Legendre transformation and Hamiltonian mechanics — Examples

(a) One label in a uniform field. Declare the one-label model of the force branch of 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 $Q={\mathbb{R}}^3$ and potential $U(t,x):=mg{x}_3$ with constant acceleration magnitude $g>0$, $[g]=\mathrm{\ell }{\tau }^{-2}$, read in $\mathrm{m}{\mathrm{s}}^{-2}$, with no residual forces; the force map is $F=(0,0,-mg)$ and the mechanical Lagrangian is $L(t,x,v)=\frac{1}{2}m\mid v{\mid }_2^2-mg{x}_3$ (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). Then:

1. the momentum of the presentation is $p=\mathsf{M}v=mv$ (The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta), the presentation is hyperregular on the Cartesian chart with inverse branch $v=p\mathrm{/}m$ (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), and the Hamiltonian is $H(t,x,p)=\frac{\mid p{\mid }_2^2}{2m}+mg{x}_3\mathrm{.}$
2. Hamilton's equations $\dot{x}=H_p$, $\dot{p}=-H_x$ read $\dot{x}=p\mathrm{/}m$, $\dot{p}=(0,0,-mg)$, so that $m\ddot{x}=\dot{p}=(0,0,-mg)$: the system is equivalent to the model's equation of motion and the potential is the uniform field (Hamilton's equations of the mechanical presentation are equivalent to the model's equation of motion, with residual forces as momentum sources and the coordinate form on every chart).
3. for every datum $(0,x_0,v_0)$, where $x_0\in {\mathbb{R}}^3$ is the initial position (metres) and $v_0\in {\mathbb{R}}^3$ the initial coordinate velocity (metres per second), the curve $x(t):=x_0+v_{0}t-\frac{1}{2}g{t}^2{e}_3,e_3=(0,0,1),$ is the maximal motion of the model through that datum, defined for all real times; its velocity is $v_0-gt{e}_3$ and its momentum is $m(v_0-gt{e}_3)$.

(b) Two labels with unequal masses. Declare instead the free two-label model $N=2$, masses $m_1,m_2>0$, $I_0=\mathbb{R}$, $Q={\mathbb{R}}^6$ and zero force map, so that $U\equiv 0$. Then $\mathsf{M}(v_1,v_2)=(m_1{v}_1,m_2{v}_2)$, the Hamiltonian is $H=\mid p_1{\mid }_2^2\mathrm{/}(2m_1)+\mid p_2{\mid }_2^2\mathrm{/}(2m_2)$, the inverse branch is $v={\mathsf{M}}^{-1}p=(p_1\mathrm{/}{m}_1,p_2\mathrm{/}{m}_2)$, and Hamilton's equations are ${\dot{x}}_i=p_i\mathrm{/}{m}_i$, ${\dot{p}}_i=0$: each label moves uniformly in a straight line, and the fibre map is a linear isomorphism whose inverse is not a scalar multiple of the identity when $m_1\mathrm{\ne }{m}_2$.

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

• The inverse branch is the mass operator's inverse. In part (a) the inverse branch is multiplication by $1\mathrm{/}m$, and in part (b) by the two scalar factors $1\mathrm{/}{m}_1$, $1\mathrm{/}{m}_2$; this is the model instance of the general formulas $v={\mathsf{M}}^{-1}p$ and $v=G(q{)}^{-1}p$ of 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 positivity of the masses is what makes both inversions global on the fibre.

• Potential normalisation. The displayed $H$ uses the representative $U=mg{x}_3$; replacing it by $U+c(t)$ shifts $H$ by $c(t)$ (The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions), so the numerical value of $H$ at a state is convention-dependent while the Hamilton system is unchanged by the shift together with the corresponding action convention. No energy value is compared with any measurement here.

Declare one particle label of mass $m>0$ in the force branch of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance, with time window $\mathbb{R}$, ambient configuration domain $Q={\mathbb{R}}^3$, potential $U(x,y,z)=mgy$, constant $g>0$, and no residual force. Thus the force is $(0,-mg,0)$. Fix $l>0$ and impose the scleronomic regular constraint $\mathcal{G}(x,y,z)=(x^2+y^2-l^2,z)=0,$ with the ideal-reaction law of Adopted ideal-reaction law for holonomic constraints: the d'Alembert condition, its scope and its limits. Its locus is the full circle $N=\{x^2+y^2=l^2,z=0\}$. Here $m$ is a mass in kilograms, $l$ a length in metres, and $g$ an acceleration magnitude in metres per second squared; their dimensions are $\mu$, $\mathrm{\ell }$, and $\mathrm{\ell }{\tau }^{-2}$. The two components of $\mathcal{G}$ have dimensions ${\mathrm{\ell }}^2$ and $\mathrm{\ell }$, respectively. Write $G_{\mathrm{k}\mathrm{i}\mathrm{n}}$ for the kinetic metric, to distinguish it from $\mathcal{G}$.

The full angular variable is an equivalence class $[\theta ]\in \mathbb{R}\mathrm{/}(2\pi \mathbb{Z})$, with $\theta \sim \theta +2\pi k$, $k\in \mathbb{Z}$. The smooth periodic map $X(\theta )=(l\sin \theta ,-l\cos \theta ,0)$ descends to the circle of constrained configurations; this identification is the rescaled and rotated circle parametrization of $[t]\mapsto (\cos 2\pi t,\sin 2\pi t)$ is a homeomorphism from $\mathbb{R}\mathrm{/}\mathbb{Z}$ to the unit circle. It is not a one-to-one map from $\mathbb{R}$ and is not used as one global generalized-coordinate parametrization.

Instead use the two angular charts ${\mathrm{\Omega }}_0=(-\pi ,\pi )$ and ${\mathrm{\Omega }}_{\pi }=(0,2\pi )$, with restrictions $X_0,X_{\pi }$ of $X$. The ambient open sets $Q_0={\mathbb{R}}^3\setminus \{(0,y,z):y\ge 0\},Q_{\pi }={\mathbb{R}}^3\setminus \{(0,y,z):y\le 0\}$ satisfy $X_a({\mathrm{\Omega }}_a)=N\cap Q_a$ for $a=0,\pi$. Restrict the ambient force model and constraint datum to $Q_a$ when applying Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces clause (d). Each restricted map is a one-to-one $C^3$ parametrization with injective derivative, covering its constrained arc exactly once. The two arcs cover $N$. On each connected component of an overlap the angular coordinates differ by $2\pi k$, with derivative $1$; no chart uses the same constrained configuration twice.

In either angular chart, the following hold, with dimensionless radian angle $\theta$:

On each phase-coordinate chart ${\mathrm{\Omega }}_a\times \mathbb{R}$ with coordinates $(\theta ,p)$, define an alternating bilinear form on coordinate increments $u=(u_{\theta },u_p)$ and $v=(v_{\theta },v_p)$ in ${\mathbb{R}}^2$ by ${\omega }_a(u,v):=u_{\theta }{v}_p-v_{\theta }{u}_p\mathrm{.}$ The notation $d\theta \wedge dp$ below denotes this explicitly defined form field.

1. $G_{\mathrm{k}\mathrm{i}\mathrm{n}}=m{l}^2$, $\tilde{U}=-mgl\cos \theta$, and $\tilde{L}(\theta ,\dot{\theta })=\frac{1}{2}m{l}^2{\dot{\theta }}^2+mgl\cos \theta \mathrm{.}$
2. The generalized momentum is $p=m{l}^2\dot{\theta }$, of dimension action $\mu {\mathrm{\ell }}^2{\tau }^{-1}$. The fibre map is hyperregular with inverse $\dot{\theta }=p\mathrm{/}(m{l}^2)$, and $\tilde{H}(\theta ,p)=\frac{p^2}{2m{l}^2}-mgl\cos \theta$ is the mechanical energy of the constrained motion.
3. Hamilton's equations are $\dot{\theta }=p\mathrm{/}(m{l}^2)$, $\dot{p}=-mgl\sin \theta$, equivalently ${\theta }^{\mathrm{\prime }\mathrm{\prime }}=-(g\mathrm{/}l)\sin \theta$. Locally these are equivalent to the ideal constrained motion, with its reaction recovered by Ideal scleronomic holonomic constraints: the substituted Lagrange equations are equivalent to the constrained model equation.
4. On overlaps ${\theta }_b={\theta }_a+2\pi k$ and $p_b=p_a$. The Lagrangian, Hamiltonian, equations, and canonical form $d\theta \wedge dp$ therefore agree, so the local Hamilton systems give one global system on the phase cylinder $(\mathbb{R}\times \mathbb{R})\mathrm{/}((\theta ,p)\sim (\theta +2\pi k,p))\mathrm{.}$ This retains the entire pendulum, including rotations crossing either chart cut. Equivalence with the ideal constrained equations is local at every time and hence holds along any constrained motion, without pretending that one real angular chart is globally injective.

All quantities are declared model data; no measurement is reported.

The angular generalized momentum $m{l}^2\dot{\theta }$ has action units; the tangential kinetic momentum $ml\dot{\theta }$ has linear-momentum units. The atlas replaces the invalid globally injective real-angle parametrization without restricting the pendulum to one arc or excluding rotations. Its static charts preserve the energy interpretation; a moving chart with a velocity-linear term has the separate interpretation given by the companion example.

Declare the one-label model of the force branch of 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; write $L(t,x,v)=\frac{1}{2}m\mid v{\mid }_2^2-U(t,x)$ for its mechanical Lagrangian (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). Fix a real parameter $\alpha$ and the smooth function $F(t,x):=\alpha t{x}_1$ on the full time-configuration projection $\mathbb{R}\times Q$, and define

$L^{\mathrm{\prime }}(t,x,v):=L(t,x,v)+F_t(t,x)+F_x(t,x)\cdot v=L(t,x,v)+\alpha x_1+\alpha t{v}_1\mathrm{.}$

The units are $[m]=\mu$, $[x]=\mathrm{\ell }$, $[t]=\tau$, $[\alpha ]=\mu \mathrm{\ell }{\tau }^{-2}$ (force, newtons), and $[F]=\mu {\mathrm{\ell }}^2{\tau }^{-1}$ (action). The free initial speed $q$ below has dimension $\mathrm{\ell }{\tau }^{-1}$ and is read in metres per second. Then:

1. $L^{\mathrm{\prime }}$ is a fibre-full $C^2$ time-dependent Lagrangian chart with velocity gradient $L_v^{\mathrm{\prime }}(t,x,v)=mv+\alpha t{e}_1$, $e_1=(1,0,0)$; the fibre map $v\mapsto L_v^{\mathrm{\prime }}(t,x,v)$ is a $C^2$ bijection of ${\mathbb{R}}^3$ onto itself with inverse $v=(p^{\mathrm{\prime }}-\alpha t{e}_1)\mathrm{/}m$, so $L^{\mathrm{\prime }}$ is hyperregular on its generic time-dependent chart (The Legendre map of a time-dependent Lagrangian: regularity and hyperregularity) and its fibre derivative is the shifted momentum $p^{\mathrm{\prime }}=p+\alpha t{e}_1,p=mv,$ which differs from the mechanical momentum $mv$ whenever $\alpha t\mathrm{\ne }0$.
2. the Hamiltonian of the branch of $L^{\mathrm{\prime }}$ is $H^{\mathrm{\prime }}(t,x,p^{\mathrm{\prime }})=\frac{\mid p^{\mathrm{\prime }}-\alpha t{e}_1{\mid }_2^2}{2m}+U(t,x)-\alpha x_1=H(t,x,p^{\mathrm{\prime }}-\alpha t{e}_1)-\alpha x_1,$ where $H$ is the Hamiltonian of the presentation (The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions, 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);
3. the Hamilton systems of $H$ and $H^{\mathrm{\prime }}$ are equivalent under the shift $p^{\mathrm{\prime }}=p+\alpha t{e}_1$: $\dot{x}=H_{p^{\mathrm{\prime }}}^{\mathrm{\prime }}=(p^{\mathrm{\prime }}-\alpha t{e}_1)\mathrm{/}m$ and ${\dot{p}}^{\mathrm{\prime }}=-H_x^{\mathrm{\prime }}=-{\mathrm{\nabla }}_{x}U+\alpha e_1$ hold exactly when $\dot{x}=p\mathrm{/}m$ and $\dot{p}=-{\mathrm{\nabla }}_{x}U$ hold, so both systems have the same $x$-curves, namely the motions of the model (Hamilton's equations of the mechanical presentation are equivalent to the model's equation of motion, with residual forces as momentum sources and the coordinate form on every chart, Total-derivative Lagrangians, the potential representative and scaling: invariant stationary curves with endpoint terms); the Euler-Lagrange expressions coincide, $E_{L^{\mathrm{\prime }}}(x)=E_L(x)$, and the actions differ by the endpoint value $\alpha (b{x}_1(b)-a{x}_1(a))$;
4. on free data $U\equiv 0$, with $m>0$, $\alpha \mathrm{\ne }0$, initial position $x(0)=0$ and initial velocity $v(0)=(q,0,0)$ with $q>0$, the curves $x(t)=(qt,0,0),p^{\mathrm{\prime }}(t)=(mq+\alpha t,0,0)$ solve the shifted Hamilton system, and along them $H^{\mathrm{\prime }}(t,x(t),p^{\mathrm{\prime }}(t))=\frac{1}{2}m{q}^2-\alpha qt,K(t)+U(t,x(t))=\frac{1}{2}m{q}^2\mathrm{.}$ Thus $H^{\mathrm{\prime }}$ differs from the mechanical energy for $t\mathrm{\ne }0$ while the motion is unchanged. For the numerical instance $m=1\mathrm{k}\mathrm{g}$ and $\alpha =1\mathrm{N}$ the same dimensionally explicit formulas apply.

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

• The shift is exactly the gauge derivative. The general pattern is $p^{\mathrm{\prime }}=p+F_x(t,x)$ and $H^{\mathrm{\prime }}(t,x,p^{\mathrm{\prime }})=H(t,x,p^{\mathrm{\prime }}-F_x(t,x))-F_t(t,x)$: the momentum is translated by the gradient of the added function and the Hamiltonian is corrected by its time derivative. The modified Lagrangian is hyperregular but is not the mechanical Lagrangian of the presentation, and $p^{\mathrm{\prime }}$ is not the momentum $\mathsf{M}v$ of The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta; it is the fibre derivative of the modified chart, which is why the Hamiltonian of the modified chart is not the mechanical energy.

• Source check. Tong's Section 2.5.7 records the invariance of the equations under addition of a total derivative, and his Section 4.1.3 electromagnetic example exhibits a canonical momentum differing from the mechanical one by a gradient term; Stewart's Section 1.2 moving-cart computation exhibits the same pattern with a constant gradient and shows that the shifted Hamiltonian can be conserved while the mechanical energy is not. The computation above is the model instance of those statements; no electromagnetic interpretation of the parameter $\alpha$ is claimed.

Take $n=1$, $J=U=\mathbb{R}$, and consider the fibre-full $C^2$ chart Lagrangian

$\mathcal{L}:\mathbb{R}\times \mathbb{R}\times \mathbb{R}⟶\mathbb{R},\mathcal{L}(t,q,v):=e^v$

(The Legendre map of a time-dependent Lagrangian: regularity and hyperregularity, A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations). Then:

1. $\mathcal{L}$ is regular on the chart: ${\mathcal{L}}_v(t,q,v)=e^v$ and ${\mathcal{L}}_{vv}(t,q,v)=e^v>0$ for every $(t,q,v)$, so the velocity Hessian is nonsingular at every state and the Legendre map is a local $C^1$ diffeomorphism at every state (Nonsingular velocity Hessian is equivalent to invertibility of the Legendre derivative, hence gives a local C^1 inverse).
2. $\mathcal{L}$ is not hyperregular on the chart: the fibre velocity map $v\mapsto e^v$ has image $(0,\mathrm{\infty })$ and is not a bijection of $\mathbb{R}$ onto $\mathbb{R}$, so there is no inverse branch defined on the whole momentum line; a momentum $p\le 0$ is not the fibre derivative of any velocity, and no Hamiltonian of a branch is defined there (The Legendre map of a time-dependent Lagrangian: regularity and hyperregularity).
3. on the momentum half-line $p>0$ the inverse branch is $v(p)=\ln p$, the unique real number with $e^{v(p)}=p$ (The exponential is a continuous bijection from $\mathbb{R}$ onto $(0,\mathrm{\infty })$), and the Hamiltonian of that branch is

$H:(0,\mathrm{\infty })⟶\mathbb{R},H(p)=pv(p)-e^{v(p)}=p\ln p-p,$

with $H^{\mathrm{\prime }}(p)=\ln p=v(p)$ for every $p>0$; in particular the differential identity $H_p=v$ of The Hamiltonian on an inverse Legendre branch: differential, equivalence with the Euler-Lagrange equation, and the energy identity clause 1 holds on this branch even though the chart is not hyperregular. 4. for every nonempty open interval $I$ and every $C^2$ curve $q:I\to \mathbb{R}$ with $p(t):=e^{q^{\mathrm{\prime }}(t)}$ one has: $\dot{q}=H^{\mathrm{\prime }}(p)$ and $\dot{p}=0$ hold for all $t\in I$ if and only if the Euler-Lagrange expression of $\mathcal{L}$ vanishes, $E_{\mathcal{L}}(q)(t)=0$, for all $t\in I$.

The example is a pure chart computation: $\mathcal{L}=e^v$ is not of the natural mechanical form $\frac{1}{2}{v}^{\mathsf{T}}Gv+A\cdot v-V$, and no physical interpretation of it is claimed.