Nature's Laws of Conservation
"The key to the relation of symmetry laws to conservation laws is Emmy Noether's celebrated Theorem. ... Before Noether's Theorem the principle of conservation of energy was shrouded in mystery, leading to the obscure physical systems of Mach and Ostwald. Noether's simple and profound mathematical formulation did much to demystify physics."  Feza Gursey ( 1921, Istanbul  1992, New Haven, Connecticut )
§ Inertial Mass:
The measure of resistance of a body or object to changes in its velocity ( acceleration ) can be expressed as the ratio of an externally applied force to a body or object and the resulting acceleration:
Inertial masses are those masses for which Newton's 1st and 2nd Laws of Motion are valid. That is, those frames of reference in which bodies or particles whose masses are not subject to external forces and hence are moving in straight  line directions at uniform rates of speed without any rotational motion are defined as inertial ( non  accelerating ) frames of reference; the masses in such non  accelerating, inertial frames of reference are inertial masses by definition.
Also,
is the equation for matter in the form of inertial mass which can be derived from a given amount of energy .
§ Gravitational Mass:
A measure of the amount of mutual attraction massive bodies or objects exert upon each other as they tend to accelerate towards one another following the shortest, non  Euclidean geodesic.
§ Conservation of Mass:
The total amount of mass contained in any aggregate of object bodies [ or mass  particles ] before any change(s) in motion will be equal to the total amount of aggregate mass after any change(s) in motion of this aggregation of object bodies [ or mass  particles ].
§ Conservation of Linear Momentum:
According to Newton's 3rd Law ( 'Principle of Equality of Action and Reaction' ) for every action there is an equal and opposite reaction.
Also according to the Law of Impulses
Therefore in accordance with Newton's Third Law of equality of action and reaction,
§ We are assuming that elasticity is perfect, 1, and therefore no kinetic energy loss in this experiment of conservation of momentum and conservation of energy^{∗}:
Newton's Metronome  first demonstrated by Abbé ( Edme ) Mariotte ( c.1620  1684 ), French physicist and priest
( aka Newton's Cradle, Newton's Balance Balls or Newton's Balls, executive ball clicker )
§ Example of conservation of linear momentum: determining resultant for converging ( coalescing ) galaxies^{∗}:
For example, say, galaxy 1 of mass 2 x 10^{9} solar masses is converging with velocity 3.8 x 10^{4} km/sec upon galaxy 2 of mass 5.2 x 10^{11} solar masses whose linear velocity is 3.5 x 10^{4} km/sec. The astronomically observed angle of convergence is 120°. Therefore,
The full derivation is obtained by clicking here.
^{∗}note 1: the law of conservation of linear momentum is demonstrated by resultant vector owing as to how it is deconstructed by vectors and ; that is, is effectively equivalent to the latter two vectors.
note 2: these examples are used in the future upcoming Relativity Science Calculator Mac application.
§ An abstract example of impulse and conservation of linear momentum^{∗∗}:
The full derivation is obtained by clicking here.
^{∗∗}note: this example is used in the future upcoming Relativity Science Calculator Mac application.
§ An abstract example of energy conservation and atomic Doppler Effect^{∗∗}:
The full derivation is obtained by clicking here.
^{∗∗}note: this example is used in the future upcoming Relativity Science Calculator Mac application.
§ An abstract example of energy conservation and astronomical Doppler Effect^{∗∗}:
The full derivation is obtained by clicking here.
^{∗∗}note: this example is used in the future upcoming Relativity Science Calculator Mac application.
§ Law of Conservation of Orbital Angular Momentum:
A simple proof for the law of the conservation of angular momentum in a closed system can be easily shown by means of vector analysis:
implies that without any additional external torque forces acting upon a closed system of particles or orbiting bodies about a central axis,
That is, if any system experiences no external unbalancing torque forces, then also the time rate of change of angular momentum is also zero which immediately means that the angular momentum itself is unchanging or constant!
§ Energy:
from classical physics.
§ Conservation of Energy:
Total energy is neither created nor destroyed only redistributed within a given domain of space  time.
§ ( Emmy Amalie ) Noether's Theorem:
All physical laws of conservation are based upon the proposition that they are invariant with respect to time and therefore can be derived for any space  time continuum.
In other words, if gravity were different from one day to the next, then no gravity field could ever be conserved. Or, even if some meta  theory of physical reality "discovered" some time  dependent aspect of nature, it nevertheless still would be possible to rediscover some greater encompassing meta  theory which indeed would be time  invariant and thereby reestablish conservation.
Hence as a consequence, all aspects of mass, energy, momentum and so forth have respective laws of conservation in both relativity and quantum theories as well as in earlier classical Galilean  Newtonian physics, all of which allows mathematical physics to derive continuously conserved symmetries in the laws of nature.
According to Einstein himself in this New York Times obituary, Emma Noether ( 1882  1935 ) was a mathematical genius whose ultimate contribution to mathematical physics was above and beyond normal understanding and contributed immeasurably to Special Relativity Theory:
Emmy Noether
Professor Einstein Writes in Appreciation of a Fellow  Mathematician.
In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance... Pure mathematics is, in its way, the poetry of logical ideas ... In this effort toward logical beauty, spiritual formulas are discovered necessary for deeper penetration into the laws of nature.
The efforts of most human  beings are consumed in the struggle for their daily bread, but most of those who are, either through fortune or some special gift, relieved of this struggle are largely absorbed in further improving their worldly lot. Beneath the effort directed toward the accumulation of worldly goods lies all too frequently the illusion that this is the most substantial and desirable end to be achieved; but there is, fortunately, a minority composed of those who recognize early in their lives that the most beautiful and satisfying experiences open to humankind are not derived from the outside, but are bound up with the development of the individual's own feeling, thinking and acting. The genuine artists, investigators and thinkers have always been persons of this kind. However inconspicuously the life of these individuals runs its course, none the less the fruits of their endeavors are the most valuable contributions which one generation can make to its successors.
Within the past few days a distinguished mathematician, Professor Emmy Noether, formerly connected with the University of Göttingen and for the past two years at Bryn Mawr College, died in her fifty  third year. In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance in the development of the presentday younger generation of mathematicians. Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships. In this effort toward logical beauty spiritual formulas are discovered necessary for the deeper penetration into the laws of nature.
Born in a Jewish family distinguished for the love of learning, Emmy Noether, who, in spite of the efforts of the great Göttingen mathematician, Hilbert, never reached the academic standing due her in her own country, none the less surrounded herself with a group of students and investigators at Göttingen, who have already become distinguished as teachers and investigators. Her unselfish, significant work over a period of many years was rewarded by the new rulers of Germany with a dismissal, which cost her the means of maintaining her simple life and the opportunity to carry on her mathematical studies. Farsighted friends of science in this country were fortunately able to make such arrangements at Bryn Mawr College and at Princeton that she found in America up to the day of her death not only colleagues who esteemed her friendship but grateful pupils whose enthusiasm made her last years the happiest and perhaps the most fruitful of her entire career.
ALBERT EINSTEIN.
Princeton University, May 1, 1935.
[New York Times May 5, 1935]
[ source: http://wwwgroups.dcs.standrews.ac.uk/~history/Obits2/Noether_Emmy_Einstein.html ]
§ THE HERITAGE OF EMMY NOETHER IN ALGEBRA, GEOMETRY, AND PHYSICS  Bar Ilan University, Tel Aviv, Israel, December 2  3, 1996:
ISRAEL MATHEMATICAL CONFERENCE PROCEEDINGS Vol. 12, 1999 > Download this proceeding in pdf
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