Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces

Adopt the force model with potential presentation $(U,d)$ and mechanical Lagrangian $L=K-U$ of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, with masses $m_1,\dots ,m_N>0$, open configuration domain $Q\subseteq {\mathbb{R}}^{3N}$, time window $I_0$, mass operator $\mathsf{M}$ and $C^2$ potential $U:I_0\times Q\to \mathbb{R}$ (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets). Dimensions are read by Dimensions and SI unit conventions of the non-relativistic classical mechanics framework; gradients and Jacobians are those of The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, transposes those of The transpose $A^{\mathsf{T}}$ of a matrix, and $C^r$ classes those of $C^k$ Euclidean maps and diffeomorphisms.

(a) Generalized-coordinate map. Let $r\ge 1$ and let $\mathrm{\Omega }\subseteq {\mathbb{R}}^r$ be open and nonempty. A generalized-coordinate map of the model is a map $X:\mathrm{\Omega }\to Q$ of class $C^3$ whose derivative $DX(q)$ is injective for every $q\in \mathrm{\Omega }$ (Submersions and immersions between Euclidean open sets: an immersion); the coordinates $q=(q_1,\dots ,q_r)$ each carry a declared dimension ${\gamma }_i$ and unit fixed as part of the model declaration. When $r=3N$, injectivity makes $DX(q):{\mathbb{R}}^{3N}\to {\mathbb{R}}^{3N}$ a bijection: its kernel is $\{0\}$, so rank-nullity gives image dimension $3N$, equal to the target dimension. The inverse is linear: if $y_1=DX(q)u_1$ and $y_2=DX(q)u_2$, then linearity and uniqueness of preimages give $DX(q{)}^{-1}(a{y}_1+b{y}_2)=a{u}_1+b{u}_2$. Thus $DX(q)$ is invertible in the sense of Invertible Euclidean linear maps. The inverse-function theorem and its higher-regularity result (The Euclidean inverse function theorem, A local inverse of a $C^k$ regular map is $C^k$) make $X$ a local $C^3$ diffeomorphism at each point. This local property does not imply that $X$ is globally one-to-one: call $X$ a global coordinate chart onto $X(\mathrm{\Omega })$ only when it is also injective. The other intended case is $r=3N-k$ for an integer $k$ with $1\le k\le 3N-1$, where $X$ may be a surjective immersion of a regular holonomic-constraint locus as in clause (d); injectivity is not required there. The admissible coordinate curves are the $C^2$ maps $q:J\to \mathrm{\Omega }$ on an interval $J\subseteq I_0$ with nonempty interior (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length); each such curve induces the configuration curve $x:=X\circ q:J\to Q$.

(b) Substituted Lagrangian and kinetic metric. For a generalized-coordinate map $X$ define

$\tilde{L}:I_0\times \mathrm{\Omega }\times {\mathbb{R}}^r⟶\mathbb{R},\tilde{L}(t,q,v):=L(t,X(q),DX(q)v)=\frac{1}{2}{v}^{\mathsf{T}}G(q)v-\tilde{U}(t,q),$

with the kinetic metric and the substituted potential

$G(q):=DX(q{)}^{\mathsf{T}}\mathsf{M}DX(q),\tilde{U}(t,q):=U(t,X(q))\mathrm{.}$

$G(q)$ is symmetric and positive definite, because $DX(q)$ is injective and $\mathsf{M}$ is symmetric positive definite (Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes, Linear map between vector spaces over the same field, The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$); its entry $G_{ij}$ has dimension $\mu {\mathrm{\ell }}^2{\gamma }_i^{-1}{\gamma }_j^{-1}$ and the quadratic quantity $\frac{1}{2}{v}^{\mathsf{T}}G(q)v$ has dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$ for every choice of coordinate dimensions. $\tilde{L}$ is of class $C^2$; the substituted action of an admissible coordinate curve $q:[a,b]\to \mathrm{\Omega }$ with $[a,b]\subseteq I_0$ is $\tilde{S}(q):={\int }_a^b\tilde{L}(t,q(t),q^{\mathrm{\prime }}(t))dt$ whenever the chart conventions of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions apply to $\tilde{L}$, and the Lagrange operator of $\tilde{L}$ is

$\mathrm{\Lambda }(\tilde{L})(q)(t):=\frac{d}{dt}\left(\frac{\mathrm{\partial }\tilde{L}}{\mathrm{\partial }v}\right(t,q(t),q^{\mathrm{\prime }}(t)\left)\right)-\frac{\mathrm{\partial }\tilde{L}}{\mathrm{\partial }q}(t,q(t),q^{\mathrm{\prime }}(t))\in {\mathbb{R}}^r\mathrm{.}$

(c) Generalized forces. Let $f:I_0\times Q\times {\mathbb{R}}^{3N}\to {\mathbb{R}}^{3N}$ be a force field of the model's standing regularity (Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance clause (iii)) — for instance the residual force $d$ of the presentation, or the model's full force map $F$. Its generalized force in the coordinates $X$ is

$Q_f(t,q,v):=DX(q{)}^{\mathsf{T}}f(t,X(q),DX(q)v)\in {\mathbb{R}}^r;$

its $i$-th component has the dimension of energy per unit of the coordinate $q_i$, namely $\mu {\mathrm{\ell }}^2{\tau }^{-2}{\gamma }_i^{-1}$: the Jacobian factor $\mathrm{\partial }{X}_a\mathrm{/}\mathrm{\partial }{q}_i$ carries the dimension $\mathrm{\ell }{\gamma }_i^{-1}$ and the force $f_a$ the dimension $\mu \mathrm{\ell }{\tau }^{-2}$. Thus $Q_f\cdot \delta q$ has energy dimension; for a length-valued coordinate $Q_{f,i}$ has force dimension, and for a dimensionless angle it has torque dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$. For the presentation $(U,d)$ write $Q^{\mathrm{r}\mathrm{e}\mathrm{s}}:=Q_d$ for the generalized force of the residual part; when $d\equiv 0$ one has $Q^{\mathrm{r}\mathrm{e}\mathrm{s}}\equiv 0$.

(d) Constraint parametrizations. Let a regular holonomic-constraint datum $(O,k,g_c)$ of Holonomic-constraint models of the force branch: constraint data, the constrained equation, virtual displacements and the scleronomic/rheonomic distinction be given, where $g_c:=G$ is the constraint map called $G$ in that definition; this separate notation distinguishes it from the kinetic matrix $G(q)$ in clause (b). Let $\mathcal{C}\subseteq {\mathbb{R}}^{3N}$ be its time-independent configuration locus, so $\mathrm{\Sigma }=\{(t,x)\in O:x\in \mathcal{C}\}$ (the scleronomic case). A parametrization of the constrained configurations is a generalized-coordinate map $X:\mathrm{\Omega }\to Q$ with $r=3N-k$, $X(\mathrm{\Omega })=\mathcal{C}\cap Q$, and $(t,X(q))\in O$ for all $(t,q)\in I_0\times \mathrm{\Omega }$. Thus $X$ covers every constrained configuration, but a configuration may have more than one coordinate representative; a covering or other surjective immersion is allowed. Injectivity is an optional additional condition for applications that require unique coordinates. Each admissible coordinate curve induces a constrained curve family with permitted velocity; the reduction theorem of this page also identifies the virtual displacements with the image of $DX$.

(e) Declared Lagrange equations. With the data of clauses (a)-(c), the model with presentation $(U,d)$ declares as its Lagrange equations in the coordinates $X$

$\mathrm{\Lambda }(\tilde{L})(q)(t)=Q^{\mathrm{r}\mathrm{e}\mathrm{s}}(t,q(t),q^{\mathrm{\prime }}(t))(t\in J),$

posed for $C^2$ admissible coordinate curves $q:J\to \mathrm{\Omega }$. These equations are part of the coordinate description: for $d\equiv 0$ they reduce to the Euler-Lagrange equation $\mathrm{\Lambda }(\tilde{L})(q)=0$ of the substituted Lagrangian. For a chart ($r=3N$), the transport identity of Coordinate covariance of the Euler-Lagrange expression under time-dependent point transformations clause 4, applied to the time-independent map $X$, gives $\mathrm{\Lambda }(\tilde{L})=D{X}^{\mathsf{T}}(\mathsf{M}{x}^{\mathrm{\prime }\mathrm{\prime }}-F+d)$ along $x=X\circ q$: here $L_v=\mathsf{M}{x}^{\mathrm{\prime }}$ and $L_x=-{\mathrm{\nabla }}_{x}U$, while $F=-{\mathrm{\nabla }}_{x}U+d$. Hence $\mathrm{\Lambda }(\tilde{L})=D{X}^{\mathsf{T}}d$ is equivalent to $\mathsf{M}{x}^{\mathrm{\prime }\mathrm{\prime }}=F$, since $D{X}^{\mathsf{T}}$ is invertible. On any interval $J$, the transport calculation applies on compact subintervals around each interior time and extends to included endpoints by continuity of these expressions. Whether the same equations describe an ideal constrained motion for a parametrization of clause (d) is a separate statement, proved as the reduction theorem of this page; it is not part of this definition.

(f) Status. This item defines the dictionary between the Cartesian and the generalized-coordinate descriptions: the coordinate map, the substituted Lagrangian, the kinetic metric, generalized forces and the declared Lagrange equations. It adopts no force law beyond the model's presentation, asserts no solution existence and makes no empirical claim; the equivalence of the declared equations with the model's Cartesian equation (for charts) and with the ideal constrained equation (for parametrizations) is proved on this page.

• Coordinate dimensions. The kinetic metric is the coordinate expression of the fixed mass operator: with length-valued coordinates it has the dimension of mass, with dimensionless angle coordinates the angular-angular entries have dimension $\mu {\mathrm{\ell }}^2$, and mixed entries have the dimensions $\mu {\mathrm{\ell }}^2{\gamma }_i^{-1}{\gamma }_j^{-1}$ recorded in clause (b). The physically meaningful objects are the quadratic form $v^{\mathsf{T}}G(q)v$ and the generalized forces paired with the coordinate variations, both of which have the dimensions displayed.

• The residual split enters the forces, not the Lagrangian. The substituted Lagrangian is built from $U$ alone; the residual force $d$ of the presentation appears only through $Q^{\mathrm{r}\mathrm{e}\mathrm{s}}$. A presentation with a different split of the same total force changes $Q^{\mathrm{r}\mathrm{e}\mathrm{s}}$ and $L$ correspondingly, and the declared equations are unchanged, as they must be for the total force map to be the model's datum.