaction of which is described by Newton’s law of universal gravitation , the Laplace – Runge – Lenz vector is the integral of motion , that is, its direction and magnitude are constant regardless of what point in the orbit they are calculated ^[1] ; They say that the Laplace – Runge – Lenz vector is preserved during the gravitational interaction of two bodies. This statement can be generalized for any problem with two bodies interacting by means of a central force , which varies inversely with the square of the distance between them. Such a problem is called the Kepler problem ^[2] .

For example, such a potential arises when considering classical orbits (without quantization) in the problem of the motion of a negatively charged electron moving in an electric field of a positively charged nucleus. If the Laplace – Runge – Lenz vector is given, then the form of their relative motion can be obtained from simple geometric considerations using the laws of conservation of this vector and energy.

According to the correspondence principle , the Laplace – Runge – Lenz vector has a quantum analogue that was used in the first derivation of the spectrum of the hydrogen atom ^[3] , even before the discovery of the Schrödinger equation .

The Kepler problem has an unusual feature: the end of the momentum vector $\mathbf {p}$ ${\ displaystyle \ mathbf {p}}$ ${\ mathbf {p}}$ always moves in a circle ^[4] ^[5] ^[6] . Due to the arrangement of these circles for a given total energy $E$ ${\ displaystyle E}$ $E$ Kepler’s problem is mathematically equivalent to a particle moving freely in a four-dimensional sphere $S_{3}$ ${\ displaystyle S_ {3}}$ $S _ {{3}}$ ^[7] . By this mathematical analogy, the conserved Laplace – Runge – Lenz vector is equivalent to the additional components of the angular momentum in four-dimensional space ^[8] .

The Laplace – Runge – Lenz vector is also known as the Laplace vector , the Runge – Lenz vector, and the Lenz vector , although none of these scientists deduced it for the first time. The Laplace – Runge – Lenz vector has been reopened several times ^[9] . It is also equivalent to the dimensionless eccentricity vector in celestial mechanics ^[10] . Similarly, there is no generally accepted designation for it, although it is commonly used $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ $\ mathbf {A}$ . For various generalizations of the Laplace - Runge - Lenz vector, which are defined below, the symbol is used ${\mathcal {A}}$ ${\ displaystyle {\ mathcal {A}}}$ $\ mathcal {A}$ .

Context

A single particle moving under the influence of any conservative central force has at least four integrals of motion (quantities conserved during motion): total energy $E$ ${\ displaystyle E}$ and three components of angular momentum (vector $\mathbf {L}$ {\ displaystyle \ mathbf {L}} ) The orbit of the particle lies in a plane that is determined by the initial momentum of the particle, $\mathbf {p}$ ${\ displaystyle \ mathbf {p}}$ (or, equivalently, speed $\mathbf {v}$ ${\ displaystyle \ mathbf {v}}$ ) and coordinates, i.e., a radius vector $\mathbf {r}$ ${\ displaystyle \ mathbf {r}}$ between the center of force and the particle (see Fig. 1). This plane is perpendicular to the constant vector. $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$ that can be expressed mathematically using the scalar product $\mathbf {r} \cdot \mathbf {L} =0$ ${\ displaystyle \ mathbf {r} \ cdot \ mathbf {L} = 0}$ .

As defined below , the Laplace-Runge-Lenz vector $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ always in the plane of movement - that is, $\mathbf {A} \cdot \mathbf {L} =0$ ${\ displaystyle \ mathbf {A} \ cdot \ mathbf {L} = 0}$ - for any central force. Also $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ is constant only for a force that is inversely proportional to the square of the distance ^[2] . If the central force is approximately dependent on the inverse square of the distance, the vector $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ is approximately constant in length, but rotates slowly. For most central forces, however, this vector $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ not permanent, but changes length and direction. The generalized conserved Laplace - Runge - Lenz vector ${\mathcal {A}}$ ${\ displaystyle {\ mathcal {A}}}$ can be defined for all central forces, but this vector is a complex function of position and is usually not analytically expressed in elementary or special functions ^[11] ^[12] .

History

Laplace Vector - Runge - Lenza $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ is a conserved quantity in the Kepler problem and is useful in describing astronomical orbits , like the movement of a planet around the sun. However, it was never widely known among physicists, perhaps because it is a less intuitive vector than momentum and angular momentum . The Laplace – Runge – Lenz vector has been independently discovered several times over the past three centuries ^[9] . Jacob German was the first to show that $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ is preserved for the special case of the central force, which is inversely proportional to the square of the distance ^[13] , and found its connection with the eccentricity of the elliptical orbit. Herman's work was generalized to its modern form by Johann Bernoulli in 1710 ^[14] . In turn, Pierre-Simon Laplace at the end of the 18th century discovered conservation $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ again, proving this analytically, and not geometrically, as his predecessors ^[15] .

In the middle of the 19th century, William Hamilton obtained the equivalent of the eccentricity vector, defined below ^[10] , using it to show that the end of the momentum vector $\mathbf {p}$ ${\ displaystyle \ mathbf {p}}$ moves in a circle under the action of a central force, which depends inversely on the square of the distance (Fig. 3) ^[4] . At the beginning of the 20th century, Willard Gibbs obtained the same vector using vector analysis ^[16] . Gibbs's conclusion was used by Karl Runge in a popular German textbook on vectors as an example ^[17] , which was referenced by Wilhelm Lenz in his article on quantum mechanical (old) consideration of the hydrogen atom ^[18] .

In 1926, Wolfgang Pauli used this vector to derive the spectrum of a hydrogen atom using modern matrix quantum mechanics , rather than the Schrödinger equation ^[3] . After the publication of Pauli, the vector became mainly known as the Runge - Lenz vector .

Mathematical Definition

Fig. 1: Laplace Vector - Runge - Lenza

\scriptstyle \mathbf {A}

{\ displaystyle \ scriptstyle \ mathbf {A}}

(shown in red) at four points (indicated by 1 , 2 , 3, and 4 ) in an elliptical orbit of a connected point particle moving under the action of a central force that is inversely proportional to the square of the distance. A small black circle indicates the center of gravity. From it begin radius vectors (highlighted in black) directed to points 1 , 2 , 3 and 4 . Angular momentum vector

\scriptstyle \mathbf {L}

{\ displaystyle \ scriptstyle \ mathbf {L}}

directed perpendicular to the orbit. Coplanar vectors

\scriptstyle \mathbf {p} \times \mathbf {L}

{\ displaystyle \ scriptstyle \ mathbf {p} \ times \ mathbf {L}}

,

\scriptstyle (mk/r)\mathbf {r}

{\ displaystyle \ scriptstyle (mk / r) \ mathbf {r}}

and

\scriptstyle \mathbf {A}

{\ displaystyle \ scriptstyle \ mathbf {A}}

depicted in blue, green and red, respectively; these variables are defined below . Vector

\scriptstyle \mathbf {A}

{\ displaystyle \ scriptstyle \ mathbf {A}}

is constant in direction and magnitude.

For a single particle moving under the action of a central force , which depends inversely on the square of the distance and described by the equation $\mathbf {F} (r)={\frac {-k}{r^{2}}}\mathbf {\hat {r}}$ ${\ displaystyle \ mathbf {F} (r) = {\ frac {-k} {r ^ {2}}} \ mathbf {\ hat {r}}}$ , Laplace-Runge-Lenza vector $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ defined mathematically by the formula ^[2]

\mathbf {A} =\mathbf {p} \times \mathbf {L} -mk\mathbf {\hat {r}} ,

{\ displaystyle \ mathbf {A} = \ mathbf {p} \ times \ mathbf {L} -mk \ mathbf {\ hat {r}},}

Where

$m$ ${\ displaystyle m}$ - mass of a point particle moving under the influence of a central force,
$\mathbf {p}$ ${\ displaystyle \ mathbf {p}}$ Is the momentum vector,
$\mathbf {L} =\mathbf {r} \times \mathbf {p}$ ${\ displaystyle \ mathbf {L} = \ mathbf {r} \ times \ mathbf {p}}$ Is the angular momentum vector,
$k$ ${\ displaystyle k}$ - a parameter describing the magnitude of the central force,
$\mathbf {\hat {r}}$ ${\ displaystyle \ mathbf {\ hat {r}}}$ Is the unit vector, i.e. $\mathbf {\hat {r}} ={\frac {\mathbf {r} }{r}}$ ${\ displaystyle \ mathbf {\ hat {r}} = {\ frac {\ mathbf {r}} {r}}}$ where $\mathbf {r}$ ${\ displaystyle \ mathbf {r}}$ Is the radius vector of the position of the particle, and $r$ ${\ displaystyle r}$ - its length.

Since we assumed that the force is conservative , the total energy $E$ ${\ displaystyle E}$ saved

E={\frac {p^{2}}{2m}}-{\frac {k}{r}}={\frac {1}{2}}mv^{2}-{\frac {k}{r}}.

{\ displaystyle E = {\ frac {p ^ {2}} {2m}} - {\ frac {k} {r}} = {\ frac {1} {2}} mv ^ {2} - {\ frac {k} {r}}.}

From the centrality of the force it follows that the angular momentum vector $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$ it also preserves and determines the plane in which the particle moves. Laplace Vector - Runge - Lenza $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ perpendicular to the angular momentum vector $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$ and, thus, is in the plane of the orbit . The equation $\mathbf {A} \cdot \mathbf {L} =0$ ${\ displaystyle \ mathbf {A} \ cdot \ mathbf {L} = 0}$ true because vectors $\mathbf {p} \times \mathbf {L}$ ${\ displaystyle \ mathbf {p} \ times \ mathbf {L}}$ and $\mathbf {r}$ ${\ displaystyle \ mathbf {r}}$ perpendicular $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$ .

This is the definition of the Laplace - Runge - Lenz vector $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ applicable for a single point particle with mass $m$ ${\ displaystyle m}$ moving in a stationary (time-independent) potential. In addition, the same definition can be extended to a two-body problem, similar to the Kepler problem, if replaced $m$ ${\ displaystyle m}$ on the reduced mass of these two bodies and $\mathbf {r}$ ${\ displaystyle \ mathbf {r}}$ by the vector between these bodies.

Circular travel time curve

Fig. 2: End of the momentum vector

\scriptstyle \mathbf {p}

{\ displaystyle \ scriptstyle \ mathbf {p}}

(shown in blue) moves in a circle when the particle moves in an ellipse. The four marked points correspond to the points in fig. 1. The center of the circle is on the axis

\scriptstyle y

{\ displaystyle \ scriptstyle y}

at the point

\scriptstyle A/L

{\ displaystyle \ scriptstyle A / L} (shown in purple), with a radius

\scriptstyle mk/L

{\ displaystyle \ scriptstyle mk / L}

(shown in green). Angle

\scriptstyle \eta

{\ displaystyle \ scriptstyle \ eta}

determines eccentricity

\scriptstyle e

{\ displaystyle \ scriptstyle e}

elliptical orbit (

\scriptstyle \cos \eta =e

{\ displaystyle \ scriptstyle \ cos \ eta = e}

) It follows from the inscribed angle theorem for the circle that

\scriptstyle \eta

{\ displaystyle \ scriptstyle \ eta}

is also the angle between any point on the circle and two points of intersection of the circle with the axis

\scriptstyle p_{x}

{\ displaystyle \ scriptstyle p_ {x}}

,

\scriptstyle p_{x}=\pm p_{0}

{\ displaystyle \ scriptstyle p_ {x} = \ pm p_ {0}}

.

Preservation of the Laplace - Runge - Lenz vector $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ and angular momentum vectors $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$ used in proving that the momentum vector $\mathbf {p}$ ${\ displaystyle \ mathbf {p}}$ moves in a circle under the action of a central force inversely proportional to the square of the distance. Computing Vector Product $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ and $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$ we arrive at the equation for $\mathbf {p}$ ${\ displaystyle \ mathbf {p}}$

L^{2}\mathbf {p} =\mathbf {L} \times \mathbf {A} -mk{\hat {\mathbf {r} }}\times \mathbf {L} .

{\ displaystyle L ^ {2} \ mathbf {p} = \ mathbf {L} \ times \ mathbf {A} -mk {\ hat {\ mathbf {r}}} \ times \ mathbf {L}.}

Directing vector $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$ along the axis $z$ ${\ displaystyle z}$ and the main axis are along the axis $x$ ${\ displaystyle x}$ we arrive at the equation

p_{x}^{2}+(p_{y}-A/L)^{2}=(mk/L)^{2}.

{\ displaystyle p_ {x} ^ {2} + (p_ {y} -A / L) ^ {2} = (mk / L) ^ {2}.}

In other words, the momentum vector $\mathbf {p}$ ${\ displaystyle \ mathbf {p}}$ limited by a circle of radius $mk/L$ ${\ displaystyle mk / L}$ whose center is located at a point with coordinates $(0,\;A/L)$ ${\ displaystyle (0, \; A / L)}$ . Eccentricity $e$ ${\ displaystyle e}$ corresponds to the cosine of the angle $\eta$ ${\ displaystyle \ eta}$ shown in fig. 2. For brevity, you can enter a variable $p_{0}={\sqrt {2m|E|}}$ ${\ displaystyle p_ {0} = {\ sqrt {2m | E |}}}$ . A circular hodograph is useful for describing the symmetry of the Kepler problem.

Integrals of motion and superintegrability

Seven scalars: energy $E$ ${\ displaystyle E}$ and components of the Laplace - Runge - Lenz vectors $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ and momentum $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$ - are connected by two relations. For vectors, the orthogonality condition is satisfied $\mathbf {A} \cdot \mathbf {L} =0$ ${\ displaystyle \ mathbf {A} \ cdot \ mathbf {L} = 0}$ , and the energy is included in the expression for the squared length of the Laplace-Runge-Lenz vector obtained above $A^{2}=m^{2}k^{2}+2mEL^{2}$ ${\ displaystyle A ^ {2} = m ^ {2} k ^ {2} + 2mEL ^ {2}}$ . Then there are five independent conserved quantities, or integrals of motion . This is compatible with six initial conditions (the initial position of the particle and its velocity are vectors with three components), which determine the orbit of the particle, since the initial time is not determined by the integrals of motion. Since the value $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ (and eccentricity $e$ ${\ displaystyle e}$ orbits) can be determined from the total angular momentum $L$ ${\ displaystyle L}$ and energy $E$ ${\ displaystyle E}$ then it is claimed that only direction $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ stored independently. In addition, the vector $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ should be perpendicular $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$ - this leads to one additional conserved value.

Mechanical system with $d$ ${\ displaystyle d}$ degrees of freedom may have a maximum $2d-1$ ${\ displaystyle 2d-1}$ integrals of motion, since $2d$ ${\ displaystyle 2d}$ initial conditions and initial time cannot be determined from the integrals of motion. System with more than $d$ ${\ displaystyle d}$ integrals of motion are called superintegrable , and a system with $2d-1$ ${\ displaystyle 2d-1}$ integrals is called maximally superintegrable ^[19] . Since the solution of the Hamilton - Jacobi equation in one coordinate system can only lead to $d$ ${\ displaystyle d}$ integrals of motion, then the variables must be separated for superintegrable systems in more than one coordinate system ^[20] . The Kepler problem is maximally superintegrable, since it has three degrees of freedom ( $d=3$ ${\ displaystyle d = 3}$ ) and five independent integrals of motion; the variables in the Hamilton – Jacobi equation are separated in spherical coordinates and parabolic coordinates ^[21] , as described below . The most superintegrable systems can be quantized using only switching relations , as shown below ^[22] .

Hamilton - Jacobi equation in parabolic coordinates

The constancy of the Laplace – Runge – Lenz vector can be derived using the Hamilton – Jacobi equation in parabolic coordinates $(\xi ,\;\eta )$ ${\ displaystyle (\ xi, \; \ eta)}$ defined as follows

\xi =r+x,

{\ displaystyle \ xi = r + x,}

\eta =r-x,

{\ displaystyle \ eta = rx,}

Where $r$ ${\ displaystyle r}$ - radius in the plane of the orbit

r={\sqrt {x^{2}+y^{2}}}.

{\ displaystyle r = {\ sqrt {x ^ {2} + y ^ {2}}}.

The inverse transformation of these coordinates is written as

x={\frac {1}{2}}(\xi -\eta ),

{\ displaystyle x = {\ frac {1} {2}} (\ xi - \ eta),}

y={\sqrt {\xi \eta }}.

{\ displaystyle y = {\ sqrt {\ xi \ eta}}.}

Separation of variables in the Hamilton – Jacobi equation in these coordinates gives two equivalent equations ^[21] ^[23]

2\xi p_{\xi }^{2}-mk-mE\xi =-\beta ,

{\ displaystyle 2 \ xi p _ {\ xi} ^ {2} -mk-mE \ xi = - \ beta,}

2\eta p_{\eta }^{2}-mk-mE\eta =\beta ,

{\ displaystyle 2 \ eta p _ {\ eta} ^ {2} -mk-mE \ eta = \ beta,}

Where $\beta$ ${\ displaystyle \ beta}$ - integral of motion . By subtracting these equations and expressions in terms of the Cartesian momentum coordinates $p_{x}$ ${\ displaystyle p_ {x}}$ and $p_{y}$ ${\ displaystyle p_ {y}}$ can show that $\beta$ ${\ displaystyle \ beta}$ equivalent to Laplace-Runge-Lenza vector

\beta =p_{y}(xp_{y}-yp_{x})-mk{\frac {x}{r}}=A_{x}.

{\ displaystyle \ beta = p_ {y} (xp_ {y} -yp_ {x}) - mk {\ frac {x} {r}} = A_ {x}.}

This Hamilton – Jacobi approach can be used to derive the conserved generalized Laplace – Runge – Lenz vector ${\mathcal {A}}$ ${\ displaystyle {\ mathcal {A}}}$ in the presence of an electric field $\mathbf {E}$ ${\ displaystyle \ mathbf {E}}$ ^[21] ^[24]

{\mathcal {A}}=\mathbf {A} +{\frac {mq}{2}}\left[(\mathbf {r} \times \mathbf {E} )\times \mathbf {r} \right],

{\ displaystyle {\ mathcal {A}} = \ mathbf {A} + {\ frac {mq} {2}} \ left [(\ mathbf {r} \ times \ mathbf {E}) \ times \ mathbf {r } \ right],}

Where $q$ ${\ displaystyle q}$ Is the charge of the circulating particle.

Alternative wording

Unlike impulse $\mathbf {p}$ ${\ displaystyle \ mathbf {p}}$ and angular momentum $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$ , the Laplace - Runge - Lenz vector does not have a generally accepted definition. In the scientific literature, several different factors and symbols are used. The most general definition is given above , but another definition arises after dividing by a constant $m k$ ${\ displaystyle mk}$ to get the dimensionless persistent eccentricity vector

\mathbf {e} ={\frac {1}{mk}}(\mathbf {p} \times \mathbf {L} )-\mathbf {\hat {r}} ={\frac {m}{k}}(\mathbf {v} \times \mathbf {r} \times \mathbf {v} )-\mathbf {\hat {r}} ,

{\ displaystyle \ mathbf {e} = {\ frac {1} {mk}} (\ mathbf {p} \ times \ mathbf {L}) - \ mathbf {\ hat {r}} = {\ frac {m} {k}} (\ mathbf {v} \ times \ mathbf {r} \ times \ mathbf {v}) - \ mathbf {\ hat {r}},}

Where $\mathbf {v}$ ${\ displaystyle \ mathbf {v}}$ Is the velocity vector. Direction of this scaled vector $\mathbf {e}$ ${\ displaystyle \ mathbf {e}}$ coincides with the direction $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ , and its amplitude is equal to the eccentricity of the orbit. We get other definitions if divided $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ on $m$ ${\ displaystyle m}$ ,

\mathbf {M} =\mathbf {v} \times \mathbf {L} -k\mathbf {\hat {r}}

{\ displaystyle \ mathbf {M} = \ mathbf {v} \ times \ mathbf {L} -k \ mathbf {\ hat {r}}}

or at $p_{0}$ ${\ displaystyle p_ {0}}$

\mathbf {D} ={\frac {\mathbf {A} }{p_{0}}}={\frac {1}{\sqrt {2m|E|}}}\{\mathbf {p} \times \mathbf {L} -mk\mathbf {\hat {r}} \},

{\ displaystyle \ mathbf {D} = {\ frac {\ mathbf {A}} {p_ {0}}} = {\ frac {1} {\ sqrt {2m | E |}}} \ {\ mathbf {p } \ times \ mathbf {L} -mk \ mathbf {\ hat {r}} \},}

which has the same dimension as the angular momentum (vector $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$ ) In rare cases, the sign of the Laplace-Runge-Lenz vector can be reversed. Other common symbols for the Laplace - Runge - Lenz vector include $\mathbf {a}$ ${\ displaystyle \ mathbf {a}}$ , $\mathbf {R}$ ${\ displaystyle \ mathbf {R}}$ , $\mathbf {F}$ ${\ displaystyle \ mathbf {F}}$ , $\mathbf {J}$ ${\ displaystyle \ mathbf {J}}$ and $\mathbf {V}$ ${\ displaystyle \ mathbf {V}}$ . However, the choice of a factor and a symbol for the Laplace – Runge – Lenz vector, of course, does not affect its conservation.

Fig. 3: Angular momentum vector

\scriptstyle \mathbf {L}

{\ displaystyle \ scriptstyle \ mathbf {L}}

, Laplace-Runge-Lenza vector

\scriptstyle \mathbf {A}

{\ displaystyle \ scriptstyle \ mathbf {A}}

and Hamilton vector, binormal

\scriptstyle \mathbf {B}

{\ displaystyle \ scriptstyle \ mathbf {B}}

are mutually perpendicular;

\scriptstyle \mathbf {A}

{\ displaystyle \ scriptstyle \ mathbf {A}}

and

\scriptstyle \mathbf {B}

{\ displaystyle \ scriptstyle \ mathbf {B}}

indicate the major and minor semiaxes, respectively, of the elliptical orbit in the Kepler problem.

Alternative persistent vector: binormal - vector $\mathbf {B}$ ${\ displaystyle \ mathbf {B}}$ studied by William Hamilton ^[10]

\mathbf {B} =\mathbf {p} -\left({\frac {mk}{L^{2}r}}\right)(\mathbf {L} \times \mathbf {r} ),

{\ displaystyle \ mathbf {B} = \ mathbf {p} - \ left ({\ frac {mk} {L ^ {2} r}} \ right) (\ mathbf {L} \ times \ mathbf {r}) ,}

which is preserved and points along the minor axis of the ellipse. Laplace Vector - Runge - Lenza $\mathbf {A} =\mathbf {B} \times \mathbf {L}$ ${\ displaystyle \ mathbf {A} = \ mathbf {B} \ times \ mathbf {L}}$ is a vector work $\mathbf {B}$ ${\ displaystyle \ mathbf {B}}$ and $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$ (fig. 3). Vector $\mathbf {B}$ ${\ displaystyle \ mathbf {B}}$ denoted as binormal , since it is perpendicular to $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ so $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$ . Like the Laplace – Runge – Lenz vector, the binormal vector can be defined with various factors.

Two conserved vectors, $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ and $\mathbf {B}$ ${\ displaystyle \ mathbf {B}}$ can be combined into a persistent two-element tensor $\mathbf {W}$ ${\ displaystyle \ mathbf {W}}$

\mathbf {W} =\alpha \mathbf {A} \otimes \mathbf {A} +\beta \mathbf {B} \otimes \mathbf {B} ,

{\ displaystyle \ mathbf {W} = \ alpha \ mathbf {A} \ otimes \ mathbf {A} + \ beta \ mathbf {B} \ otimes \ mathbf {B},}

Where $\otimes$ ${\ displaystyle \ otimes}$ denotes the tensor product , and $\alpha$ ${\ displaystyle \ alpha}$ and $\beta$ ${\ displaystyle \ beta}$ - arbitrary factors ^[11] . Written in component notation, this equation reads like this

W_{ij}=\alpha A_{i}A_{j}+\beta B_{i}B_{j}.

{\ displaystyle W_ {ij} = \ alpha A_ {i} A_ {j} + \ beta B_ {i} B_ {j}.}

Vectors $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ and $\mathbf {B}$ ${\ displaystyle \ mathbf {B}}$ are orthogonal to each other, and they can be represented as the main axes of the conserved tensor $\mathbf {W}$ ${\ displaystyle \ mathbf {W}}$ , that is, as its own vectors . $\mathbf {W}$ ${\ displaystyle \ mathbf {W}}$ perpendicular $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$

\mathbf {L} \cdot \mathbf {W} =\alpha (\mathbf {L} \cdot \mathbf {A} )\mathbf {A} +\beta (\mathbf {L} \cdot \mathbf {B} )\mathbf {B} =0,

{\ displaystyle \ mathbf {L} \ cdot \ mathbf {W} = \ alpha (\ mathbf {L} \ cdot \ mathbf {A}) \ mathbf {A} + \ beta (\ mathbf {L} \ cdot \ mathbf {B}) \ mathbf {B} = 0,}

insofar as $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ and $\mathbf {B}$ ${\ displaystyle \ mathbf {B}}$ perpendicular then $\mathbf {L} \cdot \mathbf {A} =\mathbf {L} \cdot \mathbf {B} =0$ ${\ displaystyle \ mathbf {L} \ cdot \ mathbf {A} = \ mathbf {L} \ cdot \ mathbf {B} = 0}$ .

Kepler Output

Fig. 4: A simplified version of Fig. 1. The angle is determined

\theta

{\ displaystyle \ theta}

between

\scriptstyle \mathbf {A}

{\ displaystyle \ scriptstyle \ mathbf {A}}

and

\scriptstyle \mathbf {r}

{\ displaystyle \ scriptstyle \ mathbf {r}}

at one point in the orbit.

The shape and orientation of the orbit in the Kepler problem , knowing the Laplace - Runge - Lenz vector $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ can be defined as follows. Consider the scalar product of vectors $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ and $\mathbf {r}$ ${\ displaystyle \ mathbf {r}}$ (planet position):

\mathbf {A} \cdot \mathbf {r} =Ar\cos \theta =\mathbf {r} \cdot (\mathbf {p} \times \mathbf {L} )-mkr,

{\ displaystyle \ mathbf {A} \ cdot \ mathbf {r} = Ar \ cos \ theta = \ mathbf {r} \ cdot (\ mathbf {p} \ times \ mathbf {L}) -mkr,}

Where $\theta$ ${\ displaystyle \ theta}$ is the angle between $\mathbf {r}$ ${\ displaystyle \ mathbf {r}}$ and $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ (fig. 4). Reorder the factors in the mixed product $\mathbf {r} \cdot (\mathbf {p} \times \mathbf {L} )=\mathbf {L} \cdot (\mathbf {r} \times \mathbf {p} )=\mathbf {L} \cdot \mathbf {L} =L^{2}$ ${\ displaystyle \ mathbf {r} \ cdot (\ mathbf {p} \ times \ mathbf {L}) = \ mathbf {L} \ cdot (\ mathbf {r} \ times \ mathbf {p}) = \ mathbf { L} \ cdot \ mathbf {L} = L ^ {2}}$ , and using simple transformations we get the definition for a conical section :

{\frac {1}{r}}={\frac {mk}{L^{2}}}\left(1+{\frac {A}{mk}}\cos \theta \right)

{\ displaystyle {\ frac {1} {r}} = {\ frac {mk} {L ^ {2}}} \ left (1 + {\ frac {A} {mk}} \ cos \ theta \ right) }

eccentric $e$ ${\ displaystyle e}$ given by the formula:

e={\frac {A}{mk}}={\frac {|\mathbf {A} |}{mk}}.

{\ displaystyle e = {\ frac {A} {mk}} = {\ frac {| \ mathbf {A} |} {mk}}.}

We arrive at the expression of the squared modulus of the vector $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ as

A^{2}=m^{2}k^{2}+2mEL^{2},

{\ displaystyle A ^ {2} = m ^ {2} k ^ {2} + 2mEL ^ {2},}

which can be rewritten using the eccentricity of the orbit

e^{2}-1={\frac {2L^{2}}{mk^{2}}}E.

{\ displaystyle e ^ {2} -1 = {\ frac {2L ^ {2}} {mk ^ {2}}} E.}

Thus, if the energy is negative, which corresponds to coupled orbits, the eccentricity is less than one, and the orbit is elliptical . Conversely, if the energy is positive (unbound orbits, also called scattering orbits ), the eccentricity is greater than one, and the orbit is a hyperbole . Finally, if the energy is exactly zero, the eccentricity is one, and the orbit is parabola . In all cases, the vector $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ is directed along the symmetry axis of the conical section and points to the point of the closest position of the point particle from the origin ( pericenter ).

Preservation by force inversely proportional to the square of the distance

Strength $\mathbf {F}$ ${\ displaystyle \ mathbf {F}}$ acting on the particle is assumed to be central . therefore

\mathbf {F} ={\frac {d\mathbf {p} }{dt}}=f(r){\frac {\mathbf {r} }{r}}=f(r)\mathbf {\hat {r}}

{\ displaystyle \ mathbf {F} = {\ frac {d \ mathbf {p}} {dt}} = f (r) {\ frac {\ mathbf {r}} {r}} = f (r) \ mathbf {\ hat {r}}}

for some function $f(r)$ ${\ displaystyle f (r)}$ radius $r$ ${\ displaystyle r}$ . Since the angular momentum $\mathbf {L} =\mathbf {r} \times \mathbf {p}$ ${\ displaystyle \ mathbf {L} = \ mathbf {r} \ times \ mathbf {p}}$ persists under the action of central forces, then ${\frac {d}{dt}}\mathbf {L} =0$ ${\ displaystyle {\ frac {d} {dt}} \ mathbf {L} = 0}$ and

{\frac {d}{dt}}(\mathbf {p} \times \mathbf {L} )={\frac {d\mathbf {p} }{dt}}\times \mathbf {L} =f(r)\mathbf {\hat {r}} \times \left(\mathbf {r} \times m{\frac {d\mathbf {r} }{dt}}\right)=f(r){\frac {m}{r}}\left[\mathbf {r} \left(\mathbf {r} \cdot {\frac {d\mathbf {r} }{dt}}\right)-r^{2}{\frac {d\mathbf {r} }{dt}}\right],

{\ displaystyle {\ frac {d} {dt}} (\ mathbf {p} \ times \ mathbf {L}) = {\ frac {d \ mathbf {p}} {dt}} \ times \ mathbf {L} = f (r) \ mathbf {\ hat {r}} \ times \ left (\ mathbf {r} \ times m {\ frac {d \ mathbf {r}} {dt}} \ right) = f (r) {\ frac {m} {r}} \ left [\ mathbf {r} \ left (\ mathbf {r} \ cdot {\ frac {d \ mathbf {r}} {dt}} \ right) -r ^ { 2} {\ frac {d \ mathbf {r}} {dt}} \ right],}

where the impulse is written as $\mathbf {p} =m{\frac {d\mathbf {r} }{dt}}$ ${\ displaystyle \ mathbf {p} = m {\ frac {d \ mathbf {r}} {dt}}}$ , and the triple vector product was simplified using the Lagrange formula

\mathbf {r} \times \left(\mathbf {r} \times {\frac {d\mathbf {r} }{dt}}\right)=\mathbf {r} \left(\mathbf {r} \cdot {\frac {d\mathbf {r} }{dt}}\right)-r^{2}{\frac {d\mathbf {r} }{dt}}.

{\ displaystyle \ mathbf {r} \ times \ left (\ mathbf {r} \ times {\ frac {d \ mathbf {r}} {dt}} \ right) = \ mathbf {r} \ left (\ mathbf { r} \ cdot {\ frac {d \ mathbf {r}} {dt}} \ right) -r ^ {2} {\ frac {d \ mathbf {r}} {dt}}.}

Identity

{\frac {d}{dt}}(\mathbf {r} \cdot \mathbf {r} )=2\mathbf {r} \cdot {\frac {d\mathbf {r} }{dt}}={\frac {d}{dt}}(r^{2})=2r{\frac {dr}{dt}}

{\ displaystyle {\ frac {d} {dt}} (\ mathbf {r} \ cdot \ mathbf {r}) = 2 \ mathbf {r} \ cdot {\ frac {d \ mathbf {r}} {dt} } = {\ frac {d} {dt}} (r ^ {2}) = 2r {\ frac {dr} {dt}}}

leads to the equation

{\frac {d}{dt}}(\mathbf {p} \times \mathbf {L} )=-mf(r)r^{2}\left[{\frac {1}{r}}{\frac {d\mathbf {r} }{dt}}-{\frac {\mathbf {r} }{r^{2}}}{\frac {dr}{dt}}\right]=-mf(r)r^{2}{\frac {d}{dt}}\left({\frac {\mathbf {r} }{r}}\right).

{\ displaystyle {\ frac {d} {dt}} (\ mathbf {p} \ times \ mathbf {L}) = - mf (r) r ^ {2} \ left [{\ frac {1} {r} } {\ frac {d \ mathbf {r}} {dt}} - {\ frac {\ mathbf {r}} {r ^ {2}}} {\ frac {dr} {dt}} \ right] = - mf (r) r ^ {2} {\ frac {d} {dt}} \ left ({\ frac {\ mathbf {r}} {r}} \ right).}

For a special case of central force, which is inversely proportional to the square of the distance $f(r)={\frac {-k}{r^{2}}}$ ${\ displaystyle f (r) = {\ frac {-k} {r ^ {2}}}}$ last expression is equal

{\frac {d}{dt}}(\mathbf {p} \times \mathbf {L} )=mk{\frac {d}{dt}}\left({\frac {\mathbf {r} }{r}}\right)={\frac {d}{dt}}(mk\mathbf {\hat {r}} ).

{\ displaystyle {\ frac {d} {dt}} (\ mathbf {p} \ times \ mathbf {L}) = mk {\ frac {d} {dt}} \ left ({\ frac {\ mathbf {r }} {r}} \ right) = {\ frac {d} {dt}} (mk \ mathbf {\ hat {r}}).}

Then $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ saved in this case

{\frac {d}{dt}}\mathbf {A} ={\frac {d}{dt}}(\mathbf {p} \times \mathbf {L} )-{\frac {d}{dt}}(mk\mathbf {\hat {r}} )=0.

{\ displaystyle {\ frac {d} {dt}} \ mathbf {A} = {\ frac {d} {dt}} (\ mathbf {p} \ times \ mathbf {L}) - {\ frac {d} {dt}} (mk \ mathbf {\ hat {r}}) = 0.}

As shown below , the Laplace-Runge-Lenz vector $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ is a special case of a generalized conserved vector ${\mathcal {A}}$ ${\ displaystyle {\ mathcal {A}}}$ , which can be defined for any central force ^[11] ^[12] . However, most central forces do not form closed orbits (see Bertrand’s theorem ), a similar vector ${\mathcal {A}}$ ${\ displaystyle {\ mathcal {A}}}$ rarely has a simple definition and in the general case is a multi-valued function of the angle $\theta$ ${\ displaystyle \ theta}$ between $\mathbf {r}$ ${\ displaystyle \ mathbf {r}}$ and ${\mathcal {A}}$ ${\ displaystyle {\ mathcal {A}}}$ .

Change under the influence of disturbing central forces

Fig. 5: Slowly precessing elliptical orbit, with eccentricity

\scriptstyle e=0{,}9

{\ displaystyle \ scriptstyle e = 0 {,} 9}

. Such a precession arises in the Kepler problem if the attractive central force is slightly different from Newton's law of gravity. The precession rate can be calculated using the formulas given in the paragraph.

In many practical problems, such as planetary motion, the interaction between two bodies only approximately depends inversely on the square of the distance. In such cases, the Laplace - Runge - Lenz vector $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ not permanent. However, if the disturbing potential $h(r)$ ${\ displaystyle h (r)}$ depends only on the distance, then the total energy $E$ ${\ displaystyle E}$ and angular momentum vector $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$ saved. Therefore, the trajectory is still perpendicular to $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$ planes and magnitude $A$ ${\ displaystyle A}$ stored according to the equation $A^{2}=m^{2}k^{2}+2mEL^{2}$ ${\ displaystyle A ^ {2} = m ^ {2} k ^ {2} + 2mEL ^ {2}}$ . Therefore direction $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ slowly rotates in orbit in the plane. Using the canonical perturbation theory and the action-angle coordinates , we can directly show ^[2] that $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ rotates with speed

{\frac {\partial }{\partial L}}\langle h(r)\rangle ={\frac {\partial }{\partial L}}\left\{{\frac {1}{T}}\int \limits _{0}^{T}h(r)\,dt\right\}={\frac {\partial }{\partial L}}\left\{{\frac {m}{L^{2}}}\int \limits _{0}^{2\pi }r^{2}h(r)\,d\theta \right\},

{\ displaystyle {\ frac {\ partial} {\ partial L}} \ langle h (r) \ rangle = {\ frac {\ partial} {\ partial L}} \ left \ {{\ frac {1} {T }} \ int \ limits _ {0} ^ {T} h (r) \, dt \ right \} = {\ frac {\ partial} {\ partial L}} \ left \ {{\ frac {m} { L ^ {2}}} \ int \ limits _ {0} ^ {2 \ pi} r ^ {2} h (r) \, d \ theta \ right \},}

Where $T$ ${\ displaystyle T}$ - period of orbital motion and equality $L\,dt=mr^{2}\,d\theta$ ${\ displaystyle L \, dt = mr ^ {2} \, d \ theta}$ was used to convert the time integral to the angle integral (Fig. 5). For example, taking into account the effects of the general theory of relativity , we come to an additive that, in contrast to the usual Newtonian gravitational force, depends inversely on the cube of distance ^[25] :

h(r)={\frac {kL^{2}}{m^{2}c^{2}}}\left({\frac {1}{r^{3}}}\right).

{\ displaystyle h (r) = {\ frac {kL ^ {2}} {m ^ {2} c ^ {2}}} \ left ({\ frac {1} {r ^ {3}}} \ right ).}

Substituting this function into the integral and using the equation

{\frac {1}{r}}={\frac {mk}{L^{2}}}\left(1+{\frac {A}{mk}}\cos \theta \right),

{\ displaystyle {\ frac {1} {r}} = {\ frac {mk} {L ^ {2}}} \ left (1 + {\ frac {A} {mk}} \ cos \ theta \ right) ,}

to express $r$ ${\ displaystyle r}$ in terms $\theta$ ${\ displaystyle \ theta}$ , the pericenter precession rate caused by this disturbance is written as ^[25]

{\frac {6\pi k^{2}}{TL^{2}c^{2}}}.

{\ displaystyle {\ frac {6 \ pi k ^ {2}} {TL ^ {2} c ^ {2}}}.}

which is close in value to the magnitude of the precession for Mercury by the unexplained Newtonian theory of gravity ^[26] . This expression is used to evaluate the precession associated with amendments to the general theory of relativity for double pulsars ^[27] . This agreement with experiment is a strong argument in favor of the general theory of relativity ^[28] .

Group Theory

Lee Conversion

Fig. 6: Lie transformation from which the conservation of the Laplace - Runge - Lenz vector is derived

\scriptstyle \mathbf {A}

{\ displaystyle \ scriptstyle \ mathbf {A}}

. When a scalable parameter

\scriptstyle \lambda

{\ displaystyle \ scriptstyle \ lambda}

changes, energy and angular momentum also change, but the eccentricity

\scriptstyle e

{\ displaystyle \ scriptstyle e}

and vector

\scriptstyle \mathbf {A}

{\ displaystyle \ scriptstyle \ mathbf {A}}

do not change.

There is another method for deriving the Laplace – Runge – Lenz vector, using coordinate variation without involving velocities ^[29] . Coordinate scaling $\mathbf {r}$ ${\ displaystyle \ mathbf {r}}$ and time $t$ ${\ displaystyle t}$ with varying degrees of parameter $\lambda$ ${\ displaystyle \ lambda}$ (fig. 6)

t\to \lambda ^{3}t,\;\mathbf {r} \to \lambda ^{2}\mathbf {r} ,\;\mathbf {p} \to {\frac {1}{\lambda }}\mathbf {p} .

{\ displaystyle t \ to \ lambda ^ {3} t, \; \ mathbf {r} \ to \ lambda ^ {2} \ mathbf {r}, \; \ mathbf {p} \ to {\ frac {1} {\ lambda}} \ mathbf {p}.}

This transformation changes the total angular momentum. $L$ ${\ displaystyle L}$ and energy $E$ ${\ displaystyle E}$

L\to \lambda L,\;E\to {\frac {1}{\lambda ^{2}}}E,

{\ displaystyle L \ to \ lambda L, \; E \ to {\ frac {1} {\ lambda ^ {2}}} E,}

but retains the work $EL^{2}$ ${\ displaystyle EL ^ {2}}$ . It follows that the eccentricity $e$ ${\ displaystyle e}$ and value $A$ ${\ displaystyle A}$ stored in the previously mentioned equation

A^{2}=m^{2}k^{2}e^{2}=m^{2}k^{2}+2mEL^{2}.

{\ displaystyle A ^ {2} = m ^ {2} k ^ {2} e ^ {2} = m ^ {2} k ^ {2} + 2mEL ^ {2}.}

Direction $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ also retained, since the semi-axes do not change when scaling. This transformation leaves true to Kepler’s third law , namely, that the semi-axis $a$ ${\ displaystyle a}$ and period $T$ ${\ displaystyle T}$ form a constant $T^{2}/a^{3}$ ${\ displaystyle T ^ {2} / a ^ {3}}$ .

Poisson Brackets

For three components $L_{i}$ ${\ displaystyle L_ {i}}$ angular momentum vectors $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$ can define Poisson brackets

[L_{i},\;L_{j}]=-\sum _{s=1}^{3}\varepsilon _{ijs}L_{s},

{\ displaystyle [L_ {i}, \; L_ {j}] = - \ sum _ {s = 1} ^ {3} \ varepsilon _ {ijs} L_ {s},}

where is the index $i$ ${\ displaystyle i}$ runs through the values 1, 2, 3 and $\varepsilon _{ijs}$ ${\ displaystyle \ varepsilon _ {ijs}}$ Is an absolutely antisymmetric tensor , i.e. the Levi-Civita symbol (third summation index $s$ ${\ displaystyle s}$ so as not to be confused with the power parameter $k$ ${\ displaystyle k}$ defined above ). Square brackets (rather than curly brackets) are used as Poisson brackets, as in the literature, and, in particular, to interpret them as quantum-mechanical commutation relations in the next section .

As shown above , the modified Laplace - Runge - Lenz vector $\mathbf {D}$ ${\ displaystyle \ mathbf {D}}$ can be determined with the same dimension as the angular momentum , dividing $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ on $p_{0}$ ${\ displaystyle p_ {0}}$ . Poisson bracket $\mathbf {D}$ ${\ displaystyle \ mathbf {D}}$ with angular momentum vector $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$ recorded in a similar form

[D_{i},\;L_{j}]=\sum _{s=1}^{3}\varepsilon _{ijs}D_{s}.

{\ displaystyle [D_ {i}, \; L_ {j}] = \ sum _ {s = 1} ^ {3} \ varepsilon _ {ijs} D_ {s}.}

Poisson bracket $\mathbf {D}$ ${\ displaystyle \ mathbf {D}}$ with $\mathbf {D}$ ${\ displaystyle \ mathbf {D}}$ depends on the sign $E$ ${\ displaystyle E}$ that is, when the total energy $E$ ${\ displaystyle E}$ negative (elliptical orbits under the influence of a central force, which depends inversely with the square of the distance) or positive (hyperbolic orbits). For negative energies, the Poisson brackets take the form

[D_{i},\;D_{j}]=-\sum _{s=1}^{3}\varepsilon _{ijs}L_{s}.

{\ displaystyle [D_ {i}, \; D_ {j}] = - \ sum _ {s = 1} ^ {3} \ varepsilon _ {ijs} L_ {s}.}

While for positive energies, the Poisson brackets have the opposite sign

[D_{i},\;D_{j}]=\sum _{s=1}^{3}\varepsilon _{ijs}L_{s}.

{\ displaystyle [D_ {i}, \; D_ {j}] = \ sum _ {s = 1} ^ {3} \ varepsilon _ {ijs} L_ {s}.}

Casimir invariants for negative energies are determined by the following relations

C_{1}=\mathbf {D} \cdot \mathbf {D} +\mathbf {L} \cdot \mathbf {L} ={\frac {mk^{2}}{2|E|}},

{\ displaystyle C_ {1} = \ mathbf {D} \ cdot \ mathbf {D} + \ mathbf {L} \ cdot \ mathbf {L} = {\ frac {mk ^ {2}} {2 | E |} },}

C_{2}=\mathbf {D} \cdot \mathbf {L} =0

{\ displaystyle C_ {2} = \ mathbf {D} \ cdot \ mathbf {L} = 0}

and we have zero Poisson brackets for all components $\mathbf {D}$ ${\ displaystyle \ mathbf {D}}$ and $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$

[C_{1},\;L_{i}]=[C_{1},\;D_{i}]=[C_{2},\;L_{i}]=[C_{2},\;D_{i}]=0.

{\ displaystyle [C_ {1}, \; L_ {i}] = [C_ {1}, \; D_ {i}] = [C_ {2}, \; L_ {i}] = [C_ {2} , \; D_ {i}] = 0.}

$C_{2}$ ${\ displaystyle C_ {2}}$ is zero due to the orthogonality of the vectors. However, another invariant $C_{1}$ ${\ displaystyle C_ {1}}$ nontrivial and depends only on $m$ ${\ displaystyle m}$ , $k$ ${\ displaystyle k}$ and $E$ ${\ displaystyle E}$ . This invariant can be used to derive the spectrum of the hydrogen atom using only the quantum-mechanical canonical switching relation, instead of the more complex Schrödinger equation .

Noether's theorem

Noether's theorem states that the infinitesimal variation of the generalized coordinates of a physical system

\delta q_{i}=\varepsilon g_{i}(\mathbf {q} ,\;\mathbf {\dot {q}} ,\;t)

{\ displaystyle \ delta q_ {i} = \ varepsilon g_ {i} (\ mathbf {q}, \; \ mathbf {\ dot {q}}, \; t)}

causes a change in the Lagrange function in the first order to the full time derivative

\delta L=\varepsilon {\frac {d}{dt}}G(\mathbf {q} ,\;t)

{\ displaystyle \ delta L = \ varepsilon {\ frac {d} {dt}} G (\ mathbf {q}, \; t)}

corresponds to conservation

J=-G+\sum _{i}g_{i}\left({\frac {\partial L}{\partial {\dot {q}}_{i}}}\right).

{\ displaystyle J = -G + \ sum _ {i} g_ {i} \ left ({\ frac {\ partial L} {\ partial {\ dot {q}} _ {i}}} right).}

The stored component of the Laplace - Runge - Lenz vector $A_{s}$ ${\ displaystyle A_ {s}}$ corresponds to the variation of coordinates

\delta x_{i}=\varepsilon m({\dot {x}}_{i}x_{s}-x_{i}{\dot {x}}_{s}),

{\ displaystyle \ delta x_ {i} = \ varepsilon m ({\ dot {x}} _ {i} x_ {s} -x_ {i} {\ dot {x}} _ {s}),}

Where $i$ ${\ displaystyle i}$ equals 1, 2 and 3, and $x_{i}$ ${\ displaystyle x_ {i}}$ and ${\dot {x}}_{i}$ ${\ displaystyle {\ dot {x}} _ {i}}$ - $i$ ${\ displaystyle i}$ components of position vectors $\mathbf {r}$ ${\ displaystyle \ mathbf {r}}$ and speed $\mathbf {\dot {r}}$ ${\ displaystyle \ mathbf {\ dot {r}}}$ , respectively. Lagrange function of this system

L={\frac {m{\dot {r}}^{2}}{2}}+{\frac {k}{r}}.

{\ displaystyle L = {\ frac {m {\ dot {r}} ^ {2}} {2}} + {\ frac {k} {r}}.}

The resulting change in the first order of the Lagrange function is written as

\delta L=\varepsilon mk{\frac {d}{dt}}\left({\frac {x_{s}}{r}}\right).

{\ displaystyle \ delta L = \ varepsilon mk {\ frac {d} {dt}} \ left ({\ frac {x_ {s}} {r}} \ right).}

This results in component persistence. $A_{s}$ ${\ displaystyle A_ {s}}$

A_{s}=p^{2}x_{s}-p_{s}(\mathbf {r} \cdot \mathbf {p} )-mk\left({\frac {x_{s}}{r}}\right)=\left(\mathbf {p} \times \left(\mathbf {r} \times \mathbf {p} \right)\right)_{s}-mk\left({\frac {x_{s}}{r}}\right).

{\ displaystyle A_ {s} = p ^ {2} x_ {s} -p_ {s} (\ mathbf {r} \ cdot \ mathbf {p}) -mk \ left ({\ frac {x_ {s}} {r}} \ right) = \ left (\ mathbf {p} \ times \ left (\ mathbf {r} \ times \ mathbf {p} \ right) \ right) _ {s} -mk \ left ({\ frac {x_ {s}} {r}} \ right).}

Conservation Laws and Symmetry

Variation of the coordinate preserves the length of the Laplace - Runge - Lenz vector (see Noether's theorem ). This conservation can be considered as some symmetry of the system. In classical mechanics , symmetries are continuous operations that map one orbit to another without changing the energy of the system; in quantum mechanics , symmetries are continuous operations that mix atomic orbitals without changing the total energy. For example, any central force leading to the conservation of angular momentum $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$ . In physics, conservative central forces are usually found that have the symmetry of the SO rotation group (3) . Classically, a complete rotation of the system does not affect the energy of the orbit; quantum mechanically, rotations mix spherical functions with the same quantum number $l$ ${\ displaystyle l}$ (degenerate states) without changing energy.

Fig. 7: The family of circles of the travel time curve of a pulse for a given energy

\scriptstyle l

{\ displaystyle \ scriptstyle l}

. All circles go through two points

\scriptstyle \pm p_{0}=\pm {\sqrt {2m|E|}}

{\ displaystyle \ scriptstyle \ pm p_ {0} = \ pm {\ sqrt {2m | E |}}}

on axis

\scriptstyle p_{x}

{\ displaystyle \ scriptstyle p_ {x}}

(compare with fig. 3). This family of hodographs corresponds to the family of circles of Apollonius , and

\scriptstyle \sigma

{\ displaystyle \ scriptstyle \ sigma}

isosurfaces of bipolar coordinates .

Symmetry increases for the central force inverse to the square of the distance. The specific symmetry of the Kepler problem leads to the conservation of the angular momentum as a vector $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$ , and the Laplace - Runge - Lenz vector $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ (as defined above ) and quantum mechanically ensures that the energy levels of the hydrogen atom are independent of the quantum numbers of the angular momentum $l$ ${\ displaystyle l}$ and $m$ ${\ displaystyle m}$ . Symmetry is finer, because the symmetry operation must take place in a space of greater dimension; such symmetries are often called hidden symmetries ^[29] . Classically, the higher symmetry of the Kepler problem allows for continuous changes in the orbits that conserve energy, but not the angular momentum; in other words, orbits with the same energy but different angular momenta (eccentricity) can be transformed continuously into each other. Quantum mechanically, this corresponds to the mixing of orbitals that differ in quantum numbers $l$ ${\ displaystyle l}$ and $m$ ${\ displaystyle m}$ atomic orbitals of type $s$ ${\ displaystyle s}$ ( $l=0$ ${\ displaystyle l = 0}$ ) and $p$ ${\ displaystyle p}$ ( $l=1$ ${\ displaystyle l = 1}$ ) Such mixing cannot be done with ordinary three-dimensional broadcasts or rotations, but it is equivalent to rotation in space with a higher dimension.

A coupled system with negative total energy has SO (4) symmetry, which preserves the length of four-dimensional vectors

|\mathbf {e} |^{2}=e_{1}^{2}+e_{2}^{2}+e_{3}^{2}+e_{4}^{2}.

{\ displaystyle | \ mathbf {e} | ^ {2} = e_ {1} ^ {2} + e_ {2} ^ {2} + e_ {3} ^ {2} + e_ {4} ^ {2} .}

In 1935, Vladimir Fok showed that the Kepler quantum-mechanical problem is equivalent to the problem of a free particle bounded by a four-dimensional hypersphere ^[7] . In particular, Fock showed that the wave function of the Schrödinger equation in the momentum space for the Kepler problem is a four-dimensional generalization of the stereographic projection of spherical functions from a 3-sphere into three-dimensional space. Rotation of the hypersphere and redesigning leads to a continuous transformation of elliptical orbits that does not change energy; quantum mechanically, this corresponds to mixing all the orbitals with the same principal quantum number $n$ ${\ displaystyle n}$ . Valentine Bargman subsequently noted that the Poisson brackets for the angular momentum vector $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$ and scaled Laplace - Runge - Lenz vector $\mathbf {D}$ ${\ displaystyle \ mathbf {D}}$ form a Lie algebra for $SO(4)$ ${\ displaystyle SO (4)}$ . ^[8] Simply put, these six quantities $\mathbf {D}$ ${\ displaystyle \ mathbf {D}}$ and $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$ correspond to six conserved angular momenta in four dimensions associated with six possible simple rotations in this space (there are six ways to choose two axes from four). This conclusion does not imply that our Universe is a four-dimensional hypersphere ; it just means that this specific problem of physics ( the two-body problem for the central force, which depends on the square of the distance) is mathematically equivalent to a free particle on a four-dimensional hypersphere.

A scattered system with positive total energy has the SO (3,1) symmetry, which preserves the length of the 4-vector in space with the Minkowski metric

ds^{2}=e_{1}^{2}+e_{2}^{2}+e_{3}^{2}-e_{4}^{2}.

{\ displaystyle ds ^ {2} = e_ {1} ^ {2} + e_ {2} ^ {2} + e_ {3} ^ {2} -e_ {4} ^ {2}.}

Fock ^[7] and Bargman ^[8] considered both negative and positive energies. They were also considered encyclopedically by Bender and Itzikson ^[30] ^[31] .

Symmetry of rotations in four-dimensional space

Fig. 8: Hodograph of the pulse in Fig. 7 corresponds to the stereographic projection of large circles of four-dimensional

\scriptstyle \eta

{\ displaystyle \ scriptstyle \ eta}

spheres of unit radius. All the big circles cross

\scriptstyle \eta _{x}

{\ displaystyle \ scriptstyle \ eta _ {x}}

axis that is perpendicular to the page. Projection from the North Pole (unit vector

\scriptstyle \mathbf {w}

{\ displaystyle \ scriptstyle \ mathbf {w}}

) to (

\scriptstyle \eta _{x}

{\ displaystyle \ scriptstyle \ eta _ {x}}

-

\scriptstyle \eta _{y}

{\ displaystyle \ scriptstyle \ eta _ {y}}

) planes as shown for the purple hodograph by a dashed black line. Large circle at latitude

\scriptstyle \alpha

{\ displaystyle \ scriptstyle \ alpha}

eccentricity

\scriptstyle e=\sin \alpha

{\ displaystyle \ scriptstyle e = \ sin \ alpha}

. The colors of the large circles shown here correspond to the colors of their hodographs in Fig. 7.

The connection between the Kepler problem and rotations in the four-dimensional space SO (4) can be rather simply visualized ^[30] ^[32] ^[33] . Let the Cartesian coordinates , which are denoted by $(w,\;x,\;y,\;z)$ ${\ displaystyle (w, \; x, \; y, \; z)}$ where $(x,\;y,\;z)$ ${\ displaystyle (x, \; y, \; z)}$ represent the Cartesian coordinates of the usual position of a three-dimensional vector $\mathbf {r}$ ${\ displaystyle \ mathbf {r}}$ . Three-dimensional momentum vector $\mathbf {p}$ ${\ displaystyle \ mathbf {p}}$ associated with a four-dimensional vector ${\boldsymbol {\eta }}$ ${\ displaystyle {\ boldsymbol {\ eta}}}$ on a four-dimensional unit sphere by

{\boldsymbol {\eta }}={\frac {p^{2}-p_{0}^{2}}{p^{2}+p_{0}^{2}}}\mathbf {\hat {w}} +{\frac {2p_{0}}{p^{2}+p_{0}^{2}}}\mathbf {p} ={\frac {mk-rpp_{0}}{mk}}\mathbf {\hat {w}} +{\frac {rp_{0}}{mk}}\mathbf {p} ,

{\ displaystyle {\ boldsymbol {\ eta}} = {\ frac {p ^ {2} -p_ {0} ^ {2}} {p ^ {2} + p_ {0} ^ {2}}} \ mathbf {\ hat {w}} + {\ frac {2p_ {0}} {p ^ {2} + p_ {0} ^ {2}}} \ mathbf {p} = {\ frac {mk-rpp_ {0} } {mk}} \ mathbf {\ hat {w}} + {\ frac {rp_ {0}} {mk}} \ mathbf {p},}

Where $\mathbf {\hat {w}}$ ${\ displaystyle \ mathbf {\ hat {w}}}$ Is the unit vector along the new axis $w$ ${\ displaystyle w}$ . Insofar as ${\boldsymbol {\eta }}$ ${\ displaystyle {\ boldsymbol {\ eta}}}$ has only three independent components, then this vector can be inverted by obtaining an expression for $\mathbf {p}$ ${\ displaystyle \ mathbf {p}}$ . For example, for a component $x$ ${\ displaystyle x}$

p_{x}=p_{0}{\frac {\eta _{x}}{1-\eta _{w}}}

{\ displaystyle p_ {x} = p_ {0} {\ frac {\ eta _ {x}} {1- \ eta _ {w}}}}

and similarly for $p_{y}$ ${\ displaystyle p_ {y}}$ and $p_{z}$ ${\ displaystyle p_ {z}}$ . In other words, a three-dimensional vector $\mathbf {p}$ ${\ displaystyle \ mathbf {p}}$ is a stereographic projection of a four-dimensional vector ${\boldsymbol {\eta }}$ ${\ displaystyle {\ boldsymbol {\ eta}}}$ times $p_{0}$ ${\ displaystyle p_ {0}}$ (fig. 8).

Without loss of generality, we can eliminate normal rotational symmetry by choosing Cartesian coordinates , where the axis $z$ ${\ displaystyle z}$ directed along the angular momentum vector $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$ , and the travel time curve is located as shown in Figure 7, with the centers of the circles on the axis $y$ ${\ displaystyle y}$ . Since the movement occurs in the plane, and $\mathbf {p}$ ${\ displaystyle \ mathbf {p}}$ and $L$ ${\ displaystyle L}$ orthogonal $p_{z}=\eta _{z}=0$ ${\ displaystyle p_ {z} = \ eta _ {z} = 0}$ , and you can focus on the three-dimensional vector ${\boldsymbol {\eta }}=(\eta _{w},\;\eta _{x},\;\eta _{y})$ ${\ displaystyle {\ boldsymbol {\ eta}} = (\ eta _ {w}, \; \ eta _ {x}, \; \ eta _ {y})}$ . The family of Apollonius circles of the travel time curves (Fig. 7) corresponds to the set of large circles on a three-dimensional sphere ${\boldsymbol {\eta }}$ ${\ displaystyle {\ boldsymbol {\ eta}}}$ all of which intersect the axis $\eta _{x}$ ${\ displaystyle \ eta _ {x}}$ in these two tricks $\eta _{x}=\pm 1$ ${\ displaystyle \ eta _ {x} = \ pm 1}$ corresponding to the foci of the travel time curve for $p_{x}=\pm p_{0}$ ${\ displaystyle p_ {x} = \ pm p_ {0}}$ . Large circles are connected by simple rotation around the axis. $\eta _{x}$ ${\ displaystyle \ eta _ {x}}$ (fig. 8). This rotational symmetry converts all orbits with the same energy into each other; however, such a rotation is orthogonal to ordinary three-dimensional rotations, as it transforms the fourth dimension $\eta _{w}$ ${\ displaystyle \ eta _ {w}}$ . This higher symmetry is characteristic of the Kepler problem and corresponds to the conservation of the Laplace – Runge – Lenz vector.

An elegant solution to the Kepler problem using angle-action variables can be obtained by eliminating the excessive four-dimensional coordinate ${\boldsymbol {\eta }}$ ${\ displaystyle {\ boldsymbol {\ eta}}}$ and using elliptical cylindrical coordinates $(\alpha ,\;\beta ,\;\varphi )$ ${\ displaystyle (\ alpha, \; \ beta, \; \ varphi)}$ ^[34]

\eta _{w}=\mathrm {cn} \,\alpha \,\mathrm {cn} \,\beta ,

{\ displaystyle \ eta _ {w} = \ mathrm {cn} \, \ alpha \, \ mathrm {cn} \, \ beta,}

\eta _{x}=\mathrm {sn} \,\alpha \,\mathrm {dn} \,\beta \cos \varphi ,

{\ displaystyle \ eta _ {x} = \ mathrm {sn} \, \ alpha \, \ mathrm {dn} \, \ beta \ cos \ varphi,}

\eta _{y}=\mathrm {sn} \,\alpha \,\mathrm {dn} \,\beta \sin \varphi ,

{\ displaystyle \ eta _ {y} = \ mathrm {sn} \, \ alpha \, \ mathrm {dn} \, \ beta \ sin \ varphi,}

\eta _{z}=\mathrm {dn} \,\alpha \,\mathrm {sn} \,\beta ,

{\ displaystyle \ eta _ {z} = \ mathrm {dn} \, \ alpha \, \ mathrm {sn} \, \ beta,}

where Jacobi elliptic functions are used : $\mathrm {sn}$ ${\ displaystyle \ mathrm {sn}}$ , $\mathrm {cn}$ ${\ displaystyle \ mathrm {cn}}$ and $\mathrm {dn}$ ${\ displaystyle \ mathrm {dn}}$ .

Application and generalization

Quantum Mechanics of the Hydrogen Atom

Fig. 9: Hydrogen atom energy levels predicted using the commutation relations of the angular momentum and the Laplace – Runge – Lenz vector operators; these energy levels have been tested experimentally.

Poisson brackets provide an easy way to quantize a classical system . The commutator of two quantum-mechanical operators is equal to the Poisson bracket of the corresponding classical variables, multiplied by $i\hbar$ ${\ displaystyle i \ hbar}$ ^[35] . Performing this quantization and calculating the eigenvalues $C_{1}$ ${\ displaystyle C_ {1}}$ of the Casimir operator for the Kepler problem, Wolfgang Pauli derived the energy spectrum of a hydrogen-like atom (Fig. 9) and, thus, its atomic emission spectrum ^[3] . This elegant solution was obtained before the Schrödinger equation was obtained ^[36] .

A peculiarity of the quantum-mechanical operator for the Laplace-Runge-Lenz vector $\mathbf {A}$ ${\ displaystyle \ mathbf {A}}$ lies in the fact that the momentum and angular momentum operators do not commute with each other, therefore, the vector product $\mathbf {p}$ ${\ displaystyle \ mathbf {p}}$ and $\mathbf {L}$ ${\ displaystyle \ mathbf {L}}$ to be determined carefully ^[37] . As a rule, operators in a Cartesian coordinate system $A_{s}$ ${\ displaystyle A_ {s}}$ defined using a symmetrized product

A_{s}=-mk{\hat {r}}_{s}+{\frac {1}{2}}\sum _{i=1}^{3}\sum _{j=1}^{3}\varepsilon _{sij}(p_{i}l_{j}-l_{i}p_{j}),

{\ displaystyle A_ {s} = - mk {\ hat {r}} _ {s} + {\ frac {1} {2}} \ sum _ {i = 1} ^ {3} \ sum _ {j = 1} ^ {3} \ varepsilon _ {sij} (p_ {i} l_ {j} -l_ {i} p_ {j}),}

from which the corresponding ladder operators are determined

A_{0}=A_{3},

{\ displaystyle A_ {0} = A_ {3},}

A_{\pm 1}=\mp {\frac {1}{\sqrt {2}}}(A_{1}\pm iA_{2}).

{\ displaystyle A _ {\ pm 1} = \ mp {\ frac {1} {\ sqrt {2}}} (A_ {1} \ pm iA_ {2}).}

The normalized operator of the first Casimir invariant can be defined in a similar way.

C_{1}=-{\frac {mk^{2}}{2\hbar ^{2}}}H^{-1}-I,

{\ displaystyle C_ {1} = - {\ frac {mk ^ {2}} {2 \ hbar ^ {2}}} H ^ {- 1} -I,}

Where $H^{-1}$ ${\ displaystyle H ^ {- 1}}$ Is the operator inverse to the energy operator ( Hamiltonian ) and $I$ ${\ displaystyle I}$ Is the unit operator. Applying these ladder operators to eigenstates $|lmn\rangle$ ${\ displaystyle | lmn \ rangle}$ operators of the total angular momentum, azimuthal angular momentum and energy, it can be shown that the eigenstates of the first Casimir operator are given by the formula $n^{2}-1$ ${\ displaystyle n ^ {2} -1}$ . Therefore, energy levels are given by

E_{n}=-{\frac {mk^{2}}{2\hbar ^{2}n^{2}}},

{\ displaystyle E_ {n} = - {\ frac {mk ^ {2}} {2 \ hbar ^ {2} n ^ {2}}},}

which is identical to Rydberg’s formula for the hydrogen atom (Fig. 9).

Generalization to other potentials and SRT

The Laplace – Runge – Lenz vector was generalized to other potentials and even to the special theory of relativity . The most general form of this vector can be written as ^[11]

{\mathcal {A}}=\left({\frac {\partial \xi }{\partial u}}\right)(\mathbf {p} \times \mathbf {L} )+\left[\xi -u\left({\frac {\partial \xi }{\partial u}}\right)\right]L^{2}\mathbf {\hat {r}} ,

{\ displaystyle {\ mathcal {A}} = \ left ({\ frac {\ partial \ xi} {\ partial u}} \ right) (\ mathbf {p} \ times \ mathbf {L}) + \ left [ \ xi -u \ left ({\ frac {\ partial \ xi} {\ partial u}} \ right) \ right] L ^ {2} \ mathbf {\ hat {r}},}

Where $u=1/r$ ${\ displaystyle u = 1 / r}$ (see Bertrand's theorem ) and $\xi =\cos \theta$ ${\ displaystyle \ xi = \ cos \ theta}$ with angle $\theta$ ${\ displaystyle \ theta}$ defined as

\theta =L\int \limits ^{u}{\frac {du}{\sqrt {m^{2}c^{2}(\gamma ^{2}-1)-L^{2}u^{2}}}}.

{\ displaystyle \ theta = L \ int \ limits ^ {u} {\ frac {du} {\ sqrt {m ^ {2} c ^ {2} (\ gamma ^ {2} -1) -L ^ {2 } u ^ {2}}}}.}

Here $\gamma$ ${\ displaystyle \ gamma}$ - relativistic factor . As before, you can get the persisting binormal vector $\mathbf {B}$ ${\ displaystyle \ mathbf {B}}$ taking a vector product with a conserved angular momentum vector

{\mathcal {B}}=\mathbf {L} \times {\mathcal {A}}.

{\ displaystyle {\ mathcal {B}} = \ mathbf {L} \ times {\ mathcal {A}}.}

These two vectors can be combined into a persistent two-component tensor $W$ ${\ displaystyle W}$

{\mathcal {W}}=\alpha {\mathcal {A}}\otimes {\mathcal {A}}+\beta {\mathcal {B}}\otimes {\mathcal {B}}.

{\ displaystyle {\ mathcal {W}} = \ alpha {\ mathcal {A}} \ otimes {\ mathcal {A}} + \ beta {\ mathcal {B}} \ otimes {\ mathcal {B}}.}

As an example, we calculate the Laplace – Runge – Lenz vector for a nonrelativistic isotropic harmonic oscillator. ^[11] Consider the central force:

\mathbf {F} (r)=-k\mathbf {r}

{\ displaystyle \ mathbf {F} (r) = - k \ mathbf {r}}

the angular momentum vector is conserved, and therefore the motion occurs in the plane. The persisting tensor can be rewritten in a simpler form:

\mathbf {W} ={\frac {1}{2m}}\mathbf {p} \otimes \mathbf {p} +{\frac {k}{2}}\mathbf {r} \otimes \mathbf {r} ,

{\ displaystyle \ mathbf {W} = {\ frac {1} {2m}} \ mathbf {p} \ otimes \ mathbf {p} + {\ frac {k} {2}} \ mathbf {r} \ otimes \ mathbf {r},}

although it should be noted that $p$ ${\ displaystyle p}$ and $r$ ${\ displaystyle r}$ not perpendicular like $A$ ${\ displaystyle A}$ and $B$ ${\ displaystyle B}$ . The corresponding Laplace - Runge - Lenz vector has a more complex notation

\mathbf {A} ={\frac {1}{\sqrt {mr^{2}\omega _{0}A-mr^{2}E+L^{2}}}}\{(\mathbf {p} \times \mathbf {L} )+(mr\omega _{0}A-mrE)\mathbf {\hat {r}} \},

{\ displaystyle \ mathbf {A} = {\ frac {1} {\ sqrt {mr ^ {2} \ omega _ {0} A-mr ^ {2} E + L ^ {2}}}} \ {( \ mathbf {p} \ times \ mathbf {L}) + (mr \ omega _ {0} A-mrE) \ mathbf {\ hat {r}} \},}

Where $\omega _{0}={\sqrt {\frac {k}{m}}}$ ${\ displaystyle \ omega _ {0} = {\ sqrt {\ frac {k} {m}}}}$ Is the oscillator frequency.

Literature

↑ Arnold V.I. Mathematical methods of classical mechanics. 5th ed. - M .: URSS editorial, 2003 .-- 416 p. - ISBN 5-354-00341-5 . ; online is in electronic form 3rd ed. 1988, see Appendix 8, on page 381
↑ ¹ ² ³ ⁴ Goldstein G. Classical mechanics. 2nd ed. - M .: Nauka, 1975 .-- 415 p.
↑ ¹ ² ³ Pauli, W. Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik (German) // Zeitschrift für Physik : magazin. - 1926. - Bd. 36 . - S. 336–363 .
↑ ¹ ² Hamilton, W. R. The Hodograph, or a new Method of expressing in symbolical Language the Newtonian Law of Attraction (Eng.) // Proceedings of the Royal Irish Academy : journal. - 1847. - Vol. 3 . - P. 344-353 .
↑ Hickock F.A. Space Flight Graphics. - M .: Mechanical Engineering, 1968 .-- 133 p. - Ch. 3. Trajectory analysis using polar diagrams, p. 42.
↑ Gould H., Tobochnik Ya. Computer modeling in physics. T. 1. - M .: Mir, 1990 .-- 352 p. - ISBN 5-03-001593-0 . . - Task 4.9. Properties of orbits in the velocity space, p. 88.
↑ ¹ ² ³ Fock, V. Zur Theorie des Wasserstoffatoms (German) // Zeitschrift für Physik : magazin. - 1935. - Bd. 98 . - S. 145-154 .
↑ ¹ ² ³ Bargmann, V. Zur Theorie des Wasserstoffatoms: Bemerkungen zur gleichnamigen Arbeit von V. Fock (German) // Zeitschrift für Physik : magazin. - 1936. - Bd. 99 . - S. 576-582 .
↑ ¹ ² Goldstein, H. Prehistory of the Runge-Lenz vector (English) // American Journal of Physics : journal. - 1975 .-- Vol. 43 . - P. 735-738 .
Goldstein, H. More on the prehistory of the Runge-Lenz vector (English) // American Journal of Physics : journal. - 1976. - Vol. 44 . - P. 1123-1124 .
↑ ¹ ² ³ Hamilton, W. R. On the Application of the Method of Quaternions to some Dynamical Questions (English) // Proceedings of the Royal Irish Academy : journal. - 1847. - Vol. 3 . - P. Appendix III, pp. xxxvi — l .
↑ ¹ ² ³ ⁴ ⁵ Fradkin, D. M. Existence of the Dynamic Symmetries O ₄ and SU ₃ for All Classical Central Potential Problems (English) // Progress of Theoretical Physics : journal. - 1967. - Vol. 37 . - P. 798-812 .
↑ ¹ ² Yoshida, T. Two methods of generalization of the Laplace-Runge-Lenz vector (English) // European Journal of Physics : journal. - 1987. - Vol. 8 . - P. 258—259 .
↑ Hermann, J. Metodo d'investigare l'orbite de 'pianeti (neopr.) // Giornale de Letterati D'Italia. - 1710. - T. 2 . - S. 447-467 .
Hermann, J. Extrait d'une lettre de M. Herman à M. Bernoulli datée de Padoüe le 12. Juillet 1710 (Fr.) // Histoire de l'academie royale des sciences (Paris): magazine. - 1710. - Vol. 1732 . - P. 519-521 .
↑ Bernoulli, J. Extrait de la Réponse de M. Bernoulli à M. Herman datée de Basle le 7. Octobre 1710 (Fr.) // Histoire de l'academie royale des sciences (Paris): magazine. - 1710. - Vol. 1732 . - P. 521-544 .
↑ Laplace P. S. Traité de mécanique celeste. Tome I, Premiere Partie, Livre II. - Paris, 1799. - P. 165ff.
↑ Gibbs J. W. , Gibbs E. B. Vector Analysis. - New York: Scribners, 1901. - 436 p. - P. 135.
↑ Runge C. Vektoranalysis. Bd. I. - Leipzig: Hirzel, 1919 .-- 436 p.
↑ Lenz, W. Über den Bewegungsverlauf und Quantenzustände der gestörten Keplerbewegung (German) // Zeitschrift für Physik : magazin. - 1924. - Bd. 24 . - S. 197-207 .
↑ Evans, N. W. Superintegrability in classical mechanics (English) // Physical Review A : journal. - 1990. - Vol. 41 . - P. 5666-5676 .
↑ Sommerfeld A. Atomic Structure and Spectral Lines. - London: Methuen, 1923. - 118 p.
↑ ¹ ² ³ Landau LD , Lifshitz EM Mechanics. 3 ^rd ed. - Pergamon Press, 1976. - ISBN 0-08-029141-4 . . - P. 154; Landau L.D. , Lifshits E.M. Mechanics. 5th ed. - M .: Fizmatlit, 2004 .-- 224 p. - (Course in Theoretical Physics, Volume 1). - ISBN 5-9221-0055-6 . - § 15. The Keplerov problem, “a conserved vector”, p. 56; § 52. Conditionally periodic motion, a problem with a solution in polar coordinates, p. 217.
↑ Evans, N. W. Group theory of the Smorodinsky-Winternitz system (Eng.) // Journal of Mathematical Physics : journal. - 1991. - Vol. 32 . - P. 3369–3375 .
↑ Dulock, V. A .; McIntosh H. V. On the Degeneracy of the Kepler Problem (English) // Pacific Journal of Mathematics : journal. - 1966. - Vol. 19 . - P. 39–55 .
↑ Redmond, P. J. Generalization of the Runge-Lenz Vector in the Presence of an Electric Field ( Physical ) // Physical Review : journal. - 1964. - Vol. 133 . - P. B1352 — B1353 .
↑ ¹ ² Einstein, A. Erklärung der Perihelbeivegung der Merkur aus der allgemeinen Relativitätstheorie. (German) // Sitzungsberichte der der Königlich Preußischen Akademie der Wissenschaften: magazin. - 1915. - Bd. 47 , Nr. 2 . - S. 831-839 .
↑ Le Verrier, U. J. J. Sur la théorie de Mercure et sur le mouvement du périhélie de cette planète; Lettre de M. Le Verrier à M. Faye. (Fr.) // Comptes Rendus de l'Academie de Sciences (Paris): magazine. - 1859. - Vol. 49 . - P. 379-383 . [one]
↑ Will C. M. General Relativity, an Einstein Century Survey / Ed. by S. W. Hawking and W. Israel. - Cambridge: Cambridge University Press, 1979.
↑ Pais, A. Subtle is the Lord: The Science and the Life of Albert Einstein. - Oxford University Press, 1982.
Paice, Abraham. Scientific activity and life of Albert Einstein / Ed. A. A. Logunova . - M .: Nauka, 1989 .-- 566 p. - ISBN 5-02-014028-7 . .
↑ ¹ ² Prince, G. E .; Eliezer C. J. On the Lie symmetries of the classical Kepler problem (Eng.) // Journal of Physics A: Mathematical and General : journal. - 1981. - Vol. 14 . - P. 587-596 .
↑ ¹ ² Bander, M .; Itzykson C. Group Theory and the Hydrogen Atom (I ) // Reviews of Modern Physics : journal. - 1966. - Vol. 38 . - P. 330-345 .
↑ Bander, M .; Itzykson C. Group Theory and the Hydrogen Atom (II) (eng) // Reviews of Modern Physics : journal. - 1966. - Vol. 38 . - P. 346-358 .
↑ Rogers, H. H. Symmetry transformations of the classical Kepler problem (Eng.) // Journal of Mathematical Physics : journal. - 1973. - Vol. 14 . - P. 1125-1129 .
↑ Guillemin, V. Variations on a Theme by Kepler. - American Mathematical Society Colloquium Publications, volume 42, 1990. - ISBN 0-8218-1042-1 . .
↑ Lakshmanan, M .; Hasegawa H. On the canonical equivalence of the Kepler problem in coordinate and momentum spaces (English) // Journal of Physics A : journal. - Vol. 17 . - P. L889 — L893 .
↑ Dirac P. A. M. Principles of Quantum Mechanics. 4 ^th edition. - Oxford University Press, 1958.
↑ Schrödinger, E. Quantisierung als Eigenwertproblem (neopr.) // Annalen der Physik . - 1926.- T. 384 . - S. 361-376 .
↑ Bohm A. Quantum Mechanics: Foundations and Applications. 2 ^nd edition. - Springer Verlag, 1986. - P. 208-222.

Additional Reading

Leach, PGL; GP Flessas. Generalizations of the Laplace - Runge - Lenz vector (English) // J. Nonlinear Math. Phys. : journal. - 2003. - Vol. 10 . - P. 340-423 . The article is devoted to the generalization of the Laplace - Runge - Lenz vector to potentials other than Coulomb potentials. arxiv.org

[1] Arnold V.I. Mathematical methods of classical mechanics. 5th ed. - M .: URSS editorial, 2003 .-- 416 p. - ISBN 5-354-00341-5 . ; online is in electronic form 3rd ed. 1988, see Appendix 8, on page 381

[goldstein_1980-2] ¹ ² ³ ⁴ Goldstein G. Classical mechanics. 2nd ed. - M .: Nauka, 1975 .-- 415 p.

[pauli_1926-3] ¹ ² ³ Pauli, W. Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik (German) // Zeitschrift für Physik : magazin. - 1926. - Bd. 36 . - S. 336–363 .

[hamilton_1847_hodograph-4] ¹ ² Hamilton, W. R. The Hodograph, or a new Method of expressing in symbolical Language the Newtonian Law of Attraction (Eng.) // Proceedings of the Royal Irish Academy : journal. - 1847. - Vol. 3 . - P. 344-353 .

[5] Hickock F.A. Space Flight Graphics. - M .: Mechanical Engineering, 1968 .-- 133 p. - Ch. 3. Trajectory analysis using polar diagrams, p. 42.

[6] Gould H., Tobochnik Ya. Computer modeling in physics. T. 1. - M .: Mir, 1990 .-- 352 p. - ISBN 5-03-001593-0 . . - Task 4.9. Properties of orbits in the velocity space, p. 88.

[fock_1935-7] ¹ ² ³ Fock, V. Zur Theorie des Wasserstoffatoms (German) // Zeitschrift für Physik : magazin. - 1935. - Bd. 98 . - S. 145-154 .

[bargmann_1936-8] ¹ ² ³ Bargmann, V. Zur Theorie des Wasserstoffatoms: Bemerkungen zur gleichnamigen Arbeit von V. Fock (German) // Zeitschrift für Physik : magazin. - 1936. - Bd. 99 . - S. 576-582 .

[goldstein_1975_1976-9] ¹ ² Goldstein, H. Prehistory of the Runge-Lenz vector (English) // American Journal of Physics : journal. - 1975 .-- Vol. 43 . - P. 735-738 .
Goldstein, H. More on the prehistory of the Runge-Lenz vector (English) // American Journal of Physics : journal. - 1976. - Vol. 44 . - P. 1123-1124 .

[hamilton_1847_quaternions-10] ¹ ² ³ Hamilton, W. R. On the Application of the Method of Quaternions to some Dynamical Questions (English) // Proceedings of the Royal Irish Academy : journal. - 1847. - Vol. 3 . - P. Appendix III, pp. xxxvi — l .

[fradkin_1967-11] ¹ ² ³ ⁴ ⁵ Fradkin, D. M. Existence of the Dynamic Symmetries O ₄ and SU ₃ for All Classical Central Potential Problems (English) // Progress of Theoretical Physics : journal. - 1967. - Vol. 37 . - P. 798-812 .

[yoshida_1987-12] ¹ ² Yoshida, T. Two methods of generalization of the Laplace-Runge-Lenz vector (English) // European Journal of Physics : journal. - 1987. - Vol. 8 . - P. 258—259 .

[13] Hermann, J. Metodo d'investigare l'orbite de 'pianeti (neopr.) // Giornale de Letterati D'Italia. - 1710. - T. 2 . - S. 447-467 .
Hermann, J. Extrait d'une lettre de M. Herman à M. Bernoulli datée de Padoüe le 12. Juillet 1710 (Fr.) // Histoire de l'academie royale des sciences (Paris): magazine. - 1710. - Vol. 1732 . - P. 519-521 .

[14] Bernoulli, J. Extrait de la Réponse de M. Bernoulli à M. Herman datée de Basle le 7. Octobre 1710 (Fr.) // Histoire de l'academie royale des sciences (Paris): magazine. - 1710. - Vol. 1732 . - P. 521-544 .

[15] Laplace P. S. Traité de mécanique celeste. Tome I, Premiere Partie, Livre II. - Paris, 1799. - P. 165ff.

[16] Gibbs J. W. , Gibbs E. B. Vector Analysis. - New York: Scribners, 1901. - 436 p. - P. 135.

[17] Runge C. Vektoranalysis. Bd. I. - Leipzig: Hirzel, 1919 .-- 436 p.

[18] Lenz, W. Über den Bewegungsverlauf und Quantenzustände der gestörten Keplerbewegung (German) // Zeitschrift für Physik : magazin. - 1924. - Bd. 24 . - S. 197-207 .

[19] Evans, N. W. Superintegrability in classical mechanics (English) // Physical Review A : journal. - 1990. - Vol. 41 . - P. 5666-5676 .

[20] Sommerfeld A. Atomic Structure and Spectral Lines. - London: Methuen, 1923. - 118 p.

[landau_lifshitz_1976-21] ¹ ² ³ Landau LD , Lifshitz EM Mechanics. 3 ^rd ed. - Pergamon Press, 1976. - ISBN 0-08-029141-4 . . - P. 154; Landau L.D. , Lifshits E.M. Mechanics. 5th ed. - M .: Fizmatlit, 2004 .-- 224 p. - (Course in Theoretical Physics, Volume 1). - ISBN 5-9221-0055-6 . - § 15. The Keplerov problem, “a conserved vector”, p. 56; § 52. Conditionally periodic motion, a problem with a solution in polar coordinates, p. 217.

[22] Evans, N. W. Group theory of the Smorodinsky-Winternitz system (Eng.) // Journal of Mathematical Physics : journal. - 1991. - Vol. 32 . - P. 3369–3375 .

[23] Dulock, V. A .; McIntosh H. V. On the Degeneracy of the Kepler Problem (English) // Pacific Journal of Mathematics : journal. - 1966. - Vol. 19 . - P. 39–55 .

[24] Redmond, P. J. Generalization of the Runge-Lenz Vector in the Presence of an Electric Field ( Physical ) // Physical Review : journal. - 1964. - Vol. 133 . - P. B1352 — B1353 .

[einstein_1915-25] ¹ ² Einstein, A. Erklärung der Perihelbeivegung der Merkur aus der allgemeinen Relativitätstheorie. (German) // Sitzungsberichte der der Königlich Preußischen Akademie der Wissenschaften: magazin. - 1915. - Bd. 47 , Nr. 2 . - S. 831-839 .

[26] Le Verrier, U. J. J. Sur la théorie de Mercure et sur le mouvement du périhélie de cette planète; Lettre de M. Le Verrier à M. Faye. (Fr.) // Comptes Rendus de l'Academie de Sciences (Paris): magazine. - 1859. - Vol. 49 . - P. 379-383 . [one]

[27] Will C. M. General Relativity, an Einstein Century Survey / Ed. by S. W. Hawking and W. Israel. - Cambridge: Cambridge University Press, 1979.

[28] Pais, A. Subtle is the Lord: The Science and the Life of Albert Einstein. - Oxford University Press, 1982.
Paice, Abraham. Scientific activity and life of Albert Einstein / Ed. A. A. Logunova . - M .: Nauka, 1989 .-- 566 p. - ISBN 5-02-014028-7 . .

[prince_eliezer_1981-29] ¹ ² Prince, G. E .; Eliezer C. J. On the Lie symmetries of the classical Kepler problem (Eng.) // Journal of Physics A: Mathematical and General : journal. - 1981. - Vol. 14 . - P. 587-596 .

[bander_itzykson_1966-30] ¹ ² Bander, M .; Itzykson C. Group Theory and the Hydrogen Atom (I ) // Reviews of Modern Physics : journal. - 1966. - Vol. 38 . - P. 330-345 .

[31] Bander, M .; Itzykson C. Group Theory and the Hydrogen Atom (II) (eng) // Reviews of Modern Physics : journal. - 1966. - Vol. 38 . - P. 346-358 .

[rogers_1973-32] Rogers, H. H. Symmetry transformations of the classical Kepler problem (Eng.) // Journal of Mathematical Physics : journal. - 1973. - Vol. 14 . - P. 1125-1129 .

[33] Guillemin, V. Variations on a Theme by Kepler. - American Mathematical Society Colloquium Publications, volume 42, 1990. - ISBN 0-8218-1042-1 . .

[34] Lakshmanan, M .; Hasegawa H. On the canonical equivalence of the Kepler problem in coordinate and momentum spaces (English) // Journal of Physics A : journal. - Vol. 17 . - P. L889 — L893 .

[35] Dirac P. A. M. Principles of Quantum Mechanics. 4 ^th edition. - Oxford University Press, 1958.

[36] Schrödinger, E. Quantisierung als Eigenwertproblem (neopr.) // Annalen der Physik . - 1926.- T. 384 . - S. 361-376 .

[37] Bohm A. Quantum Mechanics: Foundations and Applications. 2 ^nd edition. - Springer Verlag, 1986. - P. 208-222.