Cm Math Variations — Examples

Let $n=1$, $[a,b]=[0,\pi ]$, $U=\mathbb{R}$, and $\mathcal{L}(q,v)=\frac{1}{2}(v^2-4q^2)$, a $C^{\mathrm{\infty }}$ (hence $C^2$) Lagrangian on $\mathbb{R}\times \mathbb{R}\times \mathbb{R}$ (A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations, $C^k$ Euclidean maps and diffeomorphisms). Let

$q_{\ast }(t):=\sin 2t,\eta (t):=\sin t\mathrm{.}$

Use fixed-endpoint admissible $C^2$ curves with $q(0)=q(\pi )=0$ and the norm $\mathrm{\parallel }q{\mathrm{\parallel }}_{C^2}:={\max }_{0\le j\le 2}{\sup }_{t\in [0,\pi ]}\mathrm{\mid }{q}^{(j)}(t)\mathrm{\mid }$. A local minimum means $S(q_{\ast })\le S(q)$ for all admissible $q$ in some ball about $q_{\ast }$ in this norm. Then $q_{\ast }$ is a stationary curve of the action $S(q)={\int }_0^{\pi }\frac{1}{2}(q^{\mathrm{\prime }2}-4q^2)dt$, and for the one-parameter family $q_{\ast }+ϵ\eta$ one has the exact identity

$S(q_{\ast }+ϵ\eta )=S(q_{\ast })-\frac{3\pi }{4}{ϵ}^2(ϵ\in \mathbb{R})\mathrm{.}$

In particular $S(q_{\ast }+ϵ\eta )<S(q_{\ast })$ for every $ϵ\mathrm{\ne }0$: the stationary curve is not a local minimum of the action. The model has no minimiser at all, since $S(\lambda \eta )=-\frac{3\pi }{4}{\lambda }^2\to -\mathrm{\infty }$ as $\mathrm{\mid }\lambda \mathrm{\mid }\to \mathrm{\infty }$.

• The example is a mathematical model with an indefinite quadratic Lagrangian; it shows that the Euler–Lagrange equation is a necessary condition only, and that "principle of least action" is a misnomer in general. The stationary curve here is a saddle in the infinite-dimensional sense realised by the decreasing family $q_{\ast }+ϵ\sin t$ and the increasing family $q_{\ast }+ϵ\sin 3t$ of steps 3.1 and 4.1.

• The zero curve $q=0$ is also stationary for this model, and it is likewise not a local minimum, by the same computation with $q_{\ast }$ replaced by $0$. Nothing here depends on the interpretation of $\mathcal{L}$ as a physical energy; no positivity of the kinetic term is claimed.

Take $n=1$, $J=U=\mathbb{R}$, and consider the three fibre-full $C^2$ Lagrangians ${\mathcal{L}}_1(v)=\sqrt{1+v^2}$, ${\mathcal{L}}_2(v)=0$ and ${\mathcal{L}}_3(v)=\frac{1}{2}{v}^2$ on $J\times U\times \mathbb{R}$ (The Legendre map of a time-dependent Lagrangian: regularity and hyperregularity). Then:

1. ${\mathcal{L}}_1$ is regular on the chart but not hyperregular: its velocity Hessian is $(1+v^2{)}^{-3\mathrm{/}2}>0$ at every $v$, while the fibre Legendre map $v\mapsto {\mathcal{L}}_1^{\mathrm{\prime }}(v)=v\mathrm{/}\sqrt{1+v^2}$ has range $(-1,1)$, so it is not surjective onto $\mathbb{R}$;
2. ${\mathcal{L}}_2$ is neither regular nor hyperregular: ${\mathcal{L}}_2^{\mathrm{\prime }}\equiv 0$ and ${\mathcal{L}}_2^{\mathrm{\prime }\mathrm{\prime }}\equiv 0$, so the velocity Hessian is singular everywhere and the fibre map is constant, hence not injective;
3. ${\mathcal{L}}_3$ is both regular and hyperregular: ${\mathcal{L}}_3^{\mathrm{\prime }\mathrm{\prime }}\equiv 1$ and the fibre map $v\mapsto v$ is a $C^{\mathrm{\infty }}$ bijection of $\mathbb{R}$ onto itself.

Thus regularity does not imply hyperregularity, and neither condition is automatic.

• The first example is the standard witness that global hyperregularity is a genuine extra hypothesis: the local inverse branch exists near every velocity by Nonsingular velocity Hessian is equivalent to invertibility of the Legendre derivative, hence gives a local C^1 inverse, but no global momentum variable can be solved over the whole fibre. The third example is the model case supplied by A mechanical chart Lagrangian with positive definite kinetic matrix is hyperregular, with the explicit kinetic-plus-potential Hamiltonian with constant positive definite $G$, and the second is the degenerate case used in Scope of the Legendre map: regular versus hyperregular, the meaning of the covector datum, and the limits of the branch description.

• The choice ${\mathcal{L}}_1=\sqrt{1+v^2}$ is a mathematical example, not a claim about any physical Lagrangian; no interpretation of the velocities or of the momentum variable is asserted.

Let $m,l,g>0$ and let $\mathrm{\Omega }:=(-\pi ,\pi )\subseteq \mathbb{R}$. Consider the embedding

$X:\mathrm{\Omega }\to {\mathbb{R}}^2,X(\theta ):=(l\sin \theta ,-l\cos \theta ),$

the ambient mass $M:=m{I}_2$ and the constant force $F:=(0,-mg)$. Let $I\subseteq \mathbb{R}$ be a nonempty open interval (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), let $\theta :I\to \mathrm{\Omega }$ be a $C^2$ curve and put $x(t):=X(\theta (t))$. Then:

1. $X$ is a $C^3$ immersion with $\mathrm{\parallel }{X}^{\mathrm{\prime }}(\theta ){\mathrm{\parallel }}_2=l$ for every $\theta$; if the ambient motion satisfies $M{x}^{\mathrm{\prime }\mathrm{\prime }}=F+R$ with $R(t)\cdot X^{\mathrm{\prime }}(\theta (t))\delta =0$ for all $\delta \in \mathbb{R}$, then

$m{l}^2{\theta }^{\mathrm{\prime }\mathrm{\prime }}(t)+mgl\sin \theta (t)=0(t\in I);$

2. the reduced energy $E(\theta ,{\theta }^{\mathrm{\prime }}):=\frac{1}{2}m{l}^2{\theta }^{\mathrm{\prime }2}-mgl\cos \theta$ is constant along every solution of the equation;
3. $\theta =0$ is a strict local minimum of the reduced potential $U_{\mathrm{r}\mathrm{e}\mathrm{d}}(\theta ):=-mgl\cos \theta$ on $\mathrm{\Omega }$, and the equilibrium $(0,0)$ of the pendulum system ${\theta }^{\mathrm{\prime }}=v$, $v^{\mathrm{\prime }}=-(g\mathrm{/}l)\sin \theta$ is Lyapunov stable;
4. the linearisation at $\theta =0$ is $m{l}^2{\xi }^{\mathrm{\prime }\mathrm{\prime }}+mgl\xi =0$, a quadratic mode with angular frequency

$\omega =\sqrt{g\mathrm{/}l}\mathrm{.}$

• The example is a mathematical model of an ideal frictionless planar pendulum in a uniform downward force field; statements about a real pendulum require apparatus, friction and measurement assumptions that are not part of this item. The stability conclusion is Lyapunov stability, not asymptotic attraction, and it concerns the exact pendulum equation in the chart $\mathrm{\Omega }=(-\pi ,\pi )$.

• The frequency $\sqrt{g\mathrm{/}l}$ is the frequency of the linearised mode by [F4]; no claim is made that the nonlinear pendulum oscillates periodically with that frequency for finite amplitude.

Let $a,m,g>0$ and $\mathrm{\Omega }\in \mathbb{R}$, and let

$X:\mathbb{R}\times \mathbb{R}\to {\mathbb{R}}^3,X(\theta ,t):=(a\sin \theta \cos (\mathrm{\Omega }t),a\sin \theta \sin (\mathrm{\Omega }t),a(1-\cos \theta )),$

the position of a point of a circle of radius $a$ rotating about the vertical axis with constant angular speed $\mathrm{\Omega }$. For an open interval $I$ and a $C^2$ curve $\theta :I\to \mathbb{R}$ put $x(t):=X(\theta (t),t)$. Stationarity on $I$ means stationarity of the restricted curve on every $[t_1,t_2]\subseteq I$ with $t_1<t_2$, using fixed-endpoint variations. Then:

1. $\dot{x}=X_{\theta }{\theta }^{\mathrm{\prime }}+X_t$ and $\frac{1}{2}m\mathrm{\mid }\dot{x}{\mathrm{\mid }}^2=\frac{1}{2}m{a}^2({\theta }^{\mathrm{\prime }2}+{\mathrm{\Omega }}^2{\sin }^2\theta )$, while $\ddot{x}\cdot X_{\theta }=a^2{\theta }^{\mathrm{\prime }\mathrm{\prime }}-a^2{\mathrm{\Omega }}^2\sin \theta \cos \theta$;
2. the curves $\theta$ that are stationary for the action of the time-dependent Lagrangian chart $L(\theta ,v):=\frac{1}{2}m{a}^2(v^2+{\mathrm{\Omega }}^2{\sin }^2\theta )-mga(1-\cos \theta )$ on $O:=\mathbb{R}\times \mathbb{R}\times \mathbb{R}$ are exactly the $C^2$ solutions of

${\theta }^{\mathrm{\prime }\mathrm{\prime }}={\mathrm{\Omega }}^2\sin \theta \cos \theta -\frac{g}{a}\sin \theta ;$

3. the reduced energy $E(\theta ,v):=\frac{1}{2}m{a}^2{v}^2-\frac{1}{2}m{a}^2{\mathrm{\Omega }}^2{\sin }^2\theta -mga\cos \theta$ is constant along every $C^2$ solution, whereas the inertial kinetic-plus-potential energy $K+U_g=\frac{1}{2}m\mathrm{\mid }\dot{x}{\mathrm{\mid }}^2+mga(1-\cos \theta )$ satisfies

$\frac{d}{dt}(K+U_g)=2m{a}^2{\mathrm{\Omega }}^2\sin \theta \cos \theta {\theta }^{\mathrm{\prime }}=m{a}^2{\mathrm{\Omega }}^2\frac{d}{dt}{\sin }^2\theta$

along every solution, and equals $E+m{a}^2{\mathrm{\Omega }}^2{\sin }^2\theta +mga$; it is therefore in general not conserved, and it differs from $E$ by $m{a}^2{\mathrm{\Omega }}^2{\sin }^2\theta +mga$; 4. if the ambient motion is additionally modelled by $m\ddot{x}=(0,0,-mg)+R$ with an ideal reaction satisfying $R(t)\cdot X_{\theta }(\theta (t),t)=0$, then the same equation of part 2 holds, and the reaction power is

$R\cdot \dot{x}=\frac{d}{dt}(K+U_g)=2m{a}^2{\mathrm{\Omega }}^2\sin \theta \cos \theta {\theta }^{\mathrm{\prime }},$

generally nonzero: the prescribed rotation can do work on the point; 5. if $a{\mathrm{\Omega }}^2\le g$, then $0$ is a strict local minimum of $U_{\mathrm{e}\mathrm{f}\mathrm{f}}(\theta ):=-\frac{1}{2}m{a}^2{\mathrm{\Omega }}^2{\sin }^2\theta -mga\cos \theta$ with $U_{\mathrm{e}\mathrm{f}\mathrm{f}}^{\mathrm{\prime }}(0)=0$, and the equilibrium $(0,0)$ of ${\theta }^{\mathrm{\prime }}=v,v^{\mathrm{\prime }}={\mathrm{\Omega }}^2\sin \theta \cos \theta -(g\mathrm{/}a)\sin \theta$ is Lyapunov stable; if $a{\mathrm{\Omega }}^2>g$, then $0$ is not a local minimum of $U_{\mathrm{e}\mathrm{f}\mathrm{f}}$ and the stability criterion does not apply.