Symplectic geometry and canonical methods — Examples

Declare the one-label model of the force branch Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance with $N=1$, mass $m>0$, time window $I_0=\mathbb{R}$, configuration domain $Q={\mathbb{R}}^3$, a $C^2$ potential $U:\mathbb{R}\times Q\to \mathbb{R}$ and no residual forces (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator, Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions). By Every potential presentation is hyperregular in its charts, with inverse branch $v={\mathsf{M}}^{-1}p$ (or $G(q{)}^{-1}p$) and the explicit kinetic-plus-potential Hamiltonian the presentation is hyperregular on the Cartesian chart with $H(t,x,p)=\frac{1}{2}{m}^{-1}\mid p{\mid }^2+U(t,x)$ (The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions), the two-form is $\omega ={\sum }_{a=1}^{3}d{x}^a\wedge d{p}_a$ (The canonical one-form and the two-form of a potential presentation: $\lambda =p\cdot dx$ on the Cartesian momentum phase chart and the induced forms on each generalized-coordinate chart, The two-form of a potential presentation is closed and nondegenerate, and each generalized-coordinate momentum chart carries the same symplectic structure: ${\mathrm{\Phi }}_X^{\ast }\lambda =\sum P_{a}d{q}^a$ and ${\mathrm{\Phi }}_X^{\ast }\omega =\sum d{q}^a\wedge d{P}_a$) and the bracket is The Poisson bracket of phase functions of a potential presentation: $\{f,g\}=\omega (X_f,X_g)$ and the coordinate formula in each canonical chart. Then:

1. the canonical coordinates satisfy the fundamental brackets $\{x^a,x^b\}=0$, $\{p_a,p_b\}=0$ and $\{x^a,p_b\}={\delta }_b^a$ for all $a,b\in \{1,2,3\}$ (The bracket of a potential presentation is bilinear, skew, a derivation in each slot, satisfies the Jacobi identity, and reproduces the fundamental brackets and the Hamiltonian field clause 5);
2. the brackets of the coordinates with the Hamiltonian recover the model's velocities and forces: $\{x^a,H\}=H_{p_a}=m^{-1}{p}_a$ and $\{p_a,H\}=-H_{x^a}=-U_{x^a}(t,x)$; hence Hamilton's equations read ${\dot{x}}^a=\{x^a,H\}$, ${\dot{p}}_a=\{p_a,H\}$ (The bracket of a potential presentation is bilinear, skew, a derivation in each slot, satisfies the Jacobi identity, and reproduces the fundamental brackets and the Hamiltonian field clause 4, Cartesian model motions on the potential branch and declared generalized-coordinate Lagrange equations integrate their Hamiltonian fields; the forced Cartesian field is Hamiltonian exactly when the residual force has a velocity-independent potential clause 1);
3. along every solution of the Hamiltonian system and for every $C^1$ phase function $f$, $\frac{d}{dt}f(t,x(t),p(t))={\mathrm{\partial }}_{t}f+\{f,H\}$ (Evolution of phase functions along the model's Hamiltonian solutions: $df\mathrm{/}dt={\mathrm{\partial }}_{t}f+\{f,H\}$, the first-integral criterion, and the closed bracket algebra of autonomous first integrals clause 1);
4. on the isotropic oscillator instance $U(x)=\frac{1}{2}k\mid x{\mid }^2$ with $k>0$, one has $\{p_a,H\}=-k{x}^a$ and $\{H,H\}=0$, and every solution of the Hamiltonian system has constant $H=\frac{1}{2}{m}^{-1}\mid p{\mid }^2+\frac{1}{2}k\mid x{\mid }^2$ on its domain, the Hamiltonian form of the model's energy conservation (Evolution of phase functions along the model's Hamiltonian solutions: $df\mathrm{/}dt={\mathrm{\partial }}_{t}f+\{f,H\}$, the first-integral criterion, and the closed bracket algebra of autonomous first integrals clause 2, Every potential presentation is hyperregular in its charts, with inverse branch $v={\mathsf{M}}^{-1}p$ (or $G(q{)}^{-1}p$) and the explicit kinetic-plus-potential Hamiltonian).

All quantities are model values of the declared idealization; no measurement is reported.

Declare the one-label model of the force branch Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance with $N=1$, mass $m>0$, time window $\mathbb{R}$, configuration domain $Q={\mathbb{R}}^3$, the time-independent potential $U(x)=\frac{1}{2}k\mid x{\mid }^2$ with $k>0$ and no residual forces (Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator, Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions); this is the isotropic oscillator instance. By Every potential presentation is hyperregular in its charts, with inverse branch $v={\mathsf{M}}^{-1}p$ (or $G(q{)}^{-1}p$) and the explicit kinetic-plus-potential Hamiltonian the branch Hamiltonian is

$H(x,p)=\frac{\mid p{\mid }^2}{2m}+\frac{k\mid x{\mid }^2}{2},{\omega }_0:=\sqrt{k\mathrm{/}m}>0,$

The stiffness and angular frequency have dimensions $[k]=\mu {\tau }^{-2}$ and $[{\omega }_0]={\tau }^{-1}$. The grouped phase coordinates are $z=(x^1,x^2,x^3,p_1,p_2,p_3)$, with $[x^a]=\mathrm{\ell }$ and $[p_a]=\mu \mathrm{\ell }{\tau }^{-1}$; hence the six-dimensional phase-volume unit is $[{\omega }^{\wedge 3}\mathrm{/}3!]=(\mu {\mathrm{\ell }}^2{\tau }^{-1}{)}^3$, action cubed.

on the Cartesian phase domain ${\mathbb{R}}^6$, with the two-form $\omega ={\sum }_{a=1}^{3}d{x}^a\wedge d{p}_a$ (The canonical one-form and the two-form of a potential presentation: $\lambda =p\cdot dx$ on the Cartesian momentum phase chart and the induced forms on each generalized-coordinate chart, The two-form of a potential presentation is closed and nondegenerate, and each generalized-coordinate momentum chart carries the same symplectic structure: ${\mathrm{\Phi }}_X^{\ast }\lambda =\sum P_{a}d{q}^a$ and ${\mathrm{\Phi }}_X^{\ast }\omega =\sum d{q}^a\wedge d{P}_a$) and the Hamiltonian field $X_H=({\mathsf{M}}^{-1}p,-{\mathrm{\nabla }}_{x}U)$ (The Hamiltonian field of a phase function for a potential presentation: the unique field $X_f$ with ${\iota }_{X_f}\omega =df$, and $X_H=(H_p,-H_x)$ for the branch Hamiltonian, Cartesian model motions on the potential branch and declared generalized-coordinate Lagrange equations integrate their Hamiltonian fields; the forced Cartesian field is Hamiltonian exactly when the residual force has a velocity-independent potential). Then:

1. for every initial datum $(x,p)$ the curve

$x^a(t)=x^a\cos ({\omega }_{0}t)+\frac{p_a}{m{\omega }_0}\sin ({\omega }_{0}t),p_a(t)=-m{\omega }_0{x}^a\sin ({\omega }_{0}t)+p_a\cos ({\omega }_{0}t)(a=1,2,3)$

solves the Hamiltonian system for all $t\in \mathbb{R}$; 2. in the grouped order $z=(x,p)$ its derivative is $D{\mathrm{\Phi }}_t=\left(\begin{array}{cc}c{I}_3& (m{\omega }_0{)}^{-1}s{I}_3\\ -m{\omega }_{0}s{I}_3& c{I}_3\end{array}\right),c=\cos ({\omega }_{0}t),s=\sin ({\omega }_{0}t),$ and for $J_{\mathrm{g}\mathrm{r}}=\left(\begin{array}{cc}0& I_3\\ -I_3& 0\end{array}\right)$ direct block multiplication gives $D{\mathrm{\Phi }}_t^{\mathsf{T}}{J}_{\mathrm{g}\mathrm{r}}D{\mathrm{\Phi }}_t=J_{\mathrm{g}\mathrm{r}}$ because $c^2+s^2=1$. If $T(x^1,x^2,x^3,p_1,p_2,p_3)=(x^1,p_1,x^2,p_2,x^3,p_3)$ is the permutation to interleaved order, then $TD{\mathrm{\Phi }}_t{T}^{-1}=\mathrm{diag}(A,A,A)$ and $T{J}_{\mathrm{g}\mathrm{r}}{T}^{-1}=\mathrm{diag}(J_2,J_2,J_2)$, where $A=\left(\begin{array}{cc}c& (m{\omega }_0{)}^{-1}s\\ -m{\omega }_{0}s& c\end{array}\right)$; each block satisfies $A^{\mathsf{T}}{J}_{2}A=J_2$ and $\det A=1$. Therefore $\det D{\mathrm{\Phi }}_t=(\det A{)}^3=1$, so the time-$t$ map is canonical (The Hamiltonian evolution of a potential presentation preserves the two-form, the bracket and the determinant on its actual time-slice domains, also when the Hamiltonian depends on time, A $C^1$ map between phase domains preserves the standard two-form iff $D{C}^{\mathsf{T}}JDC=J$, and a canonical diffeomorphism carries $X_{H\circ C}$ to $X_H$); 3. explicit volume check in the numeric coordinates measured in the declared SI units: the image of the unit square $[0,1{]}^2\subseteq \{(x^1,p_1)\}$ under $A$ is the parallelogram with area $\mid \det A\mid =1$, and the image of the box $[0,1{]}^3\times [0,1{]}^3$ has numeric six-volume $\mathrm{\mid }\det D{\mathrm{\Phi }}_t\mathrm{\mid }=1$, corresponding to physical phase-volume dimension action cubed; 4. the energy is constant along the flow, $H(x(t),p(t))=H(x,p)$, and $\{H,H\}=0$ (Evolution of phase functions along the model's Hamiltonian solutions: $df\mathrm{/}dt={\mathrm{\partial }}_{t}f+\{f,H\}$, the first-integral criterion, and the closed bracket algebra of autonomous first integrals clause 2).

All quantities are model values of the declared idealization; no measurement is reported.

Declare the one-label model of the force branch Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance with $N=1$, mass $m>0$, time window $\mathbb{R}$, configuration domain $Q={\mathbb{R}}^3$, zero potential $U\equiv 0$ and no residual forces (the free particle instance, Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator, Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions). By Every potential presentation is hyperregular in its charts, with inverse branch $v={\mathsf{M}}^{-1}p$ (or $G(q{)}^{-1}p$) and the explicit kinetic-plus-potential Hamiltonian the branch Hamiltonian is $H(x,p)=\mid p{\mid }^2\mathrm{/}(2m)$ with Hamiltonian system $\dot{x}=m^{-1}p$, $\dot{p}=0$ (The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions, Cartesian model motions on the potential branch and declared generalized-coordinate Lagrange equations integrate their Hamiltonian fields; the forced Cartesian field is Hamiltonian exactly when the residual force has a velocity-independent potential). Then:

1. for every initial datum the maximal solution is

${\mathrm{\Phi }}_t(x,p)=(x+(t\mathrm{/}m)p,p)(t\in \mathbb{R}),$

the free-particle flow of the presentation; 2. for $t\mathrm{\ne }0$ the function $S_t(q,Q):=-\frac{m}{2t}\mid Q-q{\mid }^2$ is of class $C^{\mathrm{\infty }}$ on ${\mathbb{R}}^3\times {\mathbb{R}}^3$ with $[S_t]=[m][Q-q{]}^2\mathrm{/}[t]=\mu {\mathrm{\ell }}^2{\tau }^{-1}$ (action), $S_{t,qQ}=(m\mathrm{/}t)I_3$ invertible everywhere, and the first-kind map it generates is exactly ${\mathrm{\Phi }}_t$: indeed $p=S_{t,q}=m(Q-q)\mathrm{/}t$ gives $Q=q+(t\mathrm{/}m)p$, and $P=-S_{t,Q}=m(Q-q)\mathrm{/}t=p$; consequently ${\mathrm{\Phi }}_t$ is a canonical map of the presentation with $\mid \det D{\mathrm{\Phi }}_t\mid =1$ (Generating functions for a potential presentation: first- and second-kind data generate canonical maps locally, the local converse under an invertible projection, and the time-dependent Hamiltonian $K=H+{\mathrm{\partial }}_{t}S$ clauses 1, Generating functions of the first and second kind for a local canonical map, The local map generated by a first- or second-kind generating function is a canonical $C^2$ local diffeomorphism); 3. projection check: the map $(q,p)\mapsto (q,Q)=(q,q+(t\mathrm{/}m)p)$ has invertible derivative $\left(\begin{array}{cc}{I}_3& 0\\ I_3& (t\mathrm{/}m)I_3\end{array}\right)$ for $t\mathrm{\ne }0$, while the family degenerates at $t=0$: $S_t$ has no value there and the projection becomes singular, so this family does not represent the identity map; 4. the canonical one-form changes by an exact form along the flow: ${\mathrm{\Phi }}_t^{\ast }\lambda =\lambda +tdK$ with $K=\mid p{\mid }^2\mathrm{/}(2m)$, because ${\mathrm{\Phi }}_t^{\ast }\lambda =P\cdot dQ=p\cdot (dq+(t\mathrm{/}m)dp)=\lambda +(t\mathrm{/}m)d(\mid p{\mid }^2\mathrm{/}2)$; in particular ${\mathrm{\Phi }}_t^{\ast }d\lambda =d\lambda$, recovering canonicity directly (The canonical one-form and the two-form of a potential presentation: $\lambda =p\cdot dx$ on the Cartesian momentum phase chart and the induced forms on each generalized-coordinate chart, A $C^1$ map between phase domains preserves the standard two-form iff $D{C}^{\mathsf{T}}JDC=J$, and a canonical diffeomorphism carries $X_{H\circ C}$ to $X_H$).

All quantities are model values of the declared idealization; no measurement is reported.

Declare the one-label model of the force branch Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance with $N=1$, mass $m>0$, configuration domain $Q={\mathbb{R}}^3$, a $C^2$ potential and no residual forces, and let $X:\mathrm{\Omega }\to Q$ be a generalized-coordinate chart of the configuration domain: $r=3$, $X$ of class $C^3$, $DX(q)$ invertible for every $q\in \mathrm{\Omega }$ (Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces, Configuration domain of $N$ labeled particles: open admissible sets, regular holonomic constraints and the mass operator, Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions). Let the generalized coordinates have declared dimensions $[q^a]={\gamma }_a$; their conjugate generalized momenta ${\pi }_a=\mathrm{\partial }\tilde{L}\mathrm{/}\mathrm{\partial }{\dot{q}}^a$ have dimensions $[{\pi }_a]=\mu {\mathrm{\ell }}^2{\tau }^{-1}{\gamma }_a^{-1}$. The Cartesian positions $X^i(q)$ have dimension $\mathrm{\ell }$ and their conjugate Cartesian momenta $P_i$ have dimension $\mu \mathrm{\ell }{\tau }^{-1}$, by Dimensions and SI unit conventions of the non-relativistic classical mechanics framework and Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces. The induced momentum chart transition of the presentation is

${\mathrm{\Phi }}_X(q,\pi )=(X(q),\mathsf{M}DX(q)G(q{)}^{-1}\pi )=(X(q),DX(q{)}^{-\mathsf{T}}\pi ),$

the second equality because $\mathsf{M}DX{G}^{-1}=D{X}^{-\mathsf{T}}$ when $DX$ is invertible and $G=D{X}^{\mathsf{T}}\mathsf{M}DX$ (The canonical one-form and the two-form of a potential presentation: $\lambda =p\cdot dx$ on the Cartesian momentum phase chart and the induced forms on each generalized-coordinate chart, The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta, Hyperregular potential presentations and their inverse Legendre branches in the model's charts). Then:

1. the function $S(q,P):=P\cdot X(q)={\sum }_{i=1}^3{P}_i{X}^i(q)$ on $\mathrm{\Omega }\times {\mathbb{R}}^3$ is of class $C^3$ and has action dimension $[S]=[P_i][X^i]=\mu {\mathrm{\ell }}^2{\tau }^{-1}$; its mixed derivative matrix $S_{qP}$ is $DX(q{)}^{\mathsf{T}}$, whose determinant is nonzero. Here $P$ is the new Cartesian momentum, while $\pi$ is the old generalized momentum. The second-kind relations are $\pi =S_q=DX(q{)}^{\mathsf{T}}P$ and $Q=S_P=X(q)$, so solving for $P$ gives $(q,\pi )\mapsto (X(q),DX(q{)}^{-\mathsf{T}}\pi )$, exactly ${\mathrm{\Phi }}_X$; consequently ${\mathrm{\Phi }}_X$ is a canonical map of the presentation with $\mathrm{\mid }\det D{\mathrm{\Phi }}_X\mathrm{\mid }=1$ (Generating functions of the first and second kind for a local canonical map, The local map generated by a first- or second-kind generating function is a canonical $C^2$ local diffeomorphism);
2. determinant check: with the block decomposition in source coordinates $(q,\pi )$ and target coordinates $(Q,P)$ the derivative is block triangular,

$D{\mathrm{\Phi }}_X=\left(\begin{array}{cc}DX(q)& 0\\ {\mathrm{\partial }}_q\left(D{X}^{-\mathsf{T}}\pi \right)& DX(q{)}^{-\mathsf{T}}\end{array}\right),\det D{\mathrm{\Phi }}_X=\det DX(q)\cdot \det DX(q{)}^{-\mathsf{T}}=1;$

3. the $(q,Q)$-projection $(q,\pi )\mapsto (q,X(q))$ is independent of the momentum and has derivative $\left(\begin{array}{cc}{I}_3& 0\\ DX(q)& 0\end{array}\right)$ of rank $3<6$ at every point: it is nowhere invertible. Since a first-kind generating function requires the relation $p=S_q(q,Q)$ to be solvable for $Q$, i.e. the map $(q,Q)\mapsto (q,S_q(q,Q))$ to be a local diffeomorphism, the transition ${\mathrm{\Phi }}_X$ admits no first-kind generating function near any point; the second-kind representation of claim 1 is the available local description (Generating functions of the first and second kind for a local canonical map, A canonical map with invertible $(q,Q)$-projection is locally generated by a first-kind generating function);
4. contrast — the Cartesian momentum shift: for a constant $c\in {\mathbb{R}}^3$ of momentum dimension the map $C(q,p)=(q,p+c)$ of the Cartesian chart is canonical trivially ($DC=I_6$), its first-kind projection is degenerate identically, and it has the global second-kind generating function $S(q,P)=q\cdot (P-c)$, also of action dimension, since $S_q=P-c$ and $S_P=q$ (Generating functions for a potential presentation: first- and second-kind data generate canonical maps locally, the local converse under an invertible projection, and the time-dependent Hamiltonian $K=H+{\mathrm{\partial }}_{t}S$ clause 2).

All quantities are model values of the declared idealization; no measurement is reported.