Moving coordinates: the substituted Hamiltonian of a spring on a uniformly moving support, its conservation and the non-conservation of the mechanical energy

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 $U(t,x):=\frac{1}{2}k(x_1-v_{0}t-c{)}^2$ with constants $k>0$, $v_0\in \mathbb{R}$ and $c\in \mathbb{R}$, and no residual forces: the label is attached by a spring of stiffness $k$ to a support whose position is the prescribed function $v_{0}t+c$, so the force map is $F=(-k(x_1-v_{0}t-c),0,0)$ and the mechanical Lagrangian is $L=K-U$ with $K(v)=\frac{1}{2}m\mid v{\mid }_2^2$ (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). Introduce the co-moving coordinates $X(t,y):=(y_1+v_{0}t+c,y_2,y_3)$, a time-dependent coordinate map with $X_q=I_3$, and the substituted Lagrangian $L^{\mathrm{\prime }}(t,y,w):=L(t,X(t,y),X_t(t,y)+X_q(t,y)w)$. This $L^{\mathrm{\prime }}$ is a generic fibre-full time-dependent Lagrangian chart of The Legendre map of a time-dependent Lagrangian: regularity and hyperregularity; it need not be a $K-U$ presentation in a static coordinate map. The model units are $[k]=\mu {\tau }^{-2}$ (newtons per metre), $[v_0]=\mathrm{\ell }{\tau }^{-1}$ (metres per second), $[c]=[y_0]=\mathrm{\ell }$ (metres), $[q]=\mu \mathrm{\ell }{\tau }^{-1}$ (momentum, kilograms metres per second), and $[\omega ]={\tau }^{-1}$ (radians per second with dimensionless radians). Here $y_0$ and $q$ are initial co-moving position and first momentum, respectively. Then:

1. $L^{\mathrm{\prime }}(t,y,w)=\frac{1}{2}m(w_1+v_0{)}^2+\frac{1}{2}m(w_2^2+w_3^2)-\frac{1}{2}k{y}_1^2$, its fibre derivative is $p^{\mathrm{\prime }}=L_w^{\mathrm{\prime }}(t,y,w)=(m(w_1+v_0),m{w}_2,m{w}_3),$ and the coordinate change reproduces the model's equation: a $C^2$ curve $y$ satisfies the Euler-Lagrange equation of $L^{\mathrm{\prime }}$ exactly when $x(t)=X(t,y(t))$ satisfies $m{\ddot{x}}_1=-k(x_1-v_{0}t-c)$, $m{\ddot{x}}_2=m{\ddot{x}}_3=0$ (Time-dependent coordinate maps transform the potential-part equation into the substituted Euler-Lagrange equation).
2. $L^{\mathrm{\prime }}$ is hyperregular on its chart with inverse branch $w=(p_1^{\mathrm{\prime }}\mathrm{/}m-v_0,p_2^{\mathrm{\prime }}\mathrm{/}m,p_3^{\mathrm{\prime }}\mathrm{/}m)$ and Hamiltonian $H^{\mathrm{\prime }}(t,y,p^{\mathrm{\prime }})=\frac{p_1^{\mathrm{\prime }}{}^2}{2m}-v_0{p}_1^{\mathrm{\prime }}+\frac{p_2^{\mathrm{\prime }}{}^2}{2m}+\frac{p_3^{\mathrm{\prime }}{}^2}{2m}+\frac{k}{2}{y}_1^2=\frac{(p_1^{\mathrm{\prime }}-m{v}_0{)}^2}{2m}+\frac{k}{2}{y}_1^2+\frac{p_2^{\mathrm{\prime }}{}^2+p_3^{\mathrm{\prime }}{}^2}{2m}-\frac{m{v}_0^2}{2},$ so that on the first coordinate the momentum $p_1^{\mathrm{\prime }}$ equals the inertial momentum $p_1=m{\dot{x}}_1$ of The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta while the branch velocity is shifted, ${\dot{y}}_1=p_1^{\mathrm{\prime }}\mathrm{/}m-v_0$.
3. Hamilton's equations of $H^{\mathrm{\prime }}$ are ${\dot{y}}_1=p_1^{\mathrm{\prime }}\mathrm{/}m-v_0$, ${\dot{p}}_1^{\mathrm{\prime }}=-k{y}_1$, ${\dot{y}}_i=p_i^{\mathrm{\prime }}\mathrm{/}m$, ${\dot{p}}_i^{\mathrm{\prime }}=0$ for $i=2,3$; they imply $m{\ddot{y}}_1=-k{y}_1$, equivalently $m{\ddot{x}}_1=-k(x_1-v_{0}t-c)$, so the co-moving description is equivalent to the model's motion and the induced curve $x$ is a motion (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model).
4. along the solutions the Hamiltonian $H^{\mathrm{\prime }}$ is conserved — it has no explicit time dependence in the co-moving chart — while the model's mechanical energy $E=K+U$ obeys, on the first axis, $H^{\mathrm{\prime }}=E-v_0{p}_1^{\mathrm{\prime }},E^{\mathrm{\prime }}(t)=-k{v}_0{y}_1(t)\mathrm{.}$ At any time when $v_0{y}_1(t)\mathrm{\ne }0$, the energy derivative is nonzero; in particular, the displayed solution has nonconstant $E$ when $v_0{y}_0\mathrm{\ne }0$. With data $y_1(0)=y_0$, $p_1^{\mathrm{\prime }}(0)=q$ and $\omega :=\sqrt{k\mathrm{/}m}$ the explicit solution is $y_1(t)=y_0\cos \omega t+\frac{q-m{v}_0}{m\omega }\sin \omega t,p_1^{\mathrm{\prime }}(t)=-m\omega y_0\sin \omega t+(q-m{v}_0)\cos \omega t+m{v}_0\mathrm{.}$

All quantities are model values of the declared idealization; the moving support is a prescribed external mechanism, and no measurement is reported.

• Chart change, not a constraint. The co-moving coordinates are a time-dependent coordinate map of the same configuration domain, covered by the covariance theorem for rheonomic coordinate changes; they are not a generalized-coordinate map Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces of the constrained-configuration type, and no ideal reaction is involved. The support's motion is prescribed external data of the potential, so its energy exchange is not modeled here.

• The Hamiltonian depends on the chart. The co-moving Hamiltonian $H^{\mathrm{\prime }}$ is conserved because the substituted Lagrangian has no explicit time dependence in the moving chart, while the mechanical energy changes at the rate of the explicit time dependence of the inertial potential. The Hamiltonian agrees with the mechanical energy in the inertial chart and in time-independent generalized-coordinate charts; this time-dependent chart instead has $H^{\mathrm{\prime }}=E-v_0{p}_1^{\mathrm{\prime }}$, as The mechanical Hamiltonian is the model's energy in the inertial chart, its balance is the energy balance, and the substituted-chart Hamiltonian is the energy of time-independent coordinates states. This is the model instance of the momentum shift familiar from the moving-cart computation of Stewart's Section 1.2.