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Specific relative angular momentum

See also: Classical central-force problemIn celestial mechanics the specific relative angular momentum h → plays a pivotal role in the analysis of the two-body problem. One can show; this proves Kepler's second law. It's called specific angular momentum because it's not the actual angular momentum L →, but the angular momentum per mass. Thus, the word "specific" in this term is short for "mass-specific" or divided-by-mass: h → = L → m Thus the SI unit is: m2·s−1. M denotes the reduced mass 1 m = 1 m 1 + 1 m 2; the specific relative angular momentum is defined as the cross product of the relative position vector r → and the relative velocity vector v →. H → = r → × v → = L → m The h → vector is always perpendicular to the instantaneous osculating orbital plane, which coincides with the instantaneous perturbed orbit, it would not be perpendicular to an average plane which accounted for many years of perturbations. As usual in physics, the magnitude of the vector quantity h → is denoted by h: h = ‖ h → ‖ The following is only valid under the simplifications applied to Newton's law of universal gravitation.

One looks at two point masses m 1 and m 2, at the distance r from one another and with the gravitational force F → = G m 1 m 2 r 2 r → r acting between them. This force acts over any distance and is the only force present; the coordinate system is inertial. The further simplification m 1 ≫ m 2 is assumed in the following, thus m 1 is the central body in the origin of the coordinate system and m 2 is the satellite orbiting around it. Now the reduced mass is equal to m 2 and the equation of the two-body problem is r → ¨ = − μ r 2 r → r with the standard gravitational parameter μ = G m 1 and the distance vector r → that points from the origin to the satellite, because of its negligible mass, it is important not to confound the gravitational parameter μ with the reduced mass, sometimes denoted by the same letter μ. One obtains the specific relative angular momentum by multiplying the equation of the two-body problem with the distance vector r → r → × r → ¨ = − r → × μ r 2 r → r The cross product of a vector with itself is 0.

The left hand side simplifies to r → × r → ¨ = r → ˙ × r → ˙ + r → × r → ¨ = d d t = 0 according to the product rule of differentiation. This means that r

Irvin B. Nathan

Irvin Bertram Nathan is an American lawyer from Washington, DC. He served as the Attorney General of the District of Columbia from 2011 to 2015, he was appointed in 2011 by Mayor Vincent C. Gray, he served as the General Counsel of the United States House of Representatives from 2007 to 2011. Nathan announced his resignation the day after the November 2014 election, in which voters chose Karl Racine as the first elected Attorney General of D. C. Nathan grew up in Maryland, his father was a procurement officer for the city, his mother was a social worker for the state. Nathan attended nearby Johns Hopkins University for college, graduating in 1964. One of his classmates was future Mayor of Michael Bloomberg, he was interested in a career in journalism, serving as editor-in-chief of The Johns Hopkins News-Letter, as a sportscaster on the school radio station, as a summer intern at The Baltimore Sun. However, he became somewhat disillusioned with the field during his summer internships and chose to pursue law instead.

Nathan moved to New York City to attend Columbia Law School, graduating in 1967. While there, he was a member of the Columbia Law Review and the winner of the Jerome Michael prize for the moot jury trial competition. After graduation, Nathan clerked for Simon Sobeloff on the United States Court of Appeals for the Fourth Circuit, whom he cites as one of his greatest mentors, he spent most of his career at Arnold & Porter, first as an associate and as the senior litigating partner and head of its white collar criminal defense practice. On the side, he served as an Adjunct Professor at Georgetown University Law Center and the University of San Diego Law School. From 1979 to 1981, Nathan served as a Deputy Assistant Attorney General in the United States Department of Justice, where he was involved in the Abscam operation, he returned in the early 1990s under the Clinton Administration to serve as the Principal Associate Deputy Attorney General. In 2007, Nathan became the General Counsel of the United States House of Representatives, succeeding Geraldine R. Gennet.

He served in that position for four years, providing legal advice to members and institutions within the House. One of his most notable actions during this time was to compel Bush Administration official, Harriet Miers and Joshua Bolten, to comply with their subpoenas, he retired in 2011 following the Republican landslide in the 2010 midterm elections and was succeeded by Deputy General Counsel Kerry W. Kircher. In 2011, he was appointed by Mayor Vincent C. Gray to be the Attorney General of the District of Columbia. In this position, he managed an office of 700 employees, including 350 lawyers. One of his most notable acts as Attorney General was to sue Harry Thomas, Jr. a sitting member of the Council of the District of Columbia, for corruption securing a judgment and a settlement. Nathan recovered in litigation for the city over $70 million in unpaid taxes from online hotel companies. S. Department of Labor for $20 million in Davis Bacon payments in connection with the private construction of City Center in downtown D.

C.. He inaugurated the Charles Ruff fellows program by which recent law school graduates from the local law schools in the District worked for one year at the Office of the Attorney General, lending their energy and talents to the office while securing excellent experience in a quest for fulltime employment either in public service or the private sector; the D. C. Attorney General position become an elected office in 2014, Nathan declined to run announcing his resignation, effective November 17, 2014. In December, 2014, Nathan re-joined the Law Firm of Porter LLP as Senior Counsel. In 2016, Nathan was the subject of an oral history of his career by the Historical Society of the U. S. Circuit Court of Appeals for the D. C. Circuit. In July 2012, the DC council voted to postpone the election of attorney general to 2018. Council Chairman Phil Mendelson called the vote "an embarrassment." In September 2013, Paul Zukerberg filed suit against the DC Council and the city elections claiming any delay would violate the District charter —, amended through the 2010 ballot question to provide for the election of the city's top lawyer.

Nathan represented the city and argued that Zukerberg was not suffering any "meaningful hardship" from pushing back the election. In a 265-page opposition to Zukerberg's preliminary injunction, Nathan claimed that "the intent of the voters…is not relevant, for a variety of obvious reasons." In November 2013, Zukerberg announced his candidacy for Attorney General. On August 28, 2014 Washingtonian magazine reported that three anonymous staff members at the Office of the Attorney General had filed complaints with the D. C. Board of Elections alleging Nathan and Office of the Attorney General employee Timothy Thomas had violated the Hatch Act of 1939 by promoting the campaign of Attorney General candidate Karl Racine at work. Thomas circulated petition signature sheets to employees at the Office, while according to one employee Nathan "praised and recommended Karl Racine, he asked us to support him" during two July 9 meetings to discuss the implications of the election for the Office. Nathan was exonerated of all of these charges by the D.

C. Board of Ethics and Government Accountability

Edwin F. Beckenbach

Edwin Ford Beckenbach was an American mathematician. Beckenbach was born 18 July 1906 in Oak Cliff, Dallas County, the son of a leather worker and on his father's side the grandson of immigrants from Germany. In 1924, he began study at Rice University, where in 1929 he earned a master's degree and in 1931 a PhD under the direction of Lester R. Ford; as a postdoc, he was a National Research Fellow at Princeton University, Ohio State University, the University of Chicago. In 1933, he was an instructor at Rice University and from 1940 an assistant professor at the University of Michigan. In 1942, Beckenbach became an associate professor at the University of Texas and was from 1945 a professor at UCLA. At UCLA, he led the development of the graduate program in mathematics; the first mathematics PhD was granted under his direction as thesis advisor. Beckenbach was a leader in the founding of the Institute of Numerical Analysis, a branch of the National Bureau of Standards, his institute developed in 1948 and 1949 a vacuum-tube computer, which began operation in July 1950 and was for a short time the fastest computer in the world.

In 1974 he retired from UCLA as professor emeritus. From 1949 to 1963, he was a consultant for the Rand Corporation and in the academic year 1951/1952 he was a visiting professor at the Institute for Advanced Study. In the academic year 1958/59 he was a Guggenheim Fellow at ETH Zürich. With František Wolf, Beckenbach founded in 1951 the Pacific Journal of Mathematics, of which he was the first editor. In 1983, he received the Distinguished Service Award from the Mathematical Association of America. Beckenbach was famous for his work on inequalities and for this subject organized three Oberwolfach seminars, he was a co-author of several college textbooks in mathematics, including algebra and analytic geometry. From 1933 to 1960, he was married to Madelene Shelby Simons and had from this marriage a son and two daughters. In 1960, the marriage ended in divorce and in the same year he married his second wife Alice Judson Curtiss, he died on 5 September 1982 in New York. Concepts of Communication, Krieger 1971 Inequalities, Ergebnisse der Mathematik, Springer Verlag, 1st edn. 1961 with Richard E. Bellman, 2nd edn.

1965, 1971 with Richard Bellman: Introduction to Inequalities, Random House 1961 Editor: Modern Mathematics for the Engineer, McGraw Hill 1957 Editor: Applied combinatorial mathematics, Wiley 1964 M. Goldberg: In memoriam: Edwin F. Beckenbach, in General Inequalities 4, Oberwolfach 1983, Birkhäuser, 1984 O'Connor, John J.. Edwin Ford Beckenbach at the Mathematics Genealogy Project Biography Literature by and about Edwin F. Beckenbach in the German National Library catalogue