Cm Math Variations

Every d'Alembert motion of the constraint $z^{\mathrm{\prime }}=x{y}^{\mathrm{\prime }}$ with unit mass and zero applied force that satisfies the initial data

$(x,y,z)(0)=(0,0,0),(x^{\mathrm{\prime }},y^{\mathrm{\prime }},z^{\mathrm{\prime }})(0)=(1,1,0)$

is a vakonomic motion of the same constraint with the same initial data.

• The counterexample separates the two prescriptions as mathematical definitions; it does not by itself say which of them, if either, models a given physical system. The empirical question is the subject of the consuming physical items and requires its own apparatus assumptions and data.

• The specific numbers were chosen for an elementary closed-form solution; the general statement that the two dynamics need not coincide is the content of the cited treatments of vakonomic and non-holonomic mechanics, and the computation here is self-contained.

Let $n\ge 1$, let $m>0$, let $a_0<b_0$ be reals, let $\epsilon >0$, let $U\subseteq {\mathbb{R}}^n$ be open and nonempty, let $U_0:(a_0-\epsilon ,b_0+\epsilon )\times U\to \mathbb{R}$ be of class $C^2$, and let

$\mathcal{L}(t,q,v):=\frac{1}{2}m\mathrm{\mid }v{\mathrm{\mid }}^2-U_0(t,q)$

be the corresponding Lagrangian chart (A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations, The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$). Let $q:I\to U$ be a $C^2$ Euler–Lagrange solution on an open interval $[a_0,b_0]\subseteq I\subseteq (a_0-\epsilon ,b_0+\epsilon )$. Then:

1. (translation) if $a\in {\mathbb{R}}^n$ and $a\cdot {\mathrm{\nabla }}_q{U}_0(t,q)=0$ for every $(t,q)$, then $P_a(t):=ma\cdot q^{\mathrm{\prime }}(t)$ is constant on $I$;
2. (rotation) if $n=3$, $a\in {\mathbb{R}}^3$ and $(a\times q)\cdot \mathrm{\nabla }{U}_0(t,q)=0$ for every $(t,q)$, then $J_a(t):=m(q(t)\times q^{\mathrm{\prime }}(t))\cdot a$ is constant on $I$;
3. (boost) if $a\in {\mathbb{R}}^n$ and $a\cdot {\mathrm{\nabla }}_q{U}_0=0$ on the whole chart, then $K_a(t):=a\cdot (tm{q}^{\mathrm{\prime }}(t)-mq(t))$ is constant on $I$.

• Clause 3 is the Galilean boost charge: its defining Noether identity holds when $a\cdot {\mathrm{\nabla }}_q{U}_0=0$ on the chart, giving conservation along every Euler–Lagrange solution. Constancy on a particular solution alone does not imply this whole-chart symmetry. A confining potential such as $\frac{k}{2}\mathrm{\mid }q{\mathrm{\mid }}^2$ breaks the boost symmetry even though the Galilean charts of the framework still exist; this is the mathematical content of the warning that a kinematic symmetry need not be a symmetry of the chosen Lagrangian.

• All three statements are conditional deductions inside the chart model: they identify conserved quantities of the Euler–Lagrange equation under explicitly stated symmetry hypotheses of the potential. They are mathematical statements about the model and are not claims that any particular physical system possesses the required symmetry.

Let $n,m\ge 1$, let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open and nonempty, and let $X:\mathrm{\Omega }\to {\mathbb{R}}^m$ be of class $C^3$ with $DX(q)$ injective for every $q\in \mathrm{\Omega }$; fix an immersion in the sense of Submersions and immersions between Euclidean open sets (this forces $n\le m$). Let $M\in M_m(\mathbb{R})$ be symmetric positive definite, meaning $M^{\mathsf{T}}=M$ and $u^{\mathsf{T}}Mu>0$ for every $0\mathrm{\ne }u\in {\mathbb{R}}^m$, let $D\subseteq \mathbb{R}\times {\mathbb{R}}^m\times {\mathbb{R}}^m$ be open and let $F:D\to {\mathbb{R}}^m$ be continuous, let $I\subseteq \mathbb{R}$ be an open interval (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), and let $q:I\to \mathrm{\Omega }$ be a $C^2$ curve with $x(t):=X(q(t))$ and $(t,x(t),x^{\mathrm{\prime }}(t))\in D$ for all $t\in I$. Suppose there is a continuous $R:I\to {\mathbb{R}}^m$ with

$M{x}^{\mathrm{\prime }\mathrm{\prime }}(t)=F(t,x(t),x^{\mathrm{\prime }}(t))+R(t),R(t)\cdot DX(q(t))\delta =0\text{for all }\delta \in {\mathbb{R}}^n,t\in I\mathrm{.}$

Then, writing $DX(q(t){)}^{\mathsf{T}}$ for the transpose and $(X_{qq}[q^{\mathrm{\prime }},q^{\mathrm{\prime }}]{)}_k:={\sum }_{i,j}{\mathrm{\partial }}_i{\mathrm{\partial }}_j{X}_k(q)q_i^{\mathrm{\prime }}{q}_j^{\mathrm{\prime }}$, one has

$DX(q{)}^{\mathsf{T}}MDX(q)q^{\mathrm{\prime }\mathrm{\prime }}=DX(q{)}^{\mathsf{T}}F(t,x,x^{\mathrm{\prime }})-DX(q{)}^{\mathsf{T}}M{X}_{qq}[q^{\mathrm{\prime }},q^{\mathrm{\prime }}]$

on $I$. The vector $DX(q{)}^{\mathsf{T}}F(t,x,x^{\mathrm{\prime }})$ is the generalized force of the ambient force $F$ in the coordinates $q$, and $D{X}^{\mathsf{T}}M{X}_{qq}[q^{\mathrm{\prime }},q^{\mathrm{\prime }}]$ is the acceleration correction of the parametrisation; when $X$ is affine, the correction vanishes.

• The corollary is the mathematical content of the reduced d'Alembert principle: the tangent projection of the ambient equation is taken, the normal component $R$ drops out, and the applied force enters as the generalized force $D{X}^{\mathsf{T}}F$. The condition $R\cdot DX\delta =0$ is an explicit hypothesis on $R$; the corollary does not assert that any particular mechanism realises it, nor that an ambient constraint produces such an $R$.

• The matrix $D{X}^{\mathsf{T}}MDX$ is symmetric and positive definite on ${\mathbb{R}}^n$: for $0\mathrm{\ne }u\in {\mathbb{R}}^n$, injectivity gives $DXu\mathrm{\ne }0$ and $u^{\mathsf{T}}D{X}^{\mathsf{T}}MDXu=(DXu{)}^{\mathsf{T}}M(DXu)>0$; this is the reduced mass matrix used in The ideal planar pendulum: equation of motion, conserved energy, Lyapunov stability and the frequency $\sqrt{g\mathrm{/}l}$ ↗ and The planar rolling disk: the holonomic constraint $x-a\theta =\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$ and the acceleration $f\mathrm{/}(m+I\mathrm{/}{a}^2)$ ↗.

Let $n\ge 1$ and $k\ge 1$, let $O\subseteq \mathbb{R}\times {\mathbb{R}}^n$ be open and nonempty, let $G:O\to {\mathbb{R}}^k$ be of class $C^2$ ($C^k$ Euclidean maps and diffeomorphisms), and write $G_q(t,q)$ for the Jacobian matrix of $q\mapsto G(t,q)$ (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case), a $k\times n$ matrix. The constraint

$G(t,q)=0$

is regular when

$G_q(t,q)\text{ has rank }k\text{ for every }(t,q)\in O,$

equivalently when $G_q$ is surjective at every point (Submersions and immersions between Euclidean open sets). Under this hypothesis each fibre $G(t,\cdot {)}^{-1}(0)$ is a regular level set and every point of it is a regular point in the sense of Regular and critical points, regular and critical values, and level sets.

Constraint set and admissible data. Fix a symmetric positive definite matrix $M\in M_n(\mathbb{R})$, meaning $M^{\mathsf{T}}=M$ and $u^{\mathsf{T}}Mu>0$ for every $0\mathrm{\ne }u\in {\mathbb{R}}^n$, and a continuous map $f:O\times {\mathbb{R}}^n\to {\mathbb{R}}^n$, the given force. A map $q:I\to {\mathbb{R}}^n$ on an open interval $I\subseteq \mathbb{R}$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length) is a constrained curve when

$(t,q(t))\in O,G(t,q(t))=0(t\in I)\mathrm{.}$

Virtual displacements and permitted velocities. For $(t,q)\in O$ the virtual displacements at $(t,q)$ are the elements of the kernel

$V(t,q):=\ker G_q(t,q)=\{\delta \in {\mathbb{R}}^n:G_q(t,q)\delta =0\}\subseteq {\mathbb{R}}^n,$

the tangent space to the level set of $G(t,\cdot )$ at $q$ in the sense of The tangent space to a regular level set; a permitted velocity at $(t,q)$ is a vector $v\in {\mathbb{R}}^n$ with

$G_t(t,q)+G_q(t,q)v=0.$

For a $C^1$ constrained curve the differentiated constraint forces $G_t(t,q(t))+G_q(t,q(t))q^{\mathrm{\prime }}(t)=0$, so its velocity is automatically permitted; the permitted condition is stated separately because it is the correct initial condition for the first-order system below.

Ideal-constraint motions. A $C^2$ constrained curve $q:I\to {\mathbb{R}}^n$ is an ideal-constraint motion of the data $(G,M,f)$ when

1. $q^{\mathrm{\prime }}(t)$ is a permitted velocity at $(t,q(t))$ for every $t\in I$, and
2. there is a continuous map $\lambda :I\to {\mathbb{R}}^k$ with

$M{q}^{\mathrm{\prime }\mathrm{\prime }}(t)=f(t,q(t),q^{\mathrm{\prime }}(t))+G_q(t,q(t){)}^{\mathsf{T}}\lambda (t)(t\in I)\mathrm{.}$

The vector field $R(t):=M{q}^{\mathrm{\prime }\mathrm{\prime }}(t)-f(t,q(t),q^{\mathrm{\prime }}(t))$ is the reaction and $R=G_q^{\mathsf{T}}\lambda$ is its multiplier form. Since $G_q(t,q{)}^{\mathsf{T}}\lambda \cdot \delta =\lambda \cdot G_q(t,q)\delta =0$ for every $\delta \in V(t,q)$, every ideal-constraint motion satisfies the d'Alembert condition

$(M{q}^{\mathrm{\prime }\mathrm{\prime }}(t)-f(t,q(t),q^{\mathrm{\prime }}(t)))\cdot \delta =0\text{for every }\delta \in V(t,q(t)),$

i.e. the reaction performs no virtual work; the converse implication, that every curve satisfying the d'Alembert condition is an ideal-constraint motion with a unique multiplier for its given reaction, is the content of Reactions annihilating the virtual displacements are transpose multiples, with a positive definite multiplier matrix and is not assumed here. Determination of the reaction and acceleration additionally uses the twice-differentiated constraint, as in The ideal holonomic system in a possibly time-dependent chart: explicit acceleration, constraint preservation, local existence and reaction power under its stated regularity hypotheses.

Scope. The definition is chartwise and time-dependent: the constraint may depend on $t$ (rheonomic case), its velocity condition is the fixed-time linearisation recorded above, and no relation between the variational (vakonomic) treatment of the constraint and the present d'Alembert law is asserted. The construction and long-time behaviour of solutions is the subject of The ideal holonomic system in a possibly time-dependent chart: explicit acceleration, constraint preservation, local existence and reaction power. For a holonomic constraint written in a chart, the multiplier $\lambda$ is not a measured quantity: only the reaction $G_q^{\mathsf{T}}\lambda$ enters the equations of motion.

• The definition covers the classical examples: a planar pendulum is $n=2$, $k=1$, $G(q)=\mathrm{\mid }q{\mathrm{\mid }}^2-l^2$ with $l>0$ on the chart $\{(x,y):y\mathrm{\ne }0\}$ or on ${\mathbb{R}}^2\setminus \{0\}$, where $G_q=2q^{\mathsf{T}}\mathrm{\ne }0$; a rolling disk in the plane with configuration $(x,\theta )$ is $G(q)=x-a\theta$, so that in that planar special case the constraint is holonomic, while general three-dimensional rolling is not reducible to a single level set. These examples are worked out on the companion page.

• The word ideal refers to the d'Alembert condition and not to any assertion that a real contact or joint enforces it; frictionless joints, no-slip rolling with sufficient static friction, and similar mechanisms are physical idealisations modelled by this law, and the empirical qualifications of such models belong to the consuming physics items, not here.

Let $(O,\mathcal{L})$ be a time-dependent Lagrangian chart over $[a,b]\times U$ in the sense of A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations. Let $V\subseteq {\mathbb{R}}^n$ and $A\subseteq \mathbb{R}\times V$ be open sets containing respectively the configuration projection of $O$ and its time-configuration projection. Let

$\xi :V⟶{\mathbb{R}}^n,B:A⟶\mathbb{R}$

be of class $C^1$ ($C^k$ Euclidean maps and diffeomorphisms); write $D\xi (q)\in M_n(\mathbb{R})$ for the Jacobian matrix of $\xi$ at $q$ (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$), and for a $C^1$ function $\phi (t,q)$ write

$D_t\phi (t,q,v):={\phi }_t(t,q)+{\phi }_q(t,q)\cdot v\mathrm{.}$

The pair $(\xi ,B)$ is an infinitesimal point symmetry of $\mathcal{L}$ when the identity

${\mathcal{L}}_q(t,q,v)\cdot \xi (q)+{\mathcal{L}}_v(t,q,v)\cdot D\xi (q)v=B_t(t,q)+B_q(t,q)\cdot v$

holds for every $(t,q,v)\in O$. The charge of the pair is the function

$C(t,q,v):={\mathcal{L}}_v(t,q,v)\cdot \xi (q)-B(t,q)\mathrm{.}$

Reading of the identity. The left-hand side is the derivative of the Lagrangian under the infinitesimal family of point transformations

$q⟼q+s\xi (q),t⟼t$

with the velocity transformed accordingly by $v\mapsto v+sD\xi (q)v$, and the right-hand side is the total derivative $\frac{d}{dt}B(t,q(t))$ along a $C^1$ trajectory with $q^{\mathrm{\prime }}(t)=v$; thus the identity says that the infinitesimal variation of $\mathcal{L}$ is a total time derivative of the chart function $B$. In particular:

• Taking $B=0$ reduces the identity to ${\mathcal{L}}_q(t,q,v)\cdot \xi (q)+{\mathcal{L}}_v(t,q,v)\cdot D\xi (q)v=0$, the invariance of $\mathcal{L}$ under the infinitesimal family.
• For a constant field $\xi \equiv a$ one has $D\xi =0$, and the identity reads ${\mathcal{L}}_q\cdot a=B_t+B_q\cdot v$; when $B$ is independent of $q$ this is the statement that $\mathcal{L}$ is unchanged by the translation $q\mapsto q+sa$ up to a total time derivative.

Scope. The definition is an identity imposed on the chart data; it does not assert that a given Lagrangian possesses any symmetry, nor that a symmetry of the configuration space, a frame change or any kinematic structure automatically induces one. Whether a symmetry produces a conserved quantity along the Euler–Lagrange solutions is the theorem The charge of an infinitesimal point symmetry is constant along Euler-Lagrange solutions; the general reparametrisation version is Noether identity for point transformations with time reparametrisation: the conserved charge $p\cdot \xi -E\tau -B$, and the scope of the statement for physical models is recorded in Scope of point symmetries: symmetries belong to the Lagrangian, not to the configuration space, and time translation is the energy case.

• The notation $D_t\phi$ is the derivative of $\phi$ along the curve $t\mapsto (t,q(t))$ when $v=q^{\mathrm{\prime }}(t)$: for such a curve, $D_t\phi (t,q(t),q^{\mathrm{\prime }}(t))=\frac{d}{dt}\phi (t,q(t))$. This is the reason the velocity slot of the symmetry identity is filled by $v$ rather than by a fixed vector: the infinitesimal transformation moves $q$ with velocity $\xi (q)$, and along a motion the velocity changes by $D_t\xi -v{D}_t\tau$ in the general case treated later.

• The charge is a function on the chart $O$; a conserved quantity is the composite $t\mapsto C(t,q(t),q^{\mathrm{\prime }}(t))$ along a solution. No claim is made that the charge is the only conserved quantity, or that it is invariant under changes of chart.

Chart data. Let $n\ge 1$, let $J\subseteq \mathbb{R}$ and $U\subseteq {\mathbb{R}}^n$ be open and nonempty, and let

$\mathcal{L}:J\times U\times {\mathbb{R}}^n⟶\mathbb{R}$

be of class $C^2$ ($C^k$ Euclidean maps and diffeomorphisms). Such a map is a fibre-full time-dependent Lagrangian chart; it is the special case of the chart of A time-dependent Lagrangian chart, its action functional and its fixed-endpoint first variations in which the velocity block is the whole space ${\mathbb{R}}^n$ at every $(t,q)\in J\times U$. Write ${\mathcal{L}}_v(t,q,v)\in {\mathbb{R}}^n$ for the Euclidean gradient of $\mathcal{L}$ in its third slot and ${\mathcal{L}}_{vv}(t,q,v)\in M_n(\mathbb{R})$ for the matrix of second partial derivatives in the third slot (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$).

Legendre map. The Legendre map of $\mathcal{L}$ is the map

${\mathbb{F}}_{\mathcal{L}}:J\times U\times {\mathbb{R}}^n⟶J\times U\times {\mathbb{R}}^n,{\mathbb{F}}_{\mathcal{L}}(t,q,v):=(t,q,{\mathcal{L}}_v(t,q,v))\mathrm{.}$

The fibre coordinate of the target is called the momentum and is written $p:={\mathcal{L}}_v(t,q,v)$. Thus ${\mathbb{F}}_{\mathcal{L}}$ is the identity in the first two slots and the gradient in the velocity slot. Since $\mathcal{L}$ is $C^2$, ${\mathbb{F}}_{\mathcal{L}}$ is $C^1$ ($C^k$ Euclidean maps and diffeomorphisms).

Regularity. Let $(t,q,v)\in J\times U\times {\mathbb{R}}^n$. The Legendre map is regular at $(t,q,v)$ when the velocity Hessian is nonsingular,

$\det {\mathcal{L}}_{vv}(t,q,v)\mathrm{\ne }0;$

it is regular (on the chart) when it is regular at every point. At fixed $(t,q)$ this is exactly the condition under which the map $v\mapsto (q,{\mathcal{L}}_v(t,q,v))$ is locally invertible near $v$, a statement made precise and proved in Nonsingular velocity Hessian is equivalent to invertibility of the Legendre derivative, hence gives a local C^1 inverse.

Hyperregularity. The Lagrangian $\mathcal{L}$ is hyperregular (on the chart) when for every $(t,q)\in J\times U$ the map

$v⟼{\mathcal{L}}_v(t,q,v)$

is a bijection of ${\mathbb{R}}^n$ onto ${\mathbb{R}}^n$ and the inverse assignment

$(t,q,p)⟼v(t,q,p),{\mathcal{L}}_v(t,q,v(t,q,p))=p,$

is defined on all of $J\times U\times {\mathbb{R}}^n$ and is of class $C^1$. In that case $v$ is called the inverse Legendre branch of $\mathcal{L}$, and the Hamiltonian of the branch is

$H:J\times U\times {\mathbb{R}}^n⟶\mathbb{R},H(t,q,p):=p\cdot v(t,q,p)-\mathcal{L}(t,q,v(t,q,p))\mathrm{.}$

The Hamiltonian is $C^1$ on its domain, being a sum of products of $C^1$ maps.

The definition asserts nothing about solving the equations of motion: the equivalence of the Euler–Lagrange equation with Hamilton's equations on an inverse branch, and the differential of $H$, are proved in The Hamiltonian on an inverse Legendre branch: differential, equivalence with the Euler-Lagrange equation, and the energy identity.

Scope and warnings. Hyperregularity is a global-in-the-fibre hypothesis: it requires the velocity Legendre map to be bijective onto the whole momentum fibre ${\mathbb{R}}^n$, with globally defined inverse. Regularity is a local condition. Hyperregularity implies regularity: at fixed $(t,q)$, differentiating $v(t,q,{\mathcal{L}}_v(t,q,w))=w$ gives $D_{p}v{\mathcal{L}}_{vv}=I_n$, so the square velocity Hessian is invertible. The converse fails: a regular Lagrangian whose velocity gradient is not surjective is not hyperregular, and a fibre-full domain whose Legendre map fails to be injective has no inverse branch. Concrete witnesses — including the mechanical Lagrangian of A mechanical chart Lagrangian with positive definite kinetic matrix is hyperregular, with the explicit kinetic-plus-potential Hamiltonian, the bounded gradient $\mathcal{L}(v)=\sqrt{1+v^2}$ and the identically zero Lagrangian — are collected in Regular versus hyperregular: the bounded-gradient Lagrangian $\sqrt{1+v^2}$ and the degenerate zero Lagrangian ↗ and Scope of the Legendre map: regular versus hyperregular, the meaning of the covector datum, and the limits of the branch description.