Continuum Dynamics

01A calculable continuum response

ASD continuum dynamics combines the inertial response of spatial states, restoring energy, sources and boundaries. It calculates free, driven and inhomogeneous responses. Field kinetic coefficients and the translational inertia of localized matter are related through one action but have different definitions.

Continuum dynamics calculates spatial-state evolution, energy flow and potential observables after backgrounds, initial data, sources and boundaries are specified. Small displacement, projections and extended orientation are variables. Restoring coefficients, kinetic normalization, nonlinear potentials and source actions are constitutive inputs. Variation and conservation then follow from those inputs.

A model is therefore more than an equation shaped like a wave equation. It specifies the function space, constrained components, effective scales and boundary energy transfer. Such a system can calculate linear propagation, static response and candidate coupling without establishing FCC uniqueness; the microscopic model must explain its parameters and cutoff.

02Kinetic normalization and matter inertia

Chapter 4 weights length-dimensional $\mathbf u$ by $\rho_s$, Chapter 5 sectors by $Z_A$, and the local dual-field model by $Z_u$. These positive coefficients weight time derivatives. Displacement-type fields can use $\mathrm{kg\,m^{-3}}$; a dimensionless scalar can instead use $\mathrm{J\,s^2m^{-3}}$.

They describe spatial-state dynamic response, without being another name for ordinary particle mass density or constituent number density. Matter inertia follows by substituting a complete localized solution and its motion-induced adjustments into the action, then calculating collective translational energy. Chapter 5 (5.6.26)–(5.6.27) provides that definition rather than reading electron mass from a field coefficient's dimensions.

Identical symbols and units do not prove identical models; the same model may also be renormalized. Cross-chapter use must match variables, kinetic inner products, source units and boundaries.

03Matching the two chapter formulations

Chapter 4's divergence/curl restoration gives energy

Positive coefficients give nonnegative energy at this linear order. Chapter 5 uses projected sectors:

Here $A=L,T$, with curl-free longitudinal and divergence-free transverse states, and generalized force-density sources. Constant coefficients, orthogonal projections and compatible boundaries permit $Z_A=\rho_s$ and $Z_Ac_A^2=K_A$. Integrated longitudinal gradient energy equals integrated divergence energy, while integrated transverse gradient energy equals integrated curl energy.

Integration by parts is part of the correspondence. Local energy densities may differ by a divergence, so they cannot be equated pointwise with boundaries or varying coefficients. Source, boundary work and initial fields must follow the representation. Chapter 5 allows separately normalized sectors; without the stated match, their energies must be calculated within their own model.

04Derivatives act on varying coefficients

For slowly changing reciprocal links, take a real component $q$, positive $Z$ and symmetric $K_{ij}=K_{ji}$ depending on position, and possibly time. The action compatible with the link energy is

Variation with temporal and spatial boundary terms treated yields

Expansion includes $(\partial_tZ)\partial_tq$ and $(\partial_iK_{ij})\partial_jq$. Those are actual responses of the same action, not optional additions. Only constant coefficients or a controlled approximation permit $Z\partial_t^2q-K_{ij}\partial_i\partial_jq=J$.

Chapter 5's constant-coefficient treatment writes $K_{ij}/Z=c^2B_{ij}$, with dimensionless $B$. Multicomponent models also carry internal indices, and projection generally need not commute with propagation. Changing a variable-coefficient nondivergence equation into divergence form without a constitutive basis would introduce extra first-gradient terms.

05Constraints, boundaries and energy exchange

Constrained sectors require allowed variations. A source incompatible with the sector must be projected. Decay at infinity, periodic boundaries, fixed displacement or defined natural boundaries can remove surface variations, but represent different physical preparations. Boundary driving and outgoing radiation retain their energy contributions.

Multiplying the sector equation by $\partial_t\mathbf u_A$ converts kinetic terms into energy derivatives and splits gradient terms into a derivative and a flux divergence:

The source power density is $\mathbf J_A\cdot\partial_t\mathbf u_A$. Total energy is conserved for a source-free closed system without net boundary flow. External sources and boundaries change field energy; a dynamical internal source must include its own energy and the interaction energy. L–T coupling uses that reciprocal requirement.

Changing among density, potential and orientation representations must preserve kernels, gauges, constraints and sources, rather than comparing only free equations. Inverse-Laplacian density kinetics and local independent-stretch kinetics are distinct longitudinal constitutive choices.

For the stated constant-coefficient real-channel action, momentum density is $g_i=-Z\dot q\,\partial_iq$ and energy flux is $S_i=-Zc^2\dot q\,B^{ij}\partial_jq$, hence $S_i=c^2B^{ij}g_j$ in (5.2.9)–(5.2.10). In this model $\mathbf g=\mathbf S/c^2$ requires isotropic $B^{ij}=\delta^{ij}$. Here $g_i$ denotes momentum density, not gravitational acceleration $\mathbf g=c^2\nabla\varphi$; higher derivatives or velocity-cross terms require the complete Noether current to be recalculated.

06What microscopic and matter models must supply

With specified coefficients, continuum systems calculate modes, source responses, static boundary solutions and energy flux. Microscopic models must supply coarse graining, full link tensors, dynamic response, coefficients, higher corrections and valid wave-number ranges. The FCC page keeps its discrete calculation explicit for the chosen scalar or fixed-polarization channel.

The master equation must coordinate nonlinear orientation, gravitational far fields and coupled spectra under one degree-of-freedom count. Existing variational and conservation results retain their stated validity; a stable matter solution satisfying all interfaces requires a further joint calculation.