Canonical transformation

In classical mechanics , an active transformation of the phase space is called canonical if it leaves essential aspects of the dynamics invariant . The invariance of the Hamiltonian equations is a necessary but not a sufficient criterion. The invariance of the Poisson brackets is necessary and sufficient , another necessary criterion is the invariance of the phase space volume. The aim is to simplify the new Hamilton function as much as possible, ideally even to make it independent of one or more variables. In this function, canonical transformations are the starting point for the Hamilton-Jacobi formalism . Canonical transformations can be constructed from so-called generating functions .

Important examples of canonical transformations are transformations of the phase space that are induced by transformations of the configuration space - so-called point transformations - as well as the canonical flow with fixed time constants, i.e. transformations of the phase space that arise through the progression of the dynamics by a constant time difference. The generating function in the latter case is the Hamiltonian principle function and corresponds precisely to the effect between the two points in time, understood as a function of the old and new coordinates.

Time-independent case

In the following, only the simpler time-independent case is dealt with first. The time-dependent case is presented in a separate section. Furthermore, the following statements are to be regarded as a local description of the transformations in bundle maps. The use of differential form calculus is essential for understanding the global relationships . They are also presented in a separate section.

definition

Consider a Hamiltonian system with degrees of freedom and the Hamilton function , which depends on the coordinates and the momenta . The canonical equations (Hamiltonian equations of motion) are thus: ${\displaystyle f}$ ${\displaystyle H(q,p)}$ ${\displaystyle q=(q_{1},\dotsc ,q_{f})}$ ${\displaystyle p=(p_{1},\dotsc ,p_{f})}$

${\displaystyle {\dot {q}}_{i}={\frac {\partial H}{\partial p_{i}}}(q(t),p(t))\ ,\quad {\dot {p}}_{i}=-{\frac {\partial H}{\partial q_{i}}}(q(t),p(t)),}$

In the following, the arguments are briefly written for the sake of clarity . We are looking for transformations that leave the canonical equations invariant, i.e. In other words , the same dynamic is to be described by substitution in the Hamilton function : ${\displaystyle q(t),p(t)}$ ${\displaystyle q,p}$ ${\displaystyle (q,p)\mapsto (Q(q,p),P(q,p))}$ ${\displaystyle (q,p)=(q(Q,P),p(Q,P))}$ ${\displaystyle H(Q,P):=H(q(Q,P),p(Q,P))}$

${\displaystyle {\dot {Q}}_{i}\equiv {\frac {\mathrm {d} }{\mathrm {d} t}}Q_{i}(q,p)=\sum _{j}\left({\frac {\partial Q_{i}}{\partial q_{j}}}{\frac {\partial H}{\partial p_{j}}}(q,p)-{\frac {\partial Q_{i}}{\partial p_{j}}}{\frac {\partial H}{\partial q_{j}}}(q,p)\right)}$

${\displaystyle {\overset {!}{=}}{\frac {\partial H}{\partial P_{i}}}(Q(q,p),P(q,p))}$

${\displaystyle {\dot {P}}_{i}\equiv {\frac {\mathrm {d} }{\mathrm {d} t}}P_{i}(q,p)=\sum _{j}\left({\frac {\partial P_{i}}{\partial q_{j}}}{\frac {\partial H}{\partial p_{j}}}(q,p)-{\frac {\partial P_{i}}{\partial p_{j}}}{\frac {\partial H}{\partial q_{j}}}(q,p)\right)}$

${\displaystyle {\overset {!}{=}}-{\frac {\partial H}{\partial Q_{i}}}(Q(q,p),P(q,p))}$

The validity of Hamilton's equations is equivalent to Hamilton's extremal principle

${\displaystyle \delta \int _{t_{1}}^{t_{2}}\mathrm {d} t\,\left(\sum _{i}p_{i}{\dot {q}}_{i}-H\right)=0,}$

where the and are varied independently. For the simultaneous validity of this and the equivalent principle of variation for the transformed system, it is sufficient that the integrands differ only up to a constant factor (i.e. up to a scale transformation) and a total time derivative: ${\displaystyle p_{i}(t)}$ ${\displaystyle q_{i}(t)}$

${\displaystyle \lambda \left(\sum _{i}p_{i}{\dot {q}}_{i}-H(q,p)\right)=\sum _{i}P_{i}{\dot {Q}}_{i}-H(Q,P)+{\frac {\mathrm {d} F}{\mathrm {d} t}}(q,p)}$

Canonical (actually locally canonical) are precisely those transformations that also satisfy the above equation (those with other coefficients are also referred to as extended canonical transformations and can always be represented as a composition of a canonical transformation and a scale transformation). For these applies: ${\displaystyle \lambda =1}$

${\displaystyle \sum _{i}p_{i}\,\mathrm {d} q_{i}=\sum _{i}P_{i}\,\mathrm {d} Q_{i}+\mathrm {d} F}$

Other transformations that also leave the canonical form of the equations of motion invariant (those that introduce a new Hamilton function would also be conceivable, as happens in the time-dependent case anyway), have the disadvantage that they cannot be derived from a generating function and important results such as B. Liouville's theorem or the invariance of Poisson brackets do not hold. For example, the transformation also leaves the canonical equations invariant, but is not counted among the canonical transformations. ${\displaystyle {\bar {q}}=q,{\bar {p}}=\lambda p,{\bar {H}}({\bar {q}},{\bar {p}})=\lambda H({\bar {q}},{\bar {p}}/\lambda )}$

Poisson brackets

The Poisson bracket of smooth functions and on the phase space with respect to and is through ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle q}$ ${\displaystyle p}$

${\displaystyle \left\{f,g\right\}_{q,p}=\sum _{i=1}^{n}\left({\frac {\partial f}{\partial q_{i}}}{\frac {\partial g}{\partial p_{i}}}-{\frac {\partial f}{\partial p_{i}}}{\frac {\partial g}{\partial q_{i}}}\right)}$

Are defined. The Poisson brackets for old and new coordinates are the same, so it holds

${\displaystyle \left\{f,g\right\}_{q,p}\equiv \left\{f,g\right\}_{Q,P}}$

if and only if the transformation is canonical (strictly speaking, the functions on the right should be viewed as pushforward ). The following relationship between the fundamental Poisson brackets is also equivalent : ${\displaystyle (q,p)\mapsto (Q,P)}$ ${\displaystyle f(Q,P):=f(q(Q,P),p(Q,P))}$

${\displaystyle \left\{Q_{i},P_{j}\right\}_{q,p}=\delta _{ij}\;,\quad \left\{Q_{i},Q_{j}\right\}_{q,p}=\left\{P_{i},P_{j}\right\}_{q,p}=0}$

This is the Kronecker Delta . This property is also sometimes used to define canonical transformations. ${\displaystyle \delta _{ij}}$

Generating functions

Canonical transformations can be found and constructed using generating functions (also generators for short ).

The transformation is canonical if and only if ${\displaystyle (q,p)\mapsto (Q,P)}$

${\displaystyle \sum _{i}p_{i}\,\mathrm {d} q_{i}=\sum _{i}P_{i}\,\mathrm {d} Q_{i}+dF.}$

There is a smooth function on the phase space and its differential. The functional matrix of the transformation has the shape ${\displaystyle F=F(q,p)}$ ${\displaystyle \mathrm {d} F}$

${\displaystyle {\begin{pmatrix}(\partial Q_{i}/\partial q_{j})_{\vert p}&(\partial Q_{i}/\partial p_{j})_{\vert q}\\(\partial P_{i}/\partial q_{j})_{\vert p}&(\partial P_{i}/\partial p_{j})_{\vert q}\end{pmatrix}}.}$

Some of the four sub-matrices can be singular. However, among them are at least two regular ones, as they are the determinants of a block matrix

${\displaystyle \det {\begin{pmatrix}A&B\\C&D\end{pmatrix}}=\det A\cdot \det D-\det B\cdot \det C}$

applies. For the following it is initially assumed that applies. Then it can be substituted and one gets with : ${\displaystyle \det(\partial Q_{i}/\partial p_{j})_{\vert q}\neq 0}$ ${\displaystyle p=p(Q,q)}$ ${\displaystyle F_{1}:=F}$

${\displaystyle p_{i}={\frac {\partial F_{1}}{\partial q_{i}}},\quad P_{i}=-{\frac {\partial F_{1}}{\partial Q_{i}}}}$

If so, it is certain . Then it can be substituted. It is . This means: ${\displaystyle \det(\partial Q_{i}/\partial p_{j})_{\vert q}=0}$ ${\displaystyle \det(\partial Q_{i}/\partial q_{j})_{\vert p}\neq 0}$ ${\displaystyle q=q(Q,p)}$ ${\displaystyle \sum _{i}p_{i}\,\mathrm {d} q_{i}=\mathrm {d} \left(\sum _{i}p_{i}\,q_{i}\right)-\sum _{i}q_{i}\,\mathrm {d} p_{i}}$

${\displaystyle \sum _{i}P_{i}\,\mathrm {d} Q_{i}=-\sum _{i}q_{i}\,\mathrm {d} p_{i}+\mathrm {d} \left(\sum _{i}p_{i}\,q_{i}-F\right)=:-\sum _{i}q_{i}\,\mathrm {d} p_{i}+\mathrm {d} F_{3}}$

It results:

${\displaystyle P_{i}=-{\frac {\partial F_{3}}{\partial Q_{i}}},\quad q_{i}=-{\frac {\partial F_{3}}{\partial p_{i}}}}$

One of the coordinates can be selected for each index in order to provide a generating function together with the one class of independent variables. Accordingly, there are classes of generating functions for a Hamiltonian system of degrees of freedom . They each merge into one another through a Legendre transformation. ${\displaystyle (Q_{i}),(P_{i})}$ ${\displaystyle i}$ ${\displaystyle (q_{j})}$ ${\displaystyle n}$ ${\displaystyle 2^{n}}$

Generating functions of classes and can be selected in an analogous manner . The transformation rules for the four common classes of generating functions are: ${\displaystyle F_{2}(q,P)}$ ${\displaystyle F_{4}(p,P)}$

Generating function	Canonical transformation
${\displaystyle F_{1}(q,Q)=F}$	${\displaystyle p_{i}={\frac {\partial F_{1}}{\partial q_{i}}},P_{i}=-{\frac {\partial F_{1}}{\partial Q_{i}}}}$
${\displaystyle F_{2}(q,P)=F+\sum _{i}Q_{i}P_{i}}$	${\displaystyle p_{i}={\frac {\partial F_{2}}{\partial q_{i}}},Q_{i}={\frac {\partial F_{2}}{\partial P_{i}}}}$
${\displaystyle F_{3}(p,Q)=F-\sum _{i}q_{i}p_{i}}$	${\displaystyle q_{i}=-{\frac {\partial F_{3}}{\partial p_{i}}},P_{i}=-{\frac {\partial F_{3}}{\partial Q_{i}}}}$
${\displaystyle F_{4}(p,P)=F+\sum _{i}\left(Q_{i}P_{i}-q_{i}p_{i}\right)}$	${\displaystyle q_{i}=-{\frac {\partial F_{4}}{\partial p_{i}}},Q_{i}={\frac {\partial F_{4}}{\partial P_{i}}}}$

In the literature, sometimes and sometimes one of the is referred to as a canonical transformation, the two terms are the same for the class . ${\displaystyle F}$ ${\displaystyle F_{i}}$ ${\displaystyle F_{1}(q,Q)}$

An important property of generating functions of the class is their additivity when executing canonical transformations one after the other. Applies approximately ${\displaystyle F_{1}(q,Q)}$

${\displaystyle \sum _{i}p'_{i}\,\mathrm {d} q'_{i}=\sum _{i}p_{i}\,\mathrm {d} q_{i}+\mathrm {d} F'_{1},}$

${\displaystyle \sum _{i}p''_{i}\,\mathrm {d} q''_{i}=\sum _{i}p'_{i}\,\mathrm {d} q'_{i}+\mathrm {d} F''_{1},}$

so also applies

${\displaystyle \sum _{i}p''_{i}\,\mathrm {d} q''_{i}=\sum _{i}p_{i}\,\mathrm {d} q_{i}+\mathrm {d} (F'_{1}+F''_{1}).}$

Liouville's theorem

Canonical transformations leave the phase space volume invariant. ${\displaystyle \mathrm {d} q_{1}\dotsb \mathrm {d} q_{f}\,\mathrm {d} p_{1}\dotsb \mathrm {d} p_{f}}$

In the geometric formalism , the phase space volume is described by the differential form . Since the roof product is natural, it holds and Liouville's theorem is proven without much effort. ${\displaystyle \omega \wedge \dotsb \wedge \omega }$ ${\displaystyle \Psi _{*}(\omega \wedge \dotsb \wedge \omega )=\Psi _{*}\omega \wedge \dotsb \wedge \Psi _{*}\omega =\omega \wedge \dotsb \wedge \omega }$

Examples

Some canonical transformations are listed below:

• The identical transformation is trivially canonical with the generating function . ${\displaystyle (Q,P)=(q,p)}$ ${\displaystyle F(q,Q)=0}$

• The transformation is not canonical. However, it is canonical. ${\displaystyle (Q,P)=(p,q)}$ ${\displaystyle (Q,P)=(p,-q)}$

• Point transformations of the configuration space induce canonical transformations, if the impulses are transformed according to (this is the transformation behavior of cotangential vectors ). Can be used as a generating function . ${\displaystyle q\mapsto Q}$ ${\displaystyle P_{i}=\sum _{j}p_{j}{\frac {\partial q_{j}}{\partial Q_{i}}}}$ ${\displaystyle F(q,P)=\sum _{i}P_{i}\,Q_{i}(q)}$

• The time evolution induces a local canonical transformation: It is chosen to be fixed (if the Hamiltonian equations do not produce a complete flow, it must be chosen to be sufficiently small). To be an integral curve of the Hamiltonian equations with and let it be . The transformation has the generator ${\displaystyle s}$ ${\displaystyle s}$ ${\displaystyle (q,p)}$ ${\displaystyle q(t),p(t)}$ ${\displaystyle (q(0),p(0))=(q,p)}$ ${\displaystyle (Q,P):=(q(s),p(s))}$ ${\displaystyle (q,p)\mapsto (Q,P)}$

${\displaystyle F(q,Q)=\int _{0}^{t}\mathrm {d} t\,\left(\sum _{i}p_{i}(t){\frac {\partial H}{\partial p_{i}}}(q(t),p(t))-H(q(t),p(t))\right),}$

the Hamilton principle function or action function.

• Linear transformations are canonical if and only if their matrices are symplectic. Let it be summarized. Then by a canonical transformation given if and only if , in which denotes the unit matrix. Symplectic matrices always have the determinant 1. Furthermore, there is an eigenvalue of if and only if are eigenvalues, and the corresponding eigenspaces are isomorphic. ${\displaystyle (q,p)=:z}$ ${\displaystyle (z_{i})_{i}\mapsto \left(\sum _{k}L_{ik}z_{k}\right)_{i}}$ ${\displaystyle L^{T}\cdot M\cdot L=M}$ ${\displaystyle M:={\begin{pmatrix}0&E_{f}\\-E_{f}&0\end{pmatrix}}}$ ${\displaystyle E_{f}}$ ${\displaystyle \lambda }$ ${\displaystyle L}$ ${\displaystyle {\frac {1}{\lambda }},\lambda ^{*},{\frac {1}{\lambda ^{*}}}}$

Global canonical transformations

The differential form calculus is indispensable for understanding global relationships.

The configuration space of a mechanical system with degrees of freedom is modeled by a smooth -dimensional manifold . The Lagrange function is a function of the generalized coordinates and velocities, i.e. a smooth function on the tangential bundle . An isomorphism between the tangential bundle and the cotangential bundle is produced according to a Legendre transformation ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle Q}$ ${\displaystyle T(Q)=\bigcup _{q\in Q}\lbrace q\rbrace \times T_{q}(Q)=\lbrace (q,{\dot {q}})\vert q\in Q,{\dot {q}}\in T_{q}(Q)\rbrace }$

${\displaystyle {\dot {q}}\in T_{q}(Q)\longleftrightarrow p\in T_{q}^{*}(Q),\quad p_{i}={\frac {\partial L}{\partial {\dot {q}}^{i}}}.}$

Here and in the following, when we talk about coordinates , we always refer to bundle cards, that is, the cards are of the shape ${\displaystyle q^{i},{\dot {q}}^{i},p_{i}}$

${\displaystyle \alpha _{\varphi }:(q,{\dot {q}})\mapsto (\varphi (q),\Theta _{q,\varphi }({\dot {q}})),}$

${\displaystyle \beta _{\varphi }:(q,p)\mapsto (\varphi (q),(\Theta _{q,\varphi }^{-1})^{T}(p)),}$

where a map of µm is defined with a curve representing the tangential vector , and a superscript T denotes the dual mapping. This card choice has the advantage that the natural pairing of a tangential and a cotangential vector corresponds to the Euclidean scalar product ( Einstein's sum convention is used here). ${\displaystyle \varphi }$ ${\displaystyle Q}$ ${\displaystyle q}$ ${\displaystyle \Theta _{q,\varphi }:[\gamma ]_{\sim }\to (\varphi \circ \gamma )'(0)}$ ${\displaystyle [\gamma ]_{\sim }}$ ${\displaystyle \gamma (t)}$ ${\displaystyle (p\vert {\dot {q}})=p_{i}{\dot {q}}^{i}=\alpha _{\varphi }(p)\cdot \beta _{\varphi }({\dot {q}})}$

On the cotangent bundle there is a natural connection between elements and tangent : . This relationship is now to be extended to tangential vectors of the cotangential bundle: the natural projection has the derivative . The differential 1-form defined for by is called canonical 1-form, in a bundle card it has the form . Its negative differential is called the canonical 2-form (it makes the cotangent bundle a symplectic manifold ). ${\displaystyle z=(q,p)\in T^{*}(Q)}$ ${\displaystyle {\dot {q}}\in T_{q}(Q)}$ ${\displaystyle (p\vert {\dot {q}})=p_{i}{\dot {q}}^{i}}$ ${\displaystyle \Pi :T^{*}(Q)\to Q,(q,p)\mapsto q}$ ${\displaystyle T(\Pi ):T(T^{*}(Q))\to T(Q),(q,p;v^{q^{i}}\partial _{q^{i}}+w^{p_{i}}\partial _{w_{i}})\to (q;v^{q^{i}}\partial _{q^{i}})}$ ${\displaystyle z\in T^{*}(Q),v\in T(T^{*}(Q))}$ ${\displaystyle (\Theta \vert v)(z):=(z\vert T(\Pi )\cdot v)}$ ${\displaystyle T^{*}(Q)}$ ${\displaystyle \Theta =p_{i}\,\mathrm {d} q^{i}}$ ${\displaystyle \omega :=-\mathrm {d} \Theta =\mathrm {d} q^{i}\wedge \mathrm {d} p_{i}}$

Canonical transformations are diffeomorphisms that leave the canonical 2-form invariant, i.e. H. (In more general terms, such mappings between symplectic manifolds are called symplectomorphism, so they represent a generalization of canonical transformations). Corresponding local diffeomorphisms are called local canonical transformation. Thus , i. i.e., according to Poincaré's lemma, local (in star-shaped areas also global) is exact: ${\displaystyle \Psi :T^{*}(Q_{1})\to T^{*}(Q_{2})}$ ${\displaystyle \Psi _{*}\omega _{1}=\omega _{2}}$ ${\displaystyle \omega _{2}-\Psi _{*}\omega _{1}=d(\Psi _{*}\Theta _{1}-\Theta _{2})=0}$ ${\displaystyle \Psi _{*}\Theta _{1}-\Theta _{2}}$

${\displaystyle \Theta _{2}=\Psi _{*}\Theta _{1}+\mathrm {d} F}$

${\displaystyle P_{i}\,\mathrm {d} Q^{i}=p_{i}\,\mathrm {d} q^{i}+\mathrm {d} F}$

From this it follows in particular that generating functions only have to describe a canonical transformation locally.

The canonical 2-form also defines a relationship between 1-forms and vector fields according to ${\displaystyle {}^{\sharp }:{\mathcal {T}}_{1}^{0}(T^{*}(Q))\to {\mathcal {T}}_{0}^{1}(T^{*}(Q))}$

${\displaystyle \omega ({}^{\sharp }\eta ,Y):=(\eta \vert Y).}$

In particular, it is called a Hamiltonian vector field (corresponding definitions are made for any smooth functions), it just creates the canonical flow. The Poisson brackets can be passed through without any coordinates ${\displaystyle {}^{\sharp }dH=\operatorname {s-grad} H=:X_{H}}$

${\displaystyle \lbrace f,g\rbrace =\omega (X_{f},X_{g})}$

define. In this way, the connection between canonical transformations and the Poisson brackets becomes particularly clear. First it is shown that Hamiltonian vector fields transform naturally. For any vector fields and the following applies: ${\displaystyle Y\in {\mathcal {T}}_{0}^{1}(T^{*}(Q_{1}))}$ ${\displaystyle z\in T^{*}(Q_{1})}$

${\displaystyle (\mathrm {d} f\vert Y)=(\Psi _{*}\mathrm {d} f\vert \Psi _{*}Y)=(\mathrm {d} (\Psi _{*}f)\vert \Psi _{*}Y)}$

${\displaystyle \omega _{1}(X_{f}(z),Y(z))=(\Psi _{*}\omega _{1})(X_{\Psi _{*}f}(\Psi (z)),\Psi _{*}Y(\Psi (z)))}$

However is also and thus ${\displaystyle \omega _{1}(X_{f}(z),Y(z))=(\Psi _{*}\omega _{1})(\Psi _{*}X_{f}(\Psi (z)),\Psi _{*}Y(\Psi (z)))}$

${\displaystyle \Psi _{*}X_{f}=X_{\Psi _{*}f}.}$

But now it is for smooth functions ${\displaystyle f,g\in C^{\infty }(T^{*}(Q_{1}))}$

${\displaystyle \lbrace f,g\rbrace =\omega _{1}(X_{f},X_{g})=(\Psi _{*}\omega _{1})(\Psi _{*}X_{f},\Psi _{*}X_{g})=(\Psi _{*}\omega _{1})(X_{\Psi _{*}f},X_{\Psi _{*}g})}$

${\displaystyle \lbrace \Psi _{*}f,\Psi _{*}g\rbrace =\omega _{2}(X_{\Psi _{*}f},X_{\Psi _{*}g}).}$

The two expressions match if and only if

${\displaystyle \Psi _{*}\omega _{1}=\omega _{2},}$

so if is a canonical transformation. ${\displaystyle \Psi }$

Time-dependent and relativistic case

There are several ways to integrate time into the formalism. In the relativistic case, too, it is particularly advantageous to expand the configuration space by a time variable to form what is known as the extended configuration space. The extended phase space then contains two further variables, the pulse variable corresponding to the time is usually referred to as. Insofar as the Hamilton function expresses the energy in the non-relative case, the new Hamilton function can ${\displaystyle -E}$ ${\displaystyle H(q,p;t)}$

${\displaystyle {\mathcal {H}}(q,t;p,-E):=H(q,p;t)-E}$

introduced, which has no physical meaning, but provides the correct equations of motion. The canonical forms are defined without change and take coordinates the figures and on. The Hamiltonian vector field then generates the flow: ${\displaystyle \Theta =p_{i}\,\mathrm {d} q^{i}-E\,\mathrm {d} t}$ ${\displaystyle \omega =\mathrm {d} q^{i}\wedge \mathrm {d} p_{i}-\mathrm {d} t\wedge \mathrm {d} E}$ ${\displaystyle X_{\mathcal {H}}}$

${\displaystyle {\frac {\mathrm {d} q^{i}}{\mathrm {d} s}}={\frac {\partial H}{\partial p_{i}}},\quad {\frac {\mathrm {d} p_{i}}{\mathrm {d} s}}=-{\frac {\partial H}{\partial q^{i}}}}$

${\displaystyle {\frac {\mathrm {d} t}{\mathrm {d} s}}=1,\quad {\frac {\mathrm {d} E}{\mathrm {d} s}}={\frac {\partial H}{\partial t}}}$

In addition, is constant along an integral curve, so that only the case is physically relevant and can be identified with and with . ${\displaystyle {\mathcal {H}}}$ ${\displaystyle {\mathcal {H}}=0}$ ${\displaystyle H}$ ${\displaystyle E}$ ${\displaystyle s}$ ${\displaystyle t}$

For the relativistic case, canonical transformations that change the time variable are also relevant. Such transformations are of no interest for the non-relativistic case. In the following, the old coordinates are marked with a dash. It now applies

${\displaystyle p_{i}\,\mathrm {d} q^{i}-E\,\mathrm {d} t={\bar {p}}_{i}\,\mathrm {d} {\bar {q}}^{i}-{\bar {E}}\,\mathrm {d} {\bar {t}}+\mathrm {d} F.}$

It is assumed that the new coordinates and the old pulses can be used as coordinates. Then you insert and get: ${\displaystyle {\bar {p}}_{i}\,\mathrm {d} {\bar {q}}^{i}-{\bar {E}}\,\mathrm {d} {\bar {t}}=-{\bar {q}}^{i}\,\mathrm {d} {\bar {p}}_{i}+{\bar {t}}\,\mathrm {d} {\bar {E}}+\mathrm {d} ({\bar {q}}^{i}{\bar {p}}_{i}-{\bar {t}}\,{\bar {E}})}$

${\displaystyle p_{i}\,\mathrm {d} q^{i}-E\,\mathrm {d} t=-{\bar {q}}^{i}\,\mathrm {d} {\bar {p}}_{i}+{\bar {t}}\,\mathrm {d} {\bar {E}}+\mathrm {d} \left(F-{\bar {q}}^{i}{\bar {p}}_{i}+{\bar {t}}\,{\bar {E}}\right)}$

Can be used to ensure that it is transformed . Then the transformation rules read: ${\displaystyle {\bar {t}}=t}$ ${\displaystyle F=f(q,{\bar {p}},t)-{\bar {E}}t}$

${\displaystyle {\bar {t}}=t,\quad E={\bar {E}}-{\frac {\partial f}{\partial t}}}$

${\displaystyle p_{i}={\frac {\partial f}{\partial q^{i}}},\quad {\bar {q}}^{i}={\frac {\partial f}{\partial {\bar {p}}_{i}}}}$

It is left invariant, i.e. the Hamilton function is generally changed. If the Hamilton-Jacobi equation holds, i.e. H. ${\displaystyle {\mathcal {H}}=H-E={\bar {H}}-{\bar {E}}}$ ${\displaystyle H}$ ${\displaystyle f}$

${\displaystyle {\frac {\partial f}{\partial t}}+H\left(q,{\frac {\partial f}{\partial q}},t\right)=0,}$

so follows and the system is transformed into equilibrium. ${\displaystyle {\bar {H}}=0}$

Symplectic structure

The functional matrices of canonical transformations

${\displaystyle J={\begin{pmatrix}{\frac {\partial Q}{\partial q}}&{\frac {\partial Q}{\partial p}}\\{\frac {\partial P}{\partial q}}&{\frac {\partial P}{\partial p}}\\\end{pmatrix}}\in \mathbb {R} _{2f\times 2f}}$

form a symplectic group , so they have the property

${\displaystyle J^{T}MJ=M}$

With

${\displaystyle M={\begin{pmatrix}0&E_{f}\\-E_{f}&0\\\end{pmatrix}}}$

and the unit matrix . ${\displaystyle f\times f}$ ${\displaystyle E_{f}}$

literature

• VI Arnold : Mathematical Methods of Classical Mechanics. Jumper. ISBN 978-0-387-96890-2 .
• H. Goldstein : Classical Mechanics. Wiley-VCH. ISBN 978-3-527-40589-3 .
• LD Landau , EM Lifschitz : Textbook of Theoretical Physics. Vol. 1: Mechanics. Publisher Harri Deutsch. ISBN 978-3-8085-5612-2 .
• W. Thirring : Textbook of Mathematical Physics. Vol. 1: Classical Dynamic Systems. Jumper. ISBN 978-3-211-82089-6 .

Individual evidence

1. ^ Walter Thirring: Classical Mathematical Physics. New York 1997, p. 90.
2. ↑ Vladimir Arnold: Mathematical Methods of Classical Mechanics. New York 1989, p. 241.
3. ↑ Vladimir Arnold: Mathematical Methods of Classical Mechanics. New York 1989, p. 163.
4. ↑ Vladimir Arnold: Mathematical Methods of Classical Mechanics. New York 1989, p. 241.
5. ^ ^{A b} Herbert Goldstein: Classical Mechanics. Addison-Wesley 2000, p. 371.
6. ^ ^{A b} Herbert Goldstein: Classical Mechanics. Addison-Wesley 2000, p. 373.
7. ^ Walter Thirring: Classical Mathematical Physics. Berlin 1901, p. 99.
8. ^ ^{A b} Walter Thirring: Classical Mathematical Physics. New York 1997, p. 101.
9. ↑ Vladimir Arnold: Mathematical Methods of Classical Mechanics. New York 1997, p. 236.