Английская Википедия:Canonical transformation

Содержание

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Шаблон:Short description In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates Шаблон:Math that preserves the form of Hamilton's equations. This is sometimes known as form invariance. Although Hamilton's equations are preserved, it need not preserve the explicit form of the Hamiltonian itself. Canonical transformations are useful in their own right, and also form the basis for the Hamilton–Jacobi equations (a useful method for calculating conserved quantities) and Liouville's theorem (itself the basis for classical statistical mechanics).

Since Lagrangian mechanics is based on generalized coordinates, transformations of the coordinates Шаблон:Math do not affect the form of Lagrange's equations and, hence, do not affect the form of Hamilton's equations if the momentum is simultaneously changed by a Legendre transformation into <math display="block"> P_i = \frac{ \partial L }{ \partial \dot{Q}_i }\ ,</math> where <math>\left\{\ (P_1 , Q_1),\ (P_2, Q_2),\ (P_3, Q_3),\ \ldots\ \right\} </math> are the new co‑ordinates, grouped in canonical conjugate pairs of momenta <math>P_i </math> and corresponding positions <math>Q_i,</math> for <math>i = 1, 2, \ldots\ N,</math> with <math>N </math> being the number of degrees of freedom in both co‑ordinate systems.

Therefore, coordinate transformations (also called point transformations) are a type of canonical transformation. However, the class of canonical transformations is much broader, since the old generalized coordinates, momenta and even time may be combined to form the new generalized coordinates and momenta. Canonical transformations that do not include the time explicitly are called restricted canonical transformations (many textbooks consider only this type).

Modern mathematical descriptions of canonical transformations are considered under the broader topic of symplectomorphism which covers the subject with advanced mathematical prerequisites such as cotangent bundles, exterior derivatives and symplectic manifolds.

Notation

Boldface variables such as Шаблон:Math represent a list of Шаблон:Mvar generalized coordinates that need not transform like a vector under rotation and similarly Шаблон:Math represents the corresponding generalized momentum, e.g., <math display="block">\begin{align} \mathbf{q} &\equiv \left (q_{1}, q_{2}, \ldots, q_{N-1}, q_{N} \right )\\ \mathbf{p} &\equiv \left (p_{1}, p_{2}, \ldots, p_{N-1}, p_{N} \right ). \end{align}</math>

A dot over a variable or list signifies the time derivative, e.g., <math display="block">\dot{\mathbf{q}} \equiv \frac{d\mathbf{q}}{dt}</math>and the equalities are read to be satisfied for all coordinates, for example:<math display="block">\dot{\mathbf{p}} = -\frac{\partial f}{\partial \mathbf{q}}\quad \Longleftrightarrow \quad \dotШаблон:P i = -\frac{\partial f}{\partial {q_i}}\quad (\forall i). </math>

The dot product notation between two lists of the same number of coordinates is a shorthand for the sum of the products of corresponding components, e.g., <math display="block">\mathbf{p} \cdot \mathbf{q} \equiv \sum_{k=1}^{N} p_{k} q_{k}.</math>

The dot product (also known as an "inner product") maps the two coordinate lists into one variable representing a single numerical value. The coordinates after transformation are similarly labelled with Шаблон:Math for transformed generalized coordinates and Шаблон:Math for transformed generalized momentum.

Conditions for restricted canonical transformation

Restricted canonical transformations are coordinate transformations where transformed coordinates Шаблон:Math and Шаблон:Math do not have explicit time dependance, ie. <math display="inline">\mathbf Q=\mathbf Q(\mathbf q,\mathbf p)</math> and <math display="inline">\mathbf P=\mathbf P(\mathbf q,\mathbf p) </math>. The following conditions can be generalized to canonical transformation with the exception of bilinear invariance condition which is only applicable under restricted canonical transformations.

Indirect conditions

The functional form of Hamilton's equations is <math display="block">\begin{align} \dot{\mathbf{p}} &= -\frac{\partial H}{\partial \mathbf{q}} \\ \dot{\mathbf{q}} &= \frac{\partial H}{\partial \mathbf{p}} \end{align}</math>In general, a transformation Шаблон:Math does not preserve the form of Hamilton's equations but in the absence of time dependance in transformation, the transformed Hamiltonian (sometimes called the Kamiltonian) can be assumed to differ by a function of time.

<math display="block">K(\mathbf Q, \mathbf P, t)= H(q(\mathbf Q,\mathbf P),p(\mathbf Q,\mathbf P),t) + \frac{\partial G}{\partial t}(t).</math> This choice of the Kamiltonian is supported by results of canonical transformation conditions, generalized through the use of generating functions. This essentially permits the use of the following relations in the derivation:<math display="block">\begin{align}\left (\frac{\partial H}{\partial q}\right)_{\mathbf q, \mathbf p ,t} =\left(\frac{\partial K}{\partial q}\right)_{\mathbf q, \mathbf p ,t}\\ \left (\frac{\partial H}{\partial p}\right)_{\mathbf q, \mathbf p ,t} =\left(\frac{\partial K}{\partial p}\right)_{\mathbf q, \mathbf p , t} \end{align} </math>These equations, combined with the form of Hamilton's equations are sufficient to derive the indirect conditions.

By definition, the transformed coordinates have analogous dynamics

<math display="block">\begin{align} \dot{\mathbf{P}} &= -\frac{\partial K}{\partial \mathbf{Q}} \\ \dot{\mathbf{Q}} &= \frac{\partial K}{\partial \mathbf{P}} \end{align}</math>

where Шаблон:Math is the new Hamiltonian that is considered.

Since restricted transformations have no explicit time dependence (by definition), the time derivative of a new generalized coordinate Шаблон:Math is <math display="block">\begin{align} \dot{Q}_{m} &= \frac{\partial Q_{m}}{\partial \mathbf{q}} \cdot \dot{\mathbf{q}} + \frac{\partial Q_{m}}{\partial \mathbf{p}} \cdot \dot{\mathbf{p}} \\ &= \frac{\partial Q_{m}}{\partial \mathbf{q}} \cdot \frac{\partial H}{\partial \mathbf{p}} - \frac{\partial Q_{m}}{\partial \mathbf{p}} \cdot \frac{\partial H}{\partial \mathbf{q}} \\ &= \lbrace Q_m , H \rbrace \end{align}</math> where Шаблон:Math is the Poisson bracket.

We also have the identity for the conjugate momentum P_m <math display="block">\begin{align} &\frac{\partial K(\mathbf{Q}, \mathbf{P}, t)}{\partial P_{m}} \\ &= \frac{\partial K(\mathbf{Q}(\mathbf{q}, \mathbf{p}), \mathbf{P}(\mathbf{q}, \mathbf{p}), t)}{\partial \mathbf{q}} \cdot \frac{\partial \mathbf{q}}{\partial P_{m}} + \frac{\partial K(\mathbf{Q}(\mathbf{q}, \mathbf{p}), \mathbf{P}(\mathbf{q}, \mathbf{p}), t)}{\partial \mathbf{p}} \cdot \frac{\partial \mathbf{p}}{\partial P_{m}} \\ &= \frac{\partial H(\mathbf{q}, \mathbf{p}, t)}{\partial \mathbf{q}} \cdot \frac{\partial \mathbf{q}}{\partial P_{m}} + \frac{\partial H(\mathbf{q}, \mathbf{p}, t)}{\partial \mathbf{p}} \cdot \frac{\partial \mathbf{p}}{\partial P_{m}} \\ &= \frac{\partial H}{\partial \mathbf{q}} \cdot \frac{\partial \mathbf{q}}{\partial P_{m}} + \frac{\partial H}{\partial \mathbf{p}} \cdot \frac{\partial \mathbf{p}}{\partial P_{m}} \end{align}</math>

If the transformation is canonical, these two must be equal, resulting in the equations <math display="block">\begin{align} \left( \frac{\partial Q_{m}}{\partial p_{n}}\right)_{\mathbf{q}, \mathbf{p}} &= -\left( \frac{\partial q_{n}}{\partial P_{m}}\right)_{\mathbf{Q}, \mathbf{P}} \\ \left( \frac{\partial Q_{m}}{\partial q_{n}}\right)_{\mathbf{q}, \mathbf{p}} &= \left( \frac{\partial p_{n}}{\partial P_{m}}\right)_{\mathbf{Q}, \mathbf{P}} \end{align}</math>

The analogous argument for the generalized momenta P_m leads to two other sets of equations <math display="block">\begin{align} \left( \frac{\partial P_{m}}{\partial p_{n}}\right)_{\mathbf{q}, \mathbf{p}} &= \left( \frac{\partial q_{n}}{\partial Q_{m}}\right)_{\mathbf{Q}, \mathbf{P}} \\ \left( \frac{\partial P_{m}}{\partial q_{n}}\right)_{\mathbf{q}, \mathbf{p}} &= -\left( \frac{\partial p_{n}}{\partial Q_{m}}\right)_{\mathbf{Q}, \mathbf{P}} \end{align}</math>

These are the indirect conditions to check whether a given transformation is canonical.

Symplectic condition

Sometimes the Hamiltonian relations are represented as:

<math display="block">\dot{\eta}= J \nabla_\eta H </math>

Where <math display="inline">J := \begin{pmatrix} 0 & I_n \\ -I_n & 0 \\ \end{pmatrix},</math>

and <math display="inline">\mathbf{\eta} =

 \begin{bmatrix}
   q_1\\
   \vdots \\
   q_n\\
   p_1\\
   \vdots\\
   p_n\\    
 \end{bmatrix}

</math>. Similarly, let <math display="inline">\mathbf{\varepsilon} =

 \begin{bmatrix}
   Q_1\\
   \vdots \\
   Q_n\\
   P_1\\
   \vdots\\
   P_n\\    
 \end{bmatrix}

</math>.

From the relation of partial derivatives, we convert <math>\dot{\eta}= J \nabla_\eta H </math> relation in terms of partial derivatives with new variables:

<math>\dot{\eta}=J ( M^T \nabla_\varepsilon H) </math> where <math display="inline">M := \frac{\partial (\mathbf{Q}, \mathbf{P})}{\partial (\mathbf{q}, \mathbf{p})}</math>.

Similarly we find:<math display="block">\dot{\varepsilon}=M\dot{\eta} =M J M^T \nabla_\varepsilon H </math>or since <math display="inline">\nabla_\varepsilon K = \nabla_\varepsilon H </math> due to the form of Kamiltonian:<math display="block">\dot{\varepsilon}=J \nabla_\varepsilon K = J \nabla_\varepsilon H </math>

we get the symplectic condition:

<math display="block">M J M^T = J </math>The left hand side of the above is called the Poisson matrix of <math>\varepsilon </math>, denoted as <math display="inline">\mathcal P(\varepsilon) = MJM^T </math>. Similarly, a Lagrange matrix of <math>\eta </math> can be constructed as <math display="inline">\mathcal L(\eta) = M^TJM </math>. It can be shown that the symplectic condition is also equivalent to <math display="inline">M^T J M = J </math> by using <math display="inline">J^{-1}=-J </math> property. The set of all matrices <math display="inline">M </math> which satisfy symplectic conditions form a symplectic group.

Invariance of Poisson Bracket

The Poisson bracket which is defined as:<math display="block">\{u, v\}_\eta := \sum_{i=1}^{n} \left( \frac{\partial u}{\partial q_{i}} \frac{\partial v}{\partial p_{i}} - \frac{\partial u}{\partial p_i} \frac{\partial v}{\partial q_i}\right)</math>can be represented in matrix form as:<math display="block">\{u, v\}_\eta := (\nabla_\eta u)^T J (\nabla_\eta v)</math>Hence using partial derivative relations and symplectic condition, we get:<math display="block">\{u, v\}_\eta = (\nabla_\eta u)^T J (\nabla_\eta v) = (M^T \nabla_\varepsilon u)^T J (M^T \nabla_\varepsilon v) = (\nabla_\varepsilon u)^T M J M^T (\nabla_\varepsilon v) = (\nabla_\varepsilon u)^T J (\nabla_\varepsilon v) = \{u, v\}_\varepsilon</math>

The symplectic condition can also be recovered by taking <math display="inline">u=\varepsilon_i </math> and <math display="inline">v=\varepsilon_j </math> which shows that <math display="inline">(M J M^T )_{ij}= J_{i j} </math>. Thus these conditions are equivalent to symplectic conditions. Furthermore, it can be seen that <math display="inline">\mathcal P_{ij}(\varepsilon) = \{ \varepsilon_i,\varepsilon_j\}_\eta =(M J M^T )_{ij} </math>, which is also the result of explicitly calculating the matrix element by expanding it.

Invariance of Lagrange Bracket

The Lagrange bracket which is defined as:

<math display="block"> [ u, v ]_{\eta}  := \sum_{i=1}^n \left(\frac{\partial q_i}{\partial u} \frac{\partial p_i}{\partial v} - \frac{\partial p_i}{\partial u} \frac{\partial q_i}{\partial v } \right) </math>

can be represented in matrix form as:

<math display="block"> [ u, v ]_{\eta}  := \left(\frac {\partial \eta}{\partial u}\right)^T J \left(\frac {\partial \eta}{\partial v}\right) </math>

Using similar derivation, we get:

<math display="block">[u, v]_\varepsilon = (\partial_u \varepsilon )^T \,J\, (\partial_v \varepsilon) = (M \, \partial_u \eta )^T \,J \, ( M \,\partial_v \eta) = (\partial_u \eta )^T\, M^TJ M\, (\partial_v \eta) = (\partial_u \eta )^T\, J\,(\partial_v \eta) = [u, v]_\eta</math>The symplectic condition can also be recovered by taking <math display="inline">u=\eta_i </math> and <math display="inline">v=\eta_j </math> which shows that <math display="inline">(M^T J M )_{ij}= J_{i j} </math>. Thus these conditions are equivalent to symplectic conditions. Furthermore, it can be seen that <math display="inline">\mathcal L_{ij}(\eta) =[\eta_i,\eta_j]_\varepsilon=(M^T J M )_{ij} </math>, which is also the result of explicitly calculating the matrix element by expanding it.

Bilinear invariance conditions

These set of conditions only apply to restricted canonical transformations or canonical transformations that are independent of time variable.

Consider arbitrary variations of two kinds, in a single pair of generalized coordinate and the corresponding momentum:

<math display="inline"> d \varepsilon=( dq_1, dp_{1},0,0,\ldots),\quad\delta \varepsilon=(\delta q_{1},\delta p_{1},0,0,\ldots). </math>

The area of the infinitesimal parallelogram is given by:

<math display="inline"> \delta a(12)=d q_{1}\delta p_{1}-\delta q_{1} d p_{1}={(\delta\varepsilon)}^T\,J \, d \varepsilon. </math>

It follows from the <math display="inline">M^T J M = J </math> symplectic condition that the infinitesimal area is conserved under canonical transformation:

<math display="inline"> \delta a(12)={(\delta\varepsilon)}^T\,J \,d \varepsilon={(M\delta\eta)}^T\,J \,Md \eta= {(\delta\eta)}^T\,M^TJM \,d \eta = {(\delta\eta)}^T\,J \,d\eta = \delta A(12). </math>

Note that the new coordinates need not be completely oriented in one coordinate momentum plane.

Hence, the condition is more generally stated as an invariance of the form <math display="inline"> {(d\varepsilon)}^T\,J \, \delta \varepsilon </math> under canonical transformation, expanded as:<math display="block"> \sum \delta q \cdot dp - \delta p \cdot dq = \sum \delta Q \cdot dP - \delta P \cdot dQ </math>
If the above is obeyed for any arbitrary variations, it would be only possible if the indirect conditions are met.

Liouville's theorem

The indirect conditions allow us to prove Liouville's theorem, which states that the volume in phase space is conserved under canonical transformations, i.e., <math display="block"> \int \mathrm{d}\mathbf{q}\, \mathrm{d}\mathbf{p} = \int \mathrm{d}\mathbf{Q}\, \mathrm{d}\mathbf{P}</math>

By calculus, the latter integral must equal the former times the determinant of Jacobian Шаблон:Mvar<math display="block">\int \mathrm{d}\mathbf{Q}\, \mathrm{d}\mathbf{P} = \int \det (M) \, \mathrm{d}\mathbf{q}\, \mathrm{d}\mathbf{p}</math>Where <math display="inline">M := \frac{\partial (\mathbf{Q}, \mathbf{P})}{\partial (\mathbf{q}, \mathbf{p})}</math>

Exploiting the "division" property of Jacobians yields<math display="block"> M \equiv \frac{\partial (\mathbf{Q}, \mathbf{P})}{\partial (\mathbf{q}, \mathbf{P})} \left/ \frac{\partial (\mathbf{q}, \mathbf{p})}{\partial (\mathbf{q}, \mathbf{P})} \right. </math>

Eliminating the repeated variables gives<math display="block">M \equiv \frac{\partial (\mathbf{Q})}{\partial (\mathbf{q})} \left/ \frac{\partial (\mathbf{p})}{\partial (\mathbf{P})} \right.</math>

Application of the indirect conditions above yields Шаблон:Math.

Generating function approach

Шаблон:Main To guarantee a valid transformation between Шаблон:Math and Шаблон:Math, we may resort to a direct generating function approach. Both sets of variables must obey Hamilton's principle. That is the Action Integral over the Lagrangian <math>\mathcal{L}_{qp}=\mathbf{p} \cdot \dot{\mathbf{q}} - H(\mathbf{q}, \mathbf{p}, t)</math> and <math>\mathcal{L}_{QP}=\mathbf{P} \cdot \dot{\mathbf{Q}} - K(\mathbf{Q}, \mathbf{P}, t)</math> respectively, obtained by the Hamiltonian via ("inverse") Legendre transformation, both must be stationary (so that one can use the Euler–Lagrange equations to arrive at equations of the above-mentioned and designated form; as it is shown for example here): <math display="block">\begin{align} \delta \int_{t_{1}}^{t_{2}} \left[ \mathbf{p} \cdot \dot{\mathbf{q}} - H(\mathbf{q}, \mathbf{p}, t) \right] dt &= 0 \\ \delta \int_{t_{1}}^{t_{2}} \left[ \mathbf{P} \cdot \dot{\mathbf{Q}} - K(\mathbf{Q}, \mathbf{P}, t) \right] dt &= 0 \end{align}</math>

One way for both variational integral equalities to be satisfied is to have <math display="block">\lambda \left[ \mathbf{p} \cdot \dot{\mathbf{q}} - H(\mathbf{q}, \mathbf{p}, t) \right] = \mathbf{P} \cdot \dot{\mathbf{Q}} - K(\mathbf{Q}, \mathbf{P}, t) + \frac{dG}{dt} </math>

Lagrangians are not unique: one can always multiply by a constant Шаблон:Mvar and add a total time derivative Шаблон:Math and yield the same equations of motion (as discussed on Wikibooks). In general, the scaling factor Шаблон:Mvar is set equal to one; canonical transformations for which Шаблон:Math are called extended canonical transformations. Шаблон:Math is kept, otherwise the problem would be rendered trivial and there would be not much freedom for the new canonical variables to differ from the old ones.

Here Шаблон:Mvar is a generating function of one old canonical coordinate (Шаблон:Math or Шаблон:Math), one new canonical coordinate (Шаблон:Math or Шаблон:Math) and (possibly) the time Шаблон:Mvar. Thus, there are four basic types of generating functions (although mixtures of these four types can exist), depending on the choice of variables. As will be shown below, the generating function will define a transformation from old to new canonical coordinates, and any such transformation Шаблон:Math is guaranteed to be canonical.

The various generating functions and its properties tabulated below is discussed in detail:

Properties of four basic Canonical Transformations

Generating Function	Generating Function Derivatives		Transformed Hamiltonian	Trivial Cases
<math>G = G_1(q,Q,t) </math>	<math>p = \frac{\partial G_1}{\partial q} </math>	<math>P = - \frac{\partial G_1}{\partial Q} </math>	<math display="inline">K = H + \frac{\partial G}{\partial t} </math>	<math>G_1 = qQ </math>	<math>Q = p </math>	<math>P = -q </math>
<math>G = G_2(q,P,t) - QP </math>	<math>p = \frac{\partial G_2}{\partial q} </math>	<math>Q = \frac{\partial G_2}{\partial P} </math>		<math>G_2 = qP </math>	<math>Q = q </math>	<math>P = p </math>
<math>G = G_3(p,Q,t) - qp </math>	<math>q = -\frac{\partial G_3}{\partial p} </math>	<math>P = -\frac{\partial G_3}{\partial Q} </math>		<math>G_3 = pQ </math>	<math>Q = -q </math>	<math>P = -p </math>
<math>G = G_4(p,P,t) + qp - QP </math>	<math>q = -\frac{\partial G_4}{\partial p} </math>	<math>Q = \frac{\partial G_4}{\partial P} </math>		<math>G_1 = pP </math>	<math>Q = p </math>	<math>P = -q </math>

Type 1 generating function

The type 1 generating function Шаблон:Math depends only on the old and new generalized coordinates <math display="block">G \equiv G_{1}(\mathbf{q}, \mathbf{Q}, t)</math> To derive the implicit transformation, we expand the defining equation above <math display="block"> \mathbf{p} \cdot \dot{\mathbf{q}} - H(\mathbf{q}, \mathbf{p}, t) = \mathbf{P} \cdot \dot{\mathbf{Q}} - K(\mathbf{Q}, \mathbf{P}, t) + \frac{\partial G_{1}}{\partial t} + \frac{\partial G_{1}}{\partial \mathbf{q}} \cdot \dot{\mathbf{q}} + \frac{\partial G_{1}}{\partial \mathbf{Q}} \cdot \dot{\mathbf{Q}}</math>

Since the new and old coordinates are each independent, the following Шаблон:Math equations must hold <math display="block">\begin{align} \mathbf{p} &= \frac{\partial G_{1}}{\partial \mathbf{q}} \\ \mathbf{P} &= -\frac{\partial G_{1}}{\partial \mathbf{Q}} \\ K &= H + \frac{\partial G_{1}}{\partial t} \end{align}</math>

These equations define the transformation Шаблон:Math as follows: The first set of Шаблон:Mvar equations <math display="block">\ \mathbf{p} = \frac{\ \partial G_{1}\ }{ \partial \mathbf{q} }\ </math> define relations between the new generalized coordinates Шаблон:Math and the old canonical coordinates Шаблон:Math. Ideally, one can invert these relations to obtain formulae for each Шаблон:Math as a function of the old canonical coordinates. Substitution of these formulae for the Шаблон:Math coordinates into the second set of Шаблон:Mvar equations <math display="block">\mathbf{P} = -\frac{\partial G_{1}}{\partial \mathbf{Q}}</math> yields analogous formulae for the new generalized momenta Шаблон:Math in terms of the old canonical coordinates Шаблон:Math. We then invert both sets of formulae to obtain the old canonical coordinates Шаблон:Math as functions of the new canonical coordinates Шаблон:Math. Substitution of the inverted formulae into the final equation <math display="block">K = H + \frac{\partial G_{1}}{\partial t}</math> yields a formula for Шаблон:Mvar as a function of the new canonical coordinates Шаблон:Math.

In practice, this procedure is easier than it sounds, because the generating function is usually simple. For example, let <math display="block">G_{1} \equiv \mathbf{q} \cdot \mathbf{Q}</math> This results in swapping the generalized coordinates for the momenta and vice versa <math display="block">\begin{align} \mathbf{p} &= \frac{\partial G_{1}}{\partial \mathbf{q}} = \mathbf{Q} \\ \mathbf{P} &= -\frac{\partial G_{1}}{\partial \mathbf{Q}} = -\mathbf{q} \end{align}</math> and Шаблон:Math. This example illustrates how independent the coordinates and momenta are in the Hamiltonian formulation; they are equivalent variables.

Type 2 generating function

The type 2 generating function <math>G_{2}(\mathbf{q}, \mathbf{P}, t)</math> depends only on the old generalized coordinates and the new generalized momenta <math display="block">G \equiv G_{2}(\mathbf{q}, \mathbf{P}, t)-\mathbf{Q} \cdot \mathbf{P}</math> where the <math>-\mathbf{Q} \cdot \mathbf{P}</math> terms represent a Legendre transformation to change the right-hand side of the equation below. To derive the implicit transformation, we expand the defining equation above <math display="block"> \mathbf{p} \cdot \dot{\mathbf{q}} - H(\mathbf{q}, \mathbf{p}, t) = -\mathbf{Q} \cdot \dot{\mathbf{P}} - K(\mathbf{Q}, \mathbf{P}, t) + \frac{\partial G_{2}}{\partial t} + \frac{\partial G_{2}}{\partial \mathbf{q}} \cdot \dot{\mathbf{q}} + \frac{\partial G_{2}}{\partial \mathbf{P}} \cdot \dot{\mathbf{P}}</math>

Since the old coordinates and new momenta are each independent, the following Шаблон:Math equations must hold <math display="block">\begin{align} \mathbf{p} &= \frac{\partial G_{2}}{\partial \mathbf{q}} \\ \mathbf{Q} &= \frac{\partial G_{2}}{\partial \mathbf{P}} \\ K &= H + \frac{\partial G_{2}}{\partial t} \end{align}</math>

These equations define the transformation Шаблон:Math as follows: The first set of Шаблон:Mvar equations <math display="block">\mathbf{p} = \frac{\partial G_{2}}{\partial \mathbf{q}}</math> define relations between the new generalized momenta Шаблон:Math and the old canonical coordinates Шаблон:Math. Ideally, one can invert these relations to obtain formulae for each Шаблон:Math as a function of the old canonical coordinates. Substitution of these formulae for the Шаблон:Math coordinates into the second set of Шаблон:Mvar equations <math display="block">\mathbf{Q} = \frac{\partial G_{2}}{\partial \mathbf{P}}</math> yields analogous formulae for the new generalized coordinates Шаблон:Math in terms of the old canonical coordinates Шаблон:Math. We then invert both sets of formulae to obtain the old canonical coordinates Шаблон:Math as functions of the new canonical coordinates Шаблон:Math. Substitution of the inverted formulae into the final equation <math display="block">K = H + \frac{\partial G_{2}}{\partial t}</math> yields a formula for Шаблон:Mvar as a function of the new canonical coordinates Шаблон:Math.

In practice, this procedure is easier than it sounds, because the generating function is usually simple. For example, let <math display="block">G_{2} \equiv \mathbf{g}(\mathbf{q}; t) \cdot \mathbf{P}</math> where Шаблон:Math is a set of Шаблон:Mvar functions. This results in a point transformation of the generalized coordinates <math display="block">\mathbf{Q} = \frac{\partial G_{2}}{\partial \mathbf{P}} = \mathbf{g}(\mathbf{q}; t)</math>

Type 3 generating function

The type 3 generating function <math>G_{3}(\mathbf{p}, \mathbf{Q}, t)</math> depends only on the old generalized momenta and the new generalized coordinates <math display="block">G \equiv G_{3}(\mathbf{p}, \mathbf{Q}, t)+ \mathbf{q} \cdot \mathbf{p}</math> where the <math>\mathbf{q} \cdot \mathbf{p}</math> terms represent a Legendre transformation to change the left-hand side of the equation below. To derive the implicit transformation, we expand the defining equation above <math display="block">-\mathbf{q} \cdot \dot{\mathbf{p}} - H(\mathbf{q}, \mathbf{p}, t) = \mathbf{P} \cdot \dot{\mathbf{Q}} - K(\mathbf{Q}, \mathbf{P}, t) + \frac{\partial G_{3}}{\partial t} + \frac{\partial G_{3}}{\partial \mathbf{p}} \cdot \dot{\mathbf{p}} + \frac{\partial G_{3}}{\partial \mathbf{Q}} \cdot \dot{\mathbf{Q}}</math>

Since the new and old coordinates are each independent, the following Шаблон:Math equations must hold <math display="block">\begin{align} \mathbf{q} &= -\frac{\partial G_{3}}{\partial \mathbf{p}} \\ \mathbf{P} &= -\frac{\partial G_{3}}{\partial \mathbf{Q}} \\ K &= H + \frac{\partial G_{3}}{\partial t} \end{align}</math>

These equations define the transformation Шаблон:Math as follows: The first set of Шаблон:Mvar equations <math display="block"> \mathbf{q} = -\frac{\partial G_{3}}{\partial \mathbf{p}}</math> define relations between the new generalized coordinates Шаблон:Math and the old canonical coordinates Шаблон:Math. Ideally, one can invert these relations to obtain formulae for each Шаблон:Math as a function of the old canonical coordinates. Substitution of these formulae for the Шаблон:Math coordinates into the second set of Шаблон:Mvar equations <math display="block">\mathbf{P} = -\frac{\partial G_{3}}{\partial \mathbf{Q}}</math> yields analogous formulae for the new generalized momenta Шаблон:Math in terms of the old canonical coordinates Шаблон:Math. We then invert both sets of formulae to obtain the old canonical coordinates Шаблон:Math as functions of the new canonical coordinates Шаблон:Math. Substitution of the inverted formulae into the final equation <math display="block">K = H + \frac{\partial G_{3}}{\partial t}</math> yields a formula for Шаблон:Mvar as a function of the new canonical coordinates Шаблон:Math.

In practice, this procedure is easier than it sounds, because the generating function is usually simple.

Type 4 generating function

The type 4 generating function <math>G_{4}(\mathbf{p}, \mathbf{P}, t)</math> depends only on the old and new generalized momenta <math display="block">G \equiv G_{4}(\mathbf{p}, \mathbf{P}, t) +\mathbf{q} \cdot \mathbf{p} - \mathbf{Q} \cdot \mathbf{P} </math> where the <math>\mathbf{q} \cdot \mathbf{p} - \mathbf{Q} \cdot \mathbf{P}</math> terms represent a Legendre transformation to change both sides of the equation below. To derive the implicit transformation, we expand the defining equation above <math display="block">-\mathbf{q} \cdot \dot{\mathbf{p}} - H(\mathbf{q}, \mathbf{p}, t) = -\mathbf{Q} \cdot \dot{\mathbf{P}} - K(\mathbf{Q}, \mathbf{P}, t) + \frac{\partial G_{4}}{\partial t} + \frac{\partial G_{4}}{\partial \mathbf{p}} \cdot \dot{\mathbf{p}} + \frac{\partial G_{4}}{\partial \mathbf{P}} \cdot \dot{\mathbf{P}} </math>

Since the new and old coordinates are each independent, the following Шаблон:Math equations must hold <math display="block">\begin{align} \mathbf{q} &= -\frac{\partial G_{4}}{\partial \mathbf{p}} \\ \mathbf{Q} &= \frac{\partial G_{4}}{\partial \mathbf{P}} \\ K &= H + \frac{\partial G_{4}}{\partial t} \end{align}</math>

These equations define the transformation Шаблон:Math as follows: The first set of Шаблон:Mvar equations <math display="block">\mathbf{q} = -\frac{\partial G_{4}}{\partial \mathbf{p}}</math> define relations between the new generalized momenta Шаблон:Math and the old canonical coordinates Шаблон:Math. Ideally, one can invert these relations to obtain formulae for each Шаблон:Math as a function of the old canonical coordinates. Substitution of these formulae for the Шаблон:Math coordinates into the second set of Шаблон:Mvar equations <math display="block">\mathbf{Q} = \frac{\partial G_{4}}{\partial \mathbf{P}} </math> yields analogous formulae for the new generalized coordinates Шаблон:Math in terms of the old canonical coordinates Шаблон:Math. We then invert both sets of formulae to obtain the old canonical coordinates Шаблон:Math as functions of the new canonical coordinates Шаблон:Math. Substitution of the inverted formulae into the final equation <math display="block">K = H + \frac{\partial G_{4}}{\partial t}</math> yields a formula for Шаблон:Mvar as a function of the new canonical coordinates Шаблон:Math.

Restrictions on generating functions

For example, using generating function of second kind: <math display="inline">{p}_i = \frac{\partial G_{2}}{\partial {q}_i}

</math> and <math display="inline">{Q}_i = \frac{\partial G_{2}}{\partial {P}_i} </math>, the first set of equations consisting of variables <math display="inline">\mathbf{p} </math>, <math display="inline">\mathbf{q} </math> and <math display="inline">\mathbf{P} </math> has to be inverted to get <math display="inline">\mathbf{P}(\mathbf q, \mathbf p) </math>. This process is possible when the matrix defined by <math display="inline">a_{ij}=\frac{\partial {p}_i(\mathbf q,\mathbf P)}{\partial P_j} </math> is non-singular.

<math display="block">\left|\begin{array}{l l l}{{\displaystyle{\frac{\partial^{2}G_{2}}{\partial P_{1}\partial q_{1}}}}}&Шаблон:\cdots&{{\displaystyle{\frac{\partial^{2}G_{2}}{\partial P_{1}\partial q_{n}}}}}\\ {\quad \vdots} & {\ddots}&{\quad \vdots}\\{{\displaystyle{\frac{\partial^{2}G_{2}}{\partial P_{n}\partial q_{1}}}}}&Шаблон:\cdots&{{\displaystyle{\frac{\partial^{2}G_{2}}{\partial P_{n}\partial q_{n}}}}}\end{array}\right|{\neq0}</math>

Hence, restrictions are placed on generating functions to have the matrices: <math display="inline">\left[\frac{\partial^2 G_1}{\partial Q_j\partial q_i} \right] </math>, <math display="inline">\left[\frac{\partial^2 G_2}{\partial P_j\partial q_i} \right] </math>, <math display="inline">\left[\frac{\partial^2 G_3}{\partial p_j\partial Q_i} \right] </math> and <math display="inline">\left[\frac{\partial^2 G_4}{\partial p_j\partial P_i} \right] </math>, being non-singular.

Limitations of generating functions

Since <math display="inline">\left[\frac{\partial^2 G_2}{\partial P_j\partial q_i} \right] </math> is non-singular, it implies that <math display="inline">\left[\frac{\partial {p}_i(\mathbf q,\mathbf P)}{\partial P_j}\right] </math> is also non-singular. Since the matrix <math display="inline">\left[\frac{\partial P_i(\mathbf q,\mathbf p)}{\partial p_j} \right] </math> is inverse of <math display="inline">\left[\frac{\partial {p}_i(\mathbf q,\mathbf P)}{\partial P_j}\right] </math>, the transformations of type 2 generating functions always have a non-singular <math display="inline">\left[\frac{\partial P_i(\mathbf q,\mathbf p)}{\partial p_j} \right] </math> matrix. Similarly, it can be stated that type 1 and type 4 generating functions always have a non-singular <math display="inline">\left[\frac{\partial Q_i(\mathbf q,\mathbf p)}{\partial p_j} \right] </math> matrix whereas type 2 and type 3 generating functions always have a non-singular <math display="inline">\left[\frac{\partial P_i(\mathbf q,\mathbf p)}{\partial p_j} \right] </math> matrix. Hence, the canonical transformations resulting from these generating functions are not completely general.

In other words, since Шаблон:Math and Шаблон:Math are each Шаблон:Math independent functions, it follows that to have generating function of the form <math display="inline">G_{1}(\mathbf{q}, \mathbf{Q}, t) </math> and <math>G_{4}(\mathbf{p}, \mathbf{P}, t)</math> or <math>G_{2}(\mathbf{q}, \mathbf{P}, t)</math> and <math>G_{3}(\mathbf{p}, \mathbf{Q}, t)</math>, the corresponding Jacobian matrices <math display="inline">\left[\frac{\partial Q_i}{\partial p_j} \right] </math> and <math display="inline">\left[\frac{\partial P_i}{\partial p_j} \right] </math> are restricted to be non singular, ensuring that the generating function is a function of Шаблон:Math independent variables. However, as a feature of canonical transformations, it is always possible to choose Шаблон:Math such independent functions from sets Шаблон:Math or Шаблон:Math, to form a generating function representation of canonical transformations, including the time variable. Hence, it can be proved that every finite canonical transformation can be given as a closed but implicit form that is a variant of the given four simple forms.

Canonical transformation conditions

Canonical transformation relations

From: <math>K = H + \frac{\partial G}{\partial t} </math>, calculate <math display="inline">\frac{\partial (K-H)}{\partial P} </math>:

<math display="block">\begin{align} \left( \frac{\partial (K-H)}{\partial P}\right)_{Q,P,t}= \frac{\partial K}{\partial P} - \frac{\partial H}{\partial p}\frac{\partial p}{\partial P} - \frac{\partial H}{\partial q}\frac{\partial q}{\partial P} - \frac{\partial H}{\partial t}\left( \frac{\partial t}{\partial P}\right)_{Q,P,t} = \dot{Q} + \dot{p} \frac{\partial q}{\partial P} - \dot{q}\frac{\partial p}{\partial P} \\ = \frac{\partial Q}{\partial t} + \frac{\partial Q}{\partial q} \cdot \dot{q} + \frac{\partial Q}{\partial p} \cdot \dot{p} + \dot{p} \frac{\partial q}{\partial P} - \dot{q}\frac{\partial p}{\partial P} \\ =\dot{q}\left(\frac{\partial Q}{\partial q} - \frac{\partial p}{\partial P}\right)+\dot{p}\left(\frac{\partial q}{\partial P} +\frac{\partial Q}{\partial p} \right) + \frac{\partial Q}{\partial t} \end{align} </math>Since the left hand side is <math display="inline">\frac{\partial (K-H)}{\partial P} = \frac \partial {\partial P}\left( \frac{\partial G}{\partial t} \right) \bigg |_{Q,P,t} </math> which is independent of dynamics of the particles, equating coefficients of <math display="inline">\dot q </math> and <math display="inline">\dot p </math> to zero, we get canonical transformation rules. This step is equivalent to equating the left hand side as <math display="inline">\frac{\partial (K-H)}{\partial P} = \frac{\partial Q}{\partial t} </math>.

Similarly:

<math display="block">\begin{align} \left(\frac{\partial (K-H)}{\partial Q}\right)_{Q,P,t}= \frac{\partial K}{\partial Q} - \frac{\partial H}{\partial p}\frac{\partial p}{\partial Q} - \frac{\partial H}{\partial q}\frac{\partial q}{\partial Q} - \frac{\partial H}{\partial t}\left(\frac{\partial t}{\partial Q}\right)_{Q,P,t} = -\dot{P} + \dot{p} \frac{\partial q}{\partial Q} - \dot{q}\frac{\partial p}{\partial Q} \\ = -\frac{\partial P}{\partial t} -\frac{\partial P}{\partial q} \cdot \dot{q} - \frac{\partial P}{\partial p} \cdot \dot{p} + \dot{p} \frac{\partial q}{\partial Q} - \dot{q}\frac{\partial p}{\partial Q} \\ =-\left(\dot{q}\left(\frac{\partial P}{\partial q} + \frac{\partial p}{\partial Q}\right)+\dot{p}\left(\frac{\partial P}{\partial p} -\frac{\partial q}{\partial Q} \right) + \frac{\partial P}{\partial t} \right) \end{align} </math>Similarly the canonical transformation rules are obtained by equating the left hand side as <math display="inline">\frac{\partial (K-H)}{\partial Q} = - \frac{\partial P}{\partial t} </math>.

The above two relations can be combined in matrix form as: <math display="inline">J \left(\nabla_\varepsilon \frac{\partial G}{\partial t} \right) = \frac{\partial \varepsilon}{\partial t} </math> (which will also retain same form for extended canonical transformation) where we have used the result <math display="inline">\frac{\partial G}{\partial t} = K-H </math>. The canonical transformation relations are hence said to be equivalent to <math display="inline">J \left(\nabla_\varepsilon \frac{\partial G}{\partial t} \right) = \frac{\partial \varepsilon}{\partial t} </math> in this context.

The canonical transformation relations can now be restated to include time dependance:<math display="block">\begin{align} \left( \frac{\partial Q_{m}}{\partial p_{n}}\right)_{\mathbf{q}, \mathbf{p},t} &= - \left( \frac{\partial q_{n}}{\partial P_{m}}\right)_{\mathbf{Q}, \mathbf{P},t} \\ \left( \frac{\partial Q_{m}}{\partial q_{n}}\right)_{\mathbf{q}, \mathbf{p},t} &= \left( \frac{\partial p_{n}}{\partial P_{m}}\right)_{\mathbf{Q}, \mathbf{P},t} \end{align} </math><math display="block">\begin{align} \left( \frac{\partial P_{m}}{\partial p_{n}}\right)_{\mathbf{q}, \mathbf{p},t} &= \left( \frac{\partial q_{n}}{\partial Q_{m}}\right)_{\mathbf{Q}, \mathbf{P},t} \\ \left( \frac{\partial P_{m}}{\partial q_{n}}\right)_{\mathbf{q}, \mathbf{p},t} &= - \left( \frac{\partial p_{n}}{\partial Q_{m}}\right)_{\mathbf{Q}, \mathbf{P},t} \end{align}</math>We can also observe that since <math display="inline">\frac{\partial (K-H)}{\partial P} = \frac{\partial Q}{\partial t} </math> and <math display="inline">\frac{\partial (K-H)}{\partial Q} = - \frac{\partial P}{\partial t} </math>, if Шаблон:Math and Шаблон:Math do not explicitly depend on time, <math display="inline">K= H + \frac{\partial G}{\partial t}(t)</math> can be taken. The analysis of restricted canonical transformations is hence consistent with this generalization.

Symplectic Condition

From:<math display="block">\dot{\eta}=J\nabla_\eta H =J ( M^T \nabla_\varepsilon H) </math>

Similarly we find:<math display="block">\dot{\varepsilon}=M\dot{\eta} + \frac{\partial \varepsilon}{\partial t} =M J M^T \nabla_\varepsilon H + \frac{\partial \varepsilon}{\partial t} </math>or:<math display="block">\dot{\varepsilon}=J \nabla_\varepsilon K = J \nabla_\varepsilon H + J \nabla_\varepsilon

\left( \frac{\partial G}{\partial t}\right)

</math>Where the last terms of each equation cancel due to <math display="inline">J \left(\nabla_\varepsilon \frac{\partial G}{\partial t} \right) = \frac{\partial \varepsilon}{\partial t} </math> condition from canonical transformations. Hence leaving the symplectic relation: <math display="inline">M J M^T = J </math> which is also equivalent with the condition <math display="inline">M^T J M = J </math>. It follows from the above two equations that the symplectic condition implies the equation <math display="inline">J \left(\nabla_\varepsilon \frac{\partial G}{\partial t} \right) = \frac{\partial \varepsilon}{\partial t} </math>, from which the indirect conditions can be recovered. Thus, symplectic conditions and indirect conditions can be said to be equivalent in the context of using generating functions.

Invariance of Poisson and Lagrange Bracket

Since <math display="inline">\mathcal P_{ij}(\varepsilon) = \{ \varepsilon_i,\varepsilon_j\}_\eta =(M J M^T )_{ij} = J_{ij} </math> and <math display="inline">\mathcal L_{ij}(\eta) =[\eta_i,\eta_j]_\varepsilon=(M^T J M )_{ij} = J_{ij} </math> where the symplectic condition is used in the last equalities. Using <math display="inline">\{\varepsilon_i,\varepsilon_j\}_\varepsilon=[\eta_i,\eta_j]_\eta = J_{ij} </math>, the equalities <math display="inline">\{ \varepsilon_i,\varepsilon_j\}_\eta= \{ \varepsilon_i,\varepsilon_j\}_\varepsilon </math> and <math display="inline">[\eta_i,\eta_j]_\varepsilon= [\eta_i,\eta_j]_\eta </math> are obtained which imply the invariance of Poisson and Lagrange brackets.

Extended Canonical Transformation

Canonical transformation relations

By solving for:<math display="block">\lambda \left[ \mathbf{p} \cdot \dot{\mathbf{q}} - H(\mathbf{q}, \mathbf{p}, t) \right] = \mathbf{P} \cdot \dot{\mathbf{Q}} - K(\mathbf{Q}, \mathbf{P}, t) + \frac{dG}{dt} </math>with various forms of generating function, we instead get the relation between K and H as <math display="inline">\frac{\partial G}{\partial t} = K-\lambda H </math> which also applies for <math display="inline">\lambda = 1 </math> case.

All results presented below can also be obtained by replacing <math display="inline">q \rightarrow \sqrt{\lambda}q </math>, <math display="inline">p \rightarrow \sqrt{\lambda}p </math> and <math display="inline">H \rightarrow {\lambda}H </math> from known solutions, since it retains the form of Hamilton's equations. The extended canonical transformations are hence said to be result of a canonical transformation (<math display="inline">\lambda = 1 </math>) and a trivial canonical transformation (<math display="inline">\lambda \neq 1 </math>) which has <math display="inline">M J M^T = \lambda J </math> (for the given example, <math display="inline">M = \sqrt{\lambda} I </math> which satisfies the condition).

Using same steps previously used in previous generalization, with <math display="inline">\frac{\partial G}{\partial t} = K-\lambda H </math> in the general case, and retaining the equation <math display="inline">J \left(\nabla_\varepsilon \frac{\partial g}{\partial t} \right) = \frac{\partial \varepsilon}{\partial t} </math>, we get extended canonical transformation partial differential relations:<math display="block">\begin{align} \left( \frac{\partial Q_{m}}{\partial p_{n}}\right)_{\mathbf{q}, \mathbf{p},t} &= -\lambda \left( \frac{\partial q_{n}}{\partial P_{m}}\right)_{\mathbf{Q}, \mathbf{P},t} \\ \left( \frac{\partial Q_{m}}{\partial q_{n}}\right)_{\mathbf{q}, \mathbf{p},t} &= \lambda \left( \frac{\partial p_{n}}{\partial P_{m}}\right)_{\mathbf{Q}, \mathbf{P},t} \end{align}</math><math display="block">\begin{align} \left( \frac{\partial P_{m}}{\partial p_{n}}\right)_{\mathbf{q}, \mathbf{p},t} &= \lambda \left( \frac{\partial q_{n}}{\partial Q_{m}}\right)_{\mathbf{Q}, \mathbf{P},t} \\ \left( \frac{\partial P_{m}}{\partial q_{n}}\right)_{\mathbf{q}, \mathbf{p},t} &= -\lambda \left( \frac{\partial p_{n}}{\partial Q_{m}}\right)_{\mathbf{Q}, \mathbf{P},t} \end{align}</math>

Symplectic condition

From: <math display="block">\dot{\eta}=J\nabla_\eta H =J ( M^T \nabla_\varepsilon H) </math>Similarly we find:<math display="block">\dot{\varepsilon}=M\dot{\eta} + \frac{\partial \varepsilon}{\partial t} =M J M^T \nabla_\varepsilon H + \frac{\partial \varepsilon}{\partial t} </math>or using <math display="inline">\frac{\partial G}{\partial t} = K-\lambda H </math>:<math display="block">\dot{\varepsilon}=J \nabla_\varepsilon K = \lambda J \nabla_\varepsilon H + J \nabla_\varepsilon

\left( \frac{\partial G}{\partial t}\right) 

</math>The second part of each equation cancel. Hence the condition for extended canonical transformation instead becomes: <math display="inline">M J M^T = \lambda J </math>.

Poisson and Lagrange Brackets

The Poisson brackets are changed as follows:<math display="block">\{u, v\}_\eta = (\nabla_\eta u)^T J (\nabla_\eta v) = (M^T \nabla_\varepsilon u)^T J (M^T \nabla_\varepsilon v) = (\nabla_\varepsilon u)^T M J M^T (\nabla_\varepsilon v) = \lambda (\nabla_\varepsilon u)^T J (\nabla_\varepsilon v) = \lambda \{u, v\}_\varepsilon</math>whereas, the Lagrange brackets are changed as:

<math display="block">[u, v]_\varepsilon = (\partial_u \varepsilon )^T \,J\, (\partial_v \varepsilon) = (M \, \partial_u \eta )^T \,J \, ( M \,\partial_v \eta) = (\partial_u \eta )^T\, M^TJ M\, (\partial_v \eta) = \lambda (\partial_u \eta )^T\, J\,(\partial_v \eta) = \lambda [u, v]_\eta</math> Hence, the Poisson bracket scales by the inverse of <math display="inline">\lambda </math> whereas the Lagrange bracket scales by a factor of <math display="inline">\lambda </math>.

Infinitesimal canonical transformation

Consider the canonical transformation that depends on a continuous parameter <math>\alpha </math>, as follows:

<math>\begin{align} & Q(q,p,t;\alpha) \quad \quad \quad & Q(q,p,t;0)=q \\ & P(q,p,t;\alpha) \quad \quad \text{with} \quad & P(q,p,t;0)=p \\ \end{align} </math>

For infinitesimal values of <math>\alpha </math>, the corresponding transformations are called as infinitesimal canonical transformations which are also known as differential canonical transformations.

Consider the following generating function:

<math>G_2(q,P,t)= qP + \alpha G(q,P,t) </math>

Since for <math>\alpha=0 </math>, <math>G_2 = qP </math> has the resulting canonical transformation, <math>Q = q </math> and <math>P = p </math>, this type of generating function can be used for infinitesimal canonical transformation by restricting <math>\alpha </math> to an infinitesimal value. From the conditions of generators of second type:<math display="block">\begin{align} {p} &= \frac{\partial G_{2}}{\partial {q}} = P + \alpha \frac{\partial G}{\partial {q}} (q,P,t) \\ {Q} &= \frac{\partial G_{2}}{\partial {P}} = q + \alpha \frac{\partial G}{\partial {P}} (q,P,t) \\ \end{align}</math>Since <math>P = P(q,p,t;\alpha) </math>, changing the variables of the function <math>G </math> to <math>G(q,p,t) </math> and neglecting terms of higher order of <math>\alpha </math>, we get:<math display="block">\begin{align} {p} &= P + \alpha \frac{\partial G}{\partial {q}} (q,p,t) \\ {Q} &= q + \alpha \frac{\partial G}{\partial p} (q,p,t) \\ \end{align}</math>Infinitesimal canonical transformations can also be derived using the matrix form of the symplectic condition.

Active canonical transformations

Шаблон:See also In the passive view of transformations, the coordinate system is changed without the physical system changing, whereas in the active view of transformation, the coordinate system is retained and the physical system is said to undergo transformations. Thus, using the relations from infinitesimal canonical transformations, the change in the system states under active view of the canonical transformation is said to be:

<math>\begin{align} & \delta q = \alpha \frac{\partial G}{\partial p} (q,p,t) \quad \text{and} \quad \delta p = - \alpha \frac{\partial G}{\partial q} (q,p,t) , \\ \end{align} </math>

or as <math>\delta \eta = \alpha J \nabla_\eta G </math> in matrix form.

For any function <math>u(\eta) </math>, it changes under active view of the transformation according to:

<math>\delta u = u(\eta +\delta \eta)-u(\eta) = (\nabla_\eta u)^T\delta\eta=\alpha (\nabla_\eta u)^T J (\nabla_\eta G) = \alpha \{ u,G \} . </math>

Considering the change of Hamiltonians in the active view, ie. for a fixed point,<math display="block">K(Q=q_0,P=p_0,t) - H(q=q_0,p=p_0,t) = \left(H(q_0',p_0',t) + \frac{\partial G_{2}}{\partial t}\right) - H(q_0,p_0,t) = - \delta H +\alpha \frac{\partial G}{\partial t} = \alpha\left(\{ G,H\}+\frac{\partial G}{\partial t} \right)=\alpha\frac{dG}{dt} </math>where <math display="inline">(q=q_0',p=p_0') </math> are mapped to the point, <math display="inline">(Q=q_0,P=p_0) </math> by the infinitesimal canonical transformation, and similar change of variables for <math>G(q,P,t) </math> to <math>G(q,p,t) </math> is considered up-to first order of <math>\alpha </math>. Hence, if the Hamiltonian is invariant for infinitesimal canonical transformations, its generator is a constant of motion.

Examples of ICT

Time evolution

Taking <math>G(q,p,t)=H(q,p,t) </math> and <math>\alpha = dt </math>, then <math>\delta \eta = (J \nabla_\eta H) dt = \dot{\eta} dt = d\eta </math>. Thus the continuous application of such a transformation maps the coordinates <math>\eta(\tau) </math> to <math>\eta(\tau+t) </math>. Hence if the Hamiltonian is time translation invariant ie. does not have explicit time dependance, its value is conserved for the motion.

Translation

Taking <math>G(q,p,t)=p_k </math>, <math> \delta p_i = 0 </math> and <math> \delta q_i = \alpha \delta_{ik} </math>. Hence, the canonical momentum generates a shift in the corresponding generalized coordinate and if the Hamiltonian is invariant of translation, the momentum is a constant of motion.

Rotation

Consider an orthogonal system for an N-particle system:

<math>\begin{array}{l}{{\mathbf q=\left(x_{1},y_{1},z_{1},\ldots,x_{n},y_{n},z_{n}\right),}}\\ {{\mathbf p=\left(p_{1x},p_{1y},p_{1z},\ldots,p_{n x},p_{n y},p_{n z}\right).}}\end{array}</math>

Choosing the generator to be: <math>G=L_{z}=\sum_{i=1}^{n}\left(x_{i}p_{i y}-y_{i}p_{i x}\right) </math> and the infinitesimal value of <math> \alpha = \delta \phi </math>, then the change in the coordinates is given for x by:

<math>\begin{array}{c} {\delta x_{i}=\{x_{i},G\}\delta\phi=\displaystyle\sum_{j} \{x_{i},x_{j}p_{j y}-y_{j}p_{j x}\}\delta\phi=\displaystyle\sum_{j}(\underbrace{\{x_{i},x_{j}p_{j y}\}}_{=0} -{ \{x_{i},y_{j}p_{j x}\}}})\delta\phi\\ {{=\displaystyle -\sum_{j}y_{j}\underbrace{\{x_i,p_{jx}\}}_{=\delta_{ij}}\delta\phi=- y_{i} \delta \phi}} \end{array} </math>

and similarly for y:

<math>\begin{array}{c}

\delta y_{i}=\{y_{i},G\}\delta\phi=\displaystyle\sum_{j}\{y_{i},x_{j}p_{j y}-y_{j}p_{j x}\}\delta\phi=\displaystyle\sum_{j}(\{y_{i},x_{j}p_{j y}\}-\underbrace{ \{y_{i},y_{j}p_{j x}\}}_{=0})\delta \phi\\ {=\displaystyle\sum_{j}x_{j}\underbrace{\{y_i,p_{jy}\}}_{=\delta_{ij}} \delta\phi=x_{i}\delta\phi\,,} \end{array} </math>

whereas the z component of all particles is unchanged: <math> \delta z_{ i }=\left\{z_{i},G\right\}\delta\phi=\sum_{j}\left\{z_{i},x_{j}p_{j y}-y_{j}p_{j x}\right\}\delta \phi =0</math>.

These transformations correspond to rotation about z axis by angle <math>\delta \phi </math> in its first order approximation. Hence, repeated application of the infinitesimal canonical transformation generates a rotation of system of particles about the z axis. If the Hamiltonian is invariant under rotation by the z axis, the generator, the component of angular momentum along the axis of rotation, is an invariant of motion.

Motion as canonical transformation

Motion itself (or, equivalently, a shift in the time origin) is a canonical transformation. If <math>\mathbf{Q}(t) \equiv \mathbf{q}(t+\tau)</math> and <math>\mathbf{P}(t) \equiv \mathbf{p}(t+\tau)</math>, then Hamilton's principle is automatically satisfied<math display="block"> \delta \int_{t_1}^{t_2} \left[ \mathbf{P} \cdot \dot{\mathbf{Q}} - K(\mathbf{Q}, \mathbf{P}, t) \right] dt = \delta \int_{t_1 + \tau}^{t_2 + \tau} \left[ \mathbf{p} \cdot \dot{\mathbf{q}} - H(\mathbf{q}, \mathbf{p}, t+\tau) \right] dt = 0 </math>since a valid trajectory <math>(\mathbf{q}(t), \mathbf{p}(t))</math> should always satisfy Hamilton's principle, regardless of the endpoints.

Examples

• The translation <math>\mathbf{Q}(\mathbf{q}, \mathbf{p})= \mathbf{q} + \mathbf{a}, \mathbf{P}(\mathbf{q}, \mathbf{p})= \mathbf{p} + \mathbf{b}</math> where <math>\mathbf{a}, \mathbf{b}</math> are two constant vectors is a canonical transformation. Indeed, the Jacobian matrix is the identity, which is symplectic: <math>I^\text{T}JI=J</math>.
• Set <math>\mathbf{x}=(q,p)</math> and <math>\mathbf{X}=(Q,P)</math>, the transformation <math>\mathbf{X}(\mathbf{x})=R \mathbf{x}</math> where <math>R \in SO(2)</math> is a rotation matrix of order 2 is canonical. Keeping in mind that special orthogonal matrices obey <math>R^\text{T}R=I</math> it's easy to see that the Jacobian is symplectic. However, this example only works in dimension 2: <math>SO(2)</math> is the only special orthogonal group in which every matrix is symplectic. Note that the rotation here acts on <math>(q,p)</math> and not on <math>q</math> and <math>p</math> independently, so these are not the same as a physical rotation of an orthogonal spatial coordinate system.
• The transformation <math>(Q(q,p), P(q,p))=(q+f(p), p)</math>, where <math>f(p)</math> is an arbitrary function of <math>p</math>, is canonical. Jacobian matrix is indeed given by <math display="block">\frac{\partial X}{\partial x} = \begin{bmatrix} 1 & f'(p) \\ 0 & 1 \end{bmatrix}</math> which is symplectic.

Modern mathematical description

In mathematical terms, canonical coordinates are any coordinates on the phase space (cotangent bundle) of the system that allow the canonical one-form to be written as <math display="block">\sum_i p_i\,dq^i</math> up to a total differential (exact form). The change of variable between one set of canonical coordinates and another is a canonical transformation. The index of the generalized coordinates Шаблон:Math is written here as a superscript (<math>q^{i}</math>), not as a subscript as done above (<math>q_{i}</math>). The superscript conveys the contravariant transformation properties of the generalized coordinates, and does not mean that the coordinate is being raised to a power. Further details may be found at the symplectomorphism article.

History

The first major application of the canonical transformation was in 1846, by Charles Delaunay, in the study of the Earth-Moon-Sun system. This work resulted in the publication of a pair of large volumes as Mémoires by the French Academy of Sciences, in 1860 and 1867.

See also

• Symplectomorphism
• Hamilton–Jacobi equation
• Liouville's theorem (Hamiltonian)
• Mathieu transformation
• Linear canonical transformation

References

• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book