I'm studying about the finite element method in a class but I don't come from a civil engineering background. Anyways, it hasn't been made clear to me what the difference between constitutive laws and governing equations are. To me they both relate physical quantities with one another.
[Physics] What’s the difference between constitutive laws and governing equations
terminology
Related Solutions
An experimental take
Exclusive implies that you have measured the energy and momenta of all the products (well, with an exception I'll discuss below). Inclusive means that you may have left some of the products unmeasured.
This applies to scattering processes as well as decays.
Some things to note:
- Exclusive measurements allow you to nail down one, well defined physics process, while inclusive measurements may tell you about a collection of processes
- It is generally difficult to measure neutral particles
- If there are more than a couple of products it begins to require a lot of instrumentation to reliably collect them all and (crucially) to know how well you have done so
In the process you are asking about the neutrino is necessarily unobserved rendering the measurement inclusive, further an $X$ in the final state is often used to indicate unmeasured and unspecified stuff (i.e. it means the measurement is inclusive by design). Here unspecified includes case in high acceptance instruments where you consider all events with the specified products: those for which we know $X$ is empty, those for which $X$ is non-empty and well characterized, and those for which $X$ is ill-characterized.
Theoretical view
I'm less sure of how theorist use these terms, but I believe there is a parallel. Something like: exclusive means one and only one process, while inclusive means all processes that include the specified products.
Convergence of theory and experiment
Of course, we haven't really learned anything until we get theory and experiment together, which is sometimes traumatic for both communities. Still exclusive measurements and calculations are clearly talking about the same thing, and inclusivity can be made to agree with some care in building the experiment and assembling the theoretical results.
Experimenters cheating on exclusivity
Sometimes in nuclear physics we talk about scattering measurements as exclusive when there is an unmeasured, heavy, recoiling nucleus involved. The assumption being that that it carries a small fraction of the total energy and momentum involved and can be neglected, though there is some risk from this if the nucleus is left in a highly excited state.
In particular my dissertation project was on $A(e,e'p)$ reaction (elastic scattering of protons out of a stationary nuclear target where the beam was characterized and both the proton and outgoing electron were observed), and we assumed that the remnant nucleus was left largely undisturbed and recoiling with a momentum opposite the Fermi motion of the target proton.
Actually I think I disagree with the answer by BMS (the group of asymptotic symmetries of asymptotically flat spacetimes?). However I am not sure to have understood BMS'answer completely.
In my opinion, there is no difference between the definition of work in pure mechanics and work in thermodynamics (I stress that I am speaking of thermodynamics and not statistical mechanics). In both cases it is computed by the integral of ${\bf F} \cdot {\bf ds}$, taking all forces acting on the system into account. In the pure mechanical case the theorem of energy conservation says that $$W = \Delta U + \Delta K\:.\qquad (1)$$ $W$ is the work done on the system by external systems, $K$ its kinetic energy and $U$ the total potential energy of internal forces. When considering situations where the work $W'$ of the system on the external systems coincides, up to the sign, to the work $W$ done by the external system on the system (and this is not the case discussed by BMS) we can also say that: $$\Delta U + \Delta K + W' =0\:. \qquad (2)$$ In real physical systems, one has to consider the fact that a system receives energy also in terms of heat, $Q$: that is energy that cannot be described in terms of macroscopic work. In this case (1) has to be improved as $$W + Q = \Delta U + \Delta K\:.\qquad (3)\:.$$ Actually, also the definition of $U$ has to be improved in (3), since it has to encompass the thermodynamic internal energy in addition to all types of macroscopic potential energies.
Referring to standard system of thermodynamics (thermal machines), where $\Delta K$ is negligible and the work done by the external system is identical up to the sign to that done by the system, (3) simplifies to $$\Delta U = Q -W'\:,$$ that is the standard statement of the first principle of thermodynamics for elementary systems. However the general form is (3).
It is worth stressing that this picture needs a sharp distinction between macroscopic description (essentially done in terms of continuous body mechanics) and microscopic description, completely disregarded but embodied in the notions of heat and internal (thermodynamic) energy. If, instead one considers also the microscopic (molecular) structure of the physical systems, the distinction between work and heat is more difficult to understand since both are represented in terms of forces. Nevertheless exploiting the statistical approach to Hamiltonian mechanics the said distinction arises quite naturally.
Focusing on the system given by a rigid block discussed by BMS, the absolute value of the work $W$ done by the friction force acting to the block due to the ground (that eventually stops the block), is different from the absolute value of the work $W'$ done by the block on the ground. The former amounts to $W= -K$ the latter, instead, is $W' = 0$. The energy equation for the block is:
$$W + Q = \Delta U + \Delta K\:.$$
$Q$ is the non-mechanical energy entering the block during the process, responsible for the increase of its temperature. Since $W= -K$ one can simplify that equation to
$$Q= \Delta U\:.$$
The equation for the ground (for instance a table) is instead simply:
$$Q' = \Delta U'$$
Now $Q' \neq -Q$ and $W'=0 \neq -W$. The fact that $Q+Q' \neq 0$ it is important because it says that there is a heat source between the contact surfaces of the two bodies, and the total heat is not conserved (as conversely was supposed in the original theory of heat, the "flogisto" represented as a fluid verifying a conservation equation).
If referring to the overall system made of the block and the table, since no energy enters it, the equation is
$$\Delta U + \Delta U' + \Delta K =0\:.$$
That is
$$\Delta U + \Delta U' = -\Delta K >0\:.$$
It says that all the initial kinetic energy is finally transformed into internal energy producing the increase of temperature of both the block and the table.
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Best Answer
A constitutive law is generally an algebraic relation which tells you the coefficients of a differential equation, while the governing equations are the differential equations themselves.
For example, if I have a metal piston on top of a gas, I can write down the equation of motion for the piston
$$m \ddot X - PA = 0$$
Where P is the pressure in the gas and A is the area of the piston. Without knowing how the pressure depends on the piston position, this is not a closed equation--- it refers to an undetermined quantity, the pressure. But the ideal gas law, that the pressure $P=C/(V-AX)$ where C,V are constants, determines the pressure in terms of X, and gives
$$ m \ddot X -{ AC\over (V - AX)} =0$$
Now the equation is closed--- it tells you the future behavior of X knowing X alone. The ideal gas law is the constitutive relation in this case, while the differential equation is the governing equation.
Constitutive equations are algebraic, governing equations are differential.