It turns out that such inconsistencies arise from relativistic wavefunctions having a probabilistic interpretation in position space, as probability conservation is not a relativistically covariant concept.

The conservation of both relativistic and invariant mass applies even to systems of particles created by pair production, where energy for new particles may come from kinetic energy of other particles, or from a photon as part of a system.

Interestingly, Schrödinger did encounter an equation for which the wave function satisfied relativistic energy conservation before he published the non-relativistic one, but it led to unacceptable consequences for that time so he discarded it.

While virtual particles obey conservation of energy and momentum, they can have any energy and momentum, even one that is not allowed by the relativistic energy-momentum relation for the observed mass of that particle.

The relativistic mass corresponds to the energy, so conservation of energy automatically means that relativistic mass is conserved for any given observer and inertial frame.

A gyroscope uses the principle of conservation of angular momentum whereas the interferometer is affected by relativistic phenomena.

However, if one does some simple thought experiments using the Newtonian definitions of momentum and energy, one sees that these quantities are not conserved in SR. One can rescue the idea of conservation by making some small modifications to the definitions to account for relativistic velocities.

In 1789, French nobleman and scientific researcher Antoine Lavoisier discovered the law of conservation of mass and defined an element as a basic substance that could not be further broken down by the methods of chemistry.

The propagation of sound waves in a fluid ( such as water ) can be modeled by an equation of motion ( conservation of momentum ) and an equation of continuity ( conservation of mass ).

The equation for the conservation of mass in a fluid volume ( without any mass sources or sinks ) is given by

The equation for the conservation of mass for an acoustic medium can also be derived in a manner similar to that used for the conservation of momentum.

The equation for the conservation of mass can similarly be written in cylindrical coordinates as

The acoustic equations for the conservation of momentum and the conservation of mass are often expressed in time harmonic form ( at fixed frequency ).

and the fixed frequency form of the conservation of mass

Lavoisier's experiments supported the law of conservation of mass, which he was the first to state, although Mikhail Lomonosov ( 1711 – 1765 ) had previously expressed similar ideas in 1748 and proved them in experiments.

Aerodynamics allows the definition and solution of equations for the conservation of mass, momentum, and energy in air.

The conservation of momentum equations are often called the Navier-Stokes equations, while others use the term for the system that includes conversation of mass, conservation of momentum, and conservation of energy.

Fundamental physical laws such as the conservation of mass, the conservation of momentum, and the conservation of energy may be applied to such models to derive differential equations describing the behavior of such objects, and some information about the particular material studied is added through a constitutive relation.

This " post-glacial rebound " brings mass closer to the rotation axis of the Earth, which makes the Earth spin faster ( law of conservation of angular momentum ): the rate derived from models is about − 0. 6 ms / day / cy.

( Any theory that includes conservation of energy and mass – energy equivalence must include gravitational redshift.

* continuity ( conservation of mass ),

The foundational axioms of fluid dynamics are the conservation laws, specifically, conservation of mass, conservation of linear momentum ( also known as Newton's Second Law of Motion ), and conservation of energy ( also known as First Law of Thermodynamics ).

Thus, conservation of four-momentum is Lorentz-invariant and implies conservation of both mass and energy.

The fact that the density is positive definite and convected according to this continuity equation, implies that we may integrate the density over a certain domain and set the total to 1, and this condition will be maintained by the conservation law.

In general, the stretching mechanism implies thinning of the vortices in the direction perpendicular to the stretching direction due to volume conservation of fluid elements.

Noss & Cooperrider support the “ larger is better ” claim and developed a model that implies areas of habitat less than 1000ha are “ tiny ” and of low conservation value.

So that the conservation of mass implies that:

and conservation of hypercharge implies a conservation of flavour.

The principle of conservation of electric charge implies that:

The principle of conservation of energy implies that

At this point Einstein brings into play the first screen as well and argues as follows: since the incident particles have velocities ( practically ) perpendicular to the screen S < sub > 1 </ sub >, and since it is only the interaction with this screen that can cause a deflection from the original direction of propagation, by the law of conservation of impulse which implies that the sum of the impulses of two systems which interact is conserved, if the incident particle is deviated toward the top, the screen will recoil toward the bottom and vice-versa.

* Time and energy-the continuous translational symmetry of time implies the conservation of energy.

* Space and momentum-the continuous translational symmetry of space implies the conservation of momentum

* Space and angular momentum-the continuous rotational symmetry of space implies the conservation of angular momentum

* Wave function phase and electric charge-the continuous phase angle symmetry of the wave function implies the conservation of electric charge

In general the conservation of angular momentum implies full rotational symmetry

( described by the groups SO ( 3 ) and SU ( 2 )) and, conversely, spherical symmetry implies conservation of angular momentum.

and the derivative of that last identity simplifies to which implicitly implies the conservation of energy since after integration the constant is the sum of kinetic and potential energy.

Again, the second equation implies charge conservation ( in curved spacetime ):

A few other instances of context are: Dimensional homogeneity ( see below ) is the quality of an equation having quantities of same units on both sides ; Homogeneity ( in space ) implies conservation of momentum ; and homogeneity in time implies conservation of energy.

In Lagrangian formalism, homogeneity in space implies conservation of momentum, and homogeneity in time implies conservation of energy.