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151-968: Constantin Carathéodory was born in 1873 in Berlin to Greek parents and grew up in Brussels . His father Stephanos  [ tr ] , a lawyer, served as the Ottoman ambassador to Belgium , St. Petersburg and Berlin. His mother, Despina, née Petrokokkinos, was from the island of Chios . The Carathéodory family, originally from Bosnochori or Vyssa , was well established and respected in Constantinople , and its members held many important governmental positions. The Carathéodory family spent 1874–75 in Constantinople, where Constantin's paternal grandfather lived, while his father Stephanos

302-765: A 2016 terrorist attack linked to ISIL , a truck was deliberately driven into a Christmas market next to the Kaiser Wilhelm Memorial Church , leaving 13 people dead and 55 others injured. In 2018, more than 200,000 protestors took to the streets in Berlin with demonstrations of solidarity against racism, in response to the emergence of far-right politics in Germany . Berlin Brandenburg Airport (BER) opened in 2020, nine years later than planned, with Terminal 1 coming into service at

453-630: A German high school, but they also obtained daily additional instruction in Greek language and culture from a Greek priest, and at home he allowed them to speak Greek only. Carathéodory was a talented public speaker, and was often invited to give speeches. In 1936, it was he who handed out the first ever Fields Medals at the meeting of the International Congress of Mathematicians in Oslo, Norway. In 2002, in recognition of his achievements,

604-633: A German maid who taught the children to speak German. Constantin was already bilingual in French and Greek by this time. Constantin began his formal schooling at a private school in Vanderstock in 1881. He left after two years and then spent time with his father on a visit to Berlin, and also spent the winters of 1883–84 and 1884–85 on the Italian Riviera . Back in Brussels in 1885 he attended

755-501: A Slavic-derived name: Pankow , Steglitz-Zehlendorf , Marzahn-Hellersdorf , Treptow-Köpenick , and Spandau . Of Berlin's ninety-six neighborhoods, twenty-two bear a Slavic-derived name: Altglienicke , Alt-Treptow , Britz , Buch , Buckow , Gatow , Karow , Kladow , Köpenick , Lankwitz , Lübars , Malchow , Marzahn , Pankow , Prenzlauer Berg , Rudow , Schmöckwitz , Spandau , Stadtrandsiedlung Malchow , Steglitz , Tegel and Zehlendorf . The earliest human settlements in

906-481: A camp for Sinti and Romani people (see Romani Holocaust ), and the Stalag III-D prisoner-of-war camp for Allied POWs of various nationalities. During World War II, large parts of Berlin were destroyed during 1943–45 Allied air raids and the 1945 Battle of Berlin . The Allies dropped 67,607 tons of bombs on the city, destroying 6,427 acres of the built-up area. Around 125,000 civilians were killed. After

1057-480: A case, the problem devolves to N {\displaystyle N} ordinary differential equations . The separability of S depends both on the Hamiltonian and on the choice of generalized coordinates . For orthogonal coordinates and Hamiltonians that have no time dependence and are quadratic in the generalized momenta, S {\displaystyle S} will be completely separable if

1208-691: A center of the Enlightenment , but also, was briefly occupied during the Seven Years' War by the Russian army. Following France's victory in the War of the Fourth Coalition , Napoleon Bonaparte marched into Berlin in 1806 , but granted self-government to the city. In 1815, the city became part of the new Province of Brandenburg . The Industrial Revolution transformed Berlin during

1359-530: A certain generalized coordinate q k {\displaystyle q_{k}} and its derivative ∂ S ∂ q k {\displaystyle {\frac {\partial S}{\partial q_{k}}}} are assumed to appear together as a single function ψ ( q k , ∂ S ∂ q k ) {\displaystyle \psi \left(q_{k},{\frac {\partial S}{\partial q_{k}}}\right)} in

1510-782: A city district separate from Brandenburg. In the early 20th century, Berlin had become a fertile ground for the German Expressionist movement. In fields such as architecture, painting and cinema new forms of artistic styles were invented. At the end of the First World War in 1918, a republic was proclaimed by Philipp Scheidemann at the Reichstag building . In 1920, the Greater Berlin Act incorporated dozens of suburban cities, villages, and estates around Berlin into an expanded city. The act increased

1661-517: A combined state of Berlin and Brandenburg is different from other state fusion proposals. Normally, Article 29 of the Basic Law stipulates that a state fusion requires a federal law. However, a clause added to the Basic Law in 1994, Article 118a, allows Berlin and Brandenburg to unify without federal approval, requiring a referendum and a ratification by both state parliaments. In 1996, there

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1812-480: A computational tool, the partial differential equations are notoriously complicated to solve except when is it possible to separate the independent variables; in this case the HJE become computationally useful. Any canonical transformation involving a type-2 generating function G 2 ( q , P , t ) {\displaystyle G_{2}(\mathbf {q} ,\mathbf {P} ,t)} leads to

1963-582: A conservative potential U can be written H = 1 2 m [ p r 2 + p θ 2 r 2 + p ϕ 2 r 2 sin 2 ⁡ θ ] + U ( r , θ , ϕ ) . {\displaystyle H={\frac {1}{2m}}\left[p_{r}^{2}+{\frac {p_{\theta }^{2}}{r^{2}}}+{\frac {p_{\phi }^{2}}{r^{2}\sin ^{2}\theta }}\right]+U(r,\theta ,\phi ).} The Hamilton–Jacobi equation

2114-524: A fourth that does indeed guarantee such a quantity exists. Carathéodory constructed his method for deriving sufficient conditions based on the use of the Hamilton–Jacobi equation to construct a field of extremals. The ideas are closely related to light propagation in optics. The method became known as Carathéodory's method of equivalent variational problems or the royal road to the calculus of variations . A key advantage of Carathéodory's work on this topic

2265-504: A function of all the α {\displaystyle \alpha } and β {\displaystyle \beta } constants and time. The Hamilton–Jacobi equation is a single , first-order partial differential equation for the function of the N {\displaystyle N} generalized coordinates q 1 , q 2 , … , q N {\displaystyle q_{1},\,q_{2},\dots ,q_{N}} and

2416-517: A grammar school for a year where he first began to become interested in mathematics. In 1886, he entered the high school Athénée Royal d'Ixelles and studied there until his graduation in 1891. Twice during his time at this school Constantin won a prize as the best mathematics student in Belgium. At this stage Carathéodory began training as a military engineer. He attended the École Militaire de Belgique from October 1891 to May 1895 and he also studied at

2567-1862: A locally unique solution γ = γ ( τ ; t 0 , q 0 , v 0 ) . {\displaystyle \gamma =\gamma (\tau ;t_{0},\mathbf {q} _{0},\mathbf {v} _{0}).} Additionally, let there be a sufficiently small time interval ( t 0 , t 1 ) {\displaystyle (t_{0},t_{1})} such that extremals with different initial velocities v 0 {\displaystyle \mathbf {v} _{0}} would not intersect in M × ( t 0 , t 1 ) . {\displaystyle M\times (t_{0},t_{1}).} The latter means that, for any q ∈ M {\displaystyle \mathbf {q} \in M} and any t ∈ ( t 0 , t 1 ) , {\displaystyle t\in (t_{0},t_{1}),} there can be at most one extremal γ = γ ( τ ; t , t 0 , q , q 0 ) {\displaystyle \gamma =\gamma (\tau ;t,t_{0},\mathbf {q} ,\mathbf {q} _{0})} for which γ | τ = t 0 = q 0 {\displaystyle \gamma |_{\tau =t_{0}}=\mathbf {q} _{0}} and γ | τ = t = q . {\displaystyle \gamma |_{\tau =t}=\mathbf {q} .} Substituting γ = γ ( τ ; t , t 0 , q , q 0 ) {\displaystyle \gamma =\gamma (\tau ;t,t_{0},\mathbf {q} ,\mathbf {q} _{0})} into

2718-417: A microclimate, with heat stored by the city's buildings and pavement . Temperatures can be 4 °C (7 °F) higher in the city than in the surrounding areas. Annual precipitation is 570 millimeters (22 in) with moderate rainfall throughout the year. Snowfall mainly occurs from December through March. The hottest month in Berlin was July 1834, with a mean temperature of 23.0 °C (73.4 °F) and

2869-424: A monumental ensemble. The National Socialist regime embarked on monumental construction projects in Berlin as a way to express their power and authority through architecture . Adolf Hitler and Albert Speer developed architectural concepts for the conversion of the city into World Capital Germania ; these were never implemented. NSDAP rule diminished Berlin's Jewish community from 160,000 (one-third of all Jews in

3020-549: A number of others. Handwritten letters by Carathéodory to Albert Einstein and Hellmuth Kneser , as well as photographs of the Carathéodory family, are on display. Efforts to furnish the museum with more exhibits are ongoing. A complete list of Carathéodory's journal article publications can be found in his Collected Works ( Ges. Math. Schr. ). Notable publications are: Berlin Berlin ( / b ɜːr ˈ l ɪ n / bur- LIN , German: [bɛʁˈliːn] )

3171-540: A rapid population increase in Berlin. 1920s Berlin was the third-largest city in the world by population. After World War II and following Berlin's occupation, the city was split into West Berlin and East Berlin , divided by the Berlin Wall . East Berlin was declared the capital of East Germany, while Bonn became the West German capital. Following German reunification in 1990, Berlin once again became

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3322-455: A scholar of ancient Greek mathematical classics. In his doctoral dissertation, Carathéodory showed how to extend solutions to discontinuous cases and studied isoperimetric problems. Previously, between the mid-1700s to the mid-1800s, Leonhard Euler , Adrien-Marie Legendre , and Carl Gustav Jacob Jacobi were able to establish necessary but insufficient conditions for the existence of a strong relative minimum. In 1879, Karl Weierstrass added

3473-413: A small margin, largely based on support in former West Berlin , Brandenburg voters disapproved of the fusion by a large margin. It failed largely due to Brandenburg voters not wanting to take on Berlin's large and growing public debt and fearing losing identity and influence to the capital. Berlin is in northeastern Germany, in an area of low-lying marshy woodlands with a mainly flat topography , part of

3624-585: A special case of the Hamilton–Jacobi–Bellman equation from dynamic programming . The Hamilton–Jacobi equation is a first-order, non-linear partial differential equation − ∂ S ∂ t = H ( q , ∂ S ∂ q , t ) . {\displaystyle -{\frac {\partial S}{\partial t}}=H\!\!\left(\mathbf {q} ,{\frac {\partial S}{\partial \mathbf {q} }},t\right).} for

3775-445: A system of particles at coordinates ⁠ q {\displaystyle \mathbf {q} } ⁠ . The function H {\displaystyle H} is the system's Hamiltonian giving the system's energy. The solution of the equation is the action functional , ⁠ S {\displaystyle S} ⁠ , called Hamilton's principal function in older textbooks. The solution can be related to

3926-408: A time instant t 0 {\displaystyle t_{0}} and a point q 0 ∈ M {\displaystyle \mathbf {q} _{0}\in M} in the configuration space be fixed. The existence and uniqueness theorems guarantee that, for every v 0 , {\displaystyle \mathbf {v} _{0},} the initial value problem with

4077-649: A variable or list signifies the time derivative (see Newton's notation ). For example, q ˙ = d q d t . {\displaystyle {\dot {\mathbf {q} }}={\frac {d\mathbf {q} }{dt}}.} The dot product notation between two lists of the same number of coordinates is a shorthand for the sum of the products of corresponding components, such as p ⋅ q = ∑ k = 1 N p k q k . {\displaystyle \mathbf {p} \cdot \mathbf {q} =\sum _{k=1}^{N}p_{k}q_{k}.} Let

4228-2463: A variation of γ {\displaystyle \gamma } "compatible" with δ γ . {\displaystyle \delta \gamma .} In precise terms, γ ε | ε = 0 = γ , {\displaystyle \gamma _{\varepsilon }|_{\varepsilon =0}=\gamma ,} γ ˙ ε | ε = 0 = δ γ , {\displaystyle {\dot {\gamma }}_{\varepsilon }|_{\varepsilon =0}=\delta \gamma ,} γ ε | τ = t 0 = γ | τ = t 0 = q 0 . {\displaystyle \gamma _{\varepsilon }|_{\tau =t_{0}}=\gamma |_{\tau =t_{0}}=\mathbf {q} _{0}.} By definition of HPF and Gateaux derivative, δ S δ γ [ γ , t ] = def d S [ γ ε , t ] d ε | ε = 0 = d S ( γ ε ( t ) , t ) d ε | ε = 0 = ∂ S ∂ q δ γ ( t ) . {\displaystyle \delta {\cal {S}}_{\delta \gamma }[\gamma ,t]{\overset {\text{def}}{{}={}}}\left.{\frac {d{\cal {S}}[\gamma _{\varepsilon },t]}{d\varepsilon }}\right|_{\varepsilon =0}=\left.{\frac {dS(\gamma _{\varepsilon }(t),t)}{d\varepsilon }}\right|_{\varepsilon =0}={\frac {\partial S}{\mathbf {\partial q} }}\,\delta \gamma (t).} Here, we took into account that q = γ ( t ; q , q 0 , t , t 0 ) {\displaystyle \mathbf {q} =\gamma (t;\mathbf {q} ,\mathbf {q} _{0},t,t_{0})} and dropped t 0 {\displaystyle t_{0}} for compactness. Step 3. We now substitute ξ = γ {\displaystyle \xi =\gamma } and δ ξ = δ γ {\displaystyle \delta \xi =\delta \gamma } into

4379-574: A wavefront-like view of the system dynamics. This property of the Hamilton–Jacobi equation connects classical mechanics to quantum mechanics. Boldface variables such as q {\displaystyle \mathbf {q} } represent a list of N {\displaystyle N} generalized coordinates , q = ( q 1 , q 2 , … , q N − 1 , q N ) {\displaystyle \mathbf {q} =(q_{1},q_{2},\ldots ,q_{N-1},q_{N})} A dot over

4530-472: Is Germany 's capital and largest city by area and by population . With over 3.85 million inhabitants, it has more population within in its city limits than any other city in the European Union . The city is also one of the states of Germany , and is the third smallest state in the country in terms of area. Berlin is surrounded by the state of Brandenburg , and Brandenburg's capital Potsdam

4681-465: Is a formulation of mechanics in which the motion of a particle can be represented as a wave. In this sense, it fulfilled a long-held goal of theoretical physics (dating at least to Johann Bernoulli in the eighteenth century) of finding an analogy between the propagation of light and the motion of a particle. The wave equation followed by mechanical systems is similar to, but not identical with, Schrödinger's equation , as described below; for this reason,

Constantin Carathéodory - Misplaced Pages Continue

4832-468: Is an iconic landmark of Berlin and Germany; it stands as a symbol of eventful European history and of unity and peace. The Reichstag building is the traditional seat of the German Parliament. It was remodeled by British architect Norman Foster in the 1990s and features a glass dome over the session area, which allows free public access to the parliamentary proceedings and magnificent views of

4983-1168: Is fixed, then, by the same logic that was used to derive the Euler–Lagrange equations, δ ξ ( t 0 ) = 0. {\displaystyle \delta \xi (t_{0})=0.} Thus, δ S δ ξ [ ξ , t ; t 0 ] = ∂ L ∂ q ˙ | q ˙ = ξ ˙ ( t ) q = ξ ( t ) δ ξ ( t ) . {\displaystyle \delta {\cal {S}}_{\delta \xi }[\xi ,t;t_{0}]=\left.{\frac {\partial {\cal {L}}}{\partial \mathbf {\dot {q}} }}\right|_{\mathbf {\dot {q}} ={\dot {\xi }}(t)}^{\mathbf {q} =\xi (t)}\,\delta \xi (t).} Step 2. Let γ = γ ( τ ; q , q 0 , t , t 0 ) {\displaystyle \gamma =\gamma (\tau ;\mathbf {q} ,\mathbf {q} _{0},t,t_{0})} be

5134-467: Is nearby. The urban area of Berlin has a population of over 4.5 million and is therefore the most populous urban area in Germany. The Berlin-Brandenburg capital region has around 6.2 million inhabitants and is Germany's second-largest metropolitan region after the Rhine-Ruhr region, and the fifth-biggest metropolitan region by GDP in the European Union . Berlin was built along the banks of

5285-540: Is provided at the original website of the club, http://www.s-karatheodoris.gr . At the same time, Greek authorities had long since intended to create a museum honoring Karatheodoris in Komotini , a major town of the northeastern Greek region, more than 200 km away from his home town above. On 21 March 2009, the "Karatheodoris" Museum (Καραθεοδωρής) opened its gates to the public in Komotini. The coordinator of

5436-627: Is that it illuminates the relation between the calculus of variations and partial differential equations. It allows for quick and elegant derivations of conditions of sufficiency in the calculus of variations and leads directly to the Euler-Lagrange equation and the Weierstrass condition. He published his Variationsrechnung und Partielle Differentialgleichungen Erster Ordnung (Calculus of Variations and First-order Partial Differential Equations) in 1935. More recently, Carathéodory's work on

5587-875: Is the initial speed (see discussion preceding the definition of HPF), L ( ξ ( t ) , ξ ˙ ( t ) , t ) = d S ( ξ ( t ) , t ) d t = [ ∂ S ∂ q q ˙ + ∂ S ∂ t ] q ˙ = ξ ˙ ( t ) q = ξ ( t ) . {\displaystyle {\cal {L}}(\xi (t),{\dot {\xi }}(t),t)={\frac {dS(\xi (t),t)}{dt}}=\left[{\frac {\partial S}{\partial \mathbf {q} }}\mathbf {\dot {q}} +{\frac {\partial S}{\partial t}}\right]_{\mathbf {\dot {q}} ={\dot {\xi }}(t)}^{\mathbf {q} =\xi (t)}.} From

5738-673: Is the most visited zoo in Europe. Babelsberg Studio is the world's first large-scale movie studio complex and the list of films set in Berlin is long. Berlin is also home to three World Heritage Sites : Museum Island , the Palaces and Parks of Potsdam and Berlin , and the Berlin Modernism Housing Estates . Other landmarks include the Brandenburg Gate , the Reichstag building , Potsdamer Platz ,

5889-671: The Forum Fridericianum along with, the Berlin State Opera , Charlottenburg Palace , Gendarmenmarkt , Alte Kommandantur , as well as the City Palace . The tallest buildings in Berlin are spread across the urban area, with clusters at Potsdamer Platz , City West , and Alexanderplatz . Over one-third of the city's area consists of green and open-space, with the Großer Tiergarten , one of

6040-517: The Berlin Wall around West Berlin, and events escalated to a tank standoff at Checkpoint Charlie . West Berlin was now de facto a part of West Germany with a unique legal status, while East Berlin was de facto a part of East Germany. John F. Kennedy gave his " Ich bin ein Berliner " speech on 26 June 1963, in front of the Schöneberg city hall, located in the city's western part, underlining

6191-698: The Burgundians followed. In the 7th century Slavic tribes, the later known Hevelli and Sprevane , reached the region. In the 12th century the region came under German rule as part of the Margraviate of Brandenburg , founded by Albert the Bear in 1157. Early evidence of middle age settlements in the area of today's Berlin are remnants of a house foundation dated 1270 to 1290, found in excavations in Berlin Mitte . The first written records of towns in

Constantin Carathéodory - Misplaced Pages Continue

6342-477: The Carathéodory conjecture claiming that a closed convex surface admits at least two umbilic points . As of 2021, this conjecture remained unproven despite having attracted a large amount of research. He proved an existence theorem for the solution to ordinary differential equations under mild regularity conditions. Another theorem of his on the derivative of a function at a point could be used to prove

6493-521: The Chain Rule and the formula for the derivative of inverse functions . He greatly extended the theory of conformal transformation proving his theorem about the extension of conformal mapping to the boundary of Jordan domains. In studying boundary correspondence he originated the theory of prime ends . He exhibited an elementary proof of the Schwarz lemma . Carathéodory was also interested in

6644-656: The Euler–Lagrange equations form a n × n {\displaystyle n\times n} system of second-order ordinary differential equations. Inverting the matrix H L {\displaystyle H_{\cal {L}}} transforms this system into q ¨ i = F i ( q , q ˙ , t ) ,   i = 1 , … , n . {\displaystyle {\ddot {q}}^{i}=F_{i}(\mathbf {q} ,\mathbf {\dot {q}} ,t),\ i=1,\ldots ,n.} Let

6795-1775: The Hessian matrix H L ( q , q ˙ , t ) = { ∂ 2 L / ∂ q ˙ i ∂ q ˙ j } i j {\textstyle H_{\cal {L}}(\mathbf {q} ,\mathbf {\dot {q}} ,t)=\left\{\partial ^{2}{\cal {L}}/\partial {\dot {q}}^{i}\partial {\dot {q}}^{j}\right\}_{ij}} be invertible. The relation d d t ∂ L ∂ q ˙ i = ∑ j = 1 n ( ∂ 2 L ∂ q ˙ i ∂ q ˙ j q ¨ j + ∂ 2 L ∂ q ˙ i ∂ q j q ˙ j ) + ∂ 2 L ∂ q ˙ i ∂ t , i = 1 , … , n , {\displaystyle {\frac {d}{dt}}{\frac {\partial {\cal {L}}}{\partial {\dot {q}}^{i}}}=\sum _{j=1}^{n}\left({\frac {\partial ^{2}{\cal {L}}}{\partial {\dot {q}}^{i}\partial {\dot {q}}^{j}}}{\ddot {q}}^{j}+{\frac {\partial ^{2}{\cal {L}}}{\partial {\dot {q}}^{i}\partial {q}^{j}}}{\dot {q}}^{j}\right)+{\frac {\partial ^{2}{\cal {L}}}{\partial {\dot {q}}^{i}\partial t}},\qquad i=1,\ldots ,n,} shows that

6946-750: The Karl-Marx-Allee heads east, an avenue lined by monumental residential buildings, designed in the Socialist Classicism style. Adjacent to this area is the Rotes Rathaus (City Hall), with its distinctive red-brick architecture. In front of it is the Neptunbrunnen , a fountain featuring a mythological group of Tritons , personifications of the four main Prussian rivers, and Neptune on top of it. The Brandenburg Gate

7097-949: The Memorial to the Murdered Jews of Europe , and the Berlin Wall Memorial . Berlin has numerous museums , galleries, and libraries. [REDACTED] Margraviate of Brandenburg 1237–1618 [REDACTED] Brandenburg-Prussia 1618–1701 [REDACTED]   Kingdom of Prussia 1701–1867 [REDACTED] North German Confederation 1867–1871 [REDACTED]   German Empire 1871–1918 [REDACTED]   Weimar Republic 1918–1933 [REDACTED]   Nazi Germany 1933–1945 [REDACTED]   Allied-occupied Germany 1945–1949 [REDACTED]   West Germany 1949–1990 [REDACTED]   East Germany 1949–1990 [REDACTED]   Germany 1990–present Berlin lies in northeastern Germany. Most of

7248-511: The Spree river, which flows into the Havel in the western borough of Spandau . The city incorporates lakes in the western and southeastern boroughs, the largest of which is Müggelsee . About one-third of the city's area is composed of forests, parks and gardens , rivers, canals, and lakes. First documented in the 13th century and at the crossing of two important historic trade routes , Berlin

7399-525: The War in Asia Minor which ended in the Great Fire of Smyrna . Carathéodory managed to save books from the library and was only rescued at the last moment by a journalist who took him by rowboat to the battleship Naxos which was standing by. Carathéodory brought to Athens some of the university library and stayed there, teaching at the university and technical school until 1924. In 1924 Carathéodory

7550-667: The Wayback Machine and still impacts urban planning decisions. Walter Momper, the mayor of West Berlin, became the first mayor of the reunified city in the interim. City-wide elections in December 1990 resulted in the first "all Berlin" mayor being elected to take office in January 1991, with the separate offices of mayors in East and West Berlin expiring by that time, and Eberhard Diepgen (a former mayor of West Berlin) became

7701-758: The action functional results in the Hamilton's principal function (HPF) S ( q , t ; q 0 , t 0 )   = def ∫ t 0 t L ( γ ( τ ; ⋅ ) , γ ˙ ( τ ; ⋅ ) , τ ) d τ , {\displaystyle S(\mathbf {q} ,t;\mathbf {q} _{0},t_{0})\ {\stackrel {\text{def}}{=}}\int _{t_{0}}^{t}{\mathcal {L}}(\gamma (\tau ;\cdot ),{\dot {\gamma }}(\tau ;\cdot ),\tau )\,d\tau ,} where The momenta are defined as

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7852-832: The end of World War II in Europe in May 1945, Berlin received large numbers of refugees from the Eastern provinces. The victorious powers divided the city into four sectors, analogous to Allied-occupied Germany the sectors of the Allies of World War II (the United States, the United Kingdom, and France) formed West Berlin , while the Soviet Union formed East Berlin . All four Allies of World War II shared administrative responsibilities for Berlin. However, in 1948, when

8003-624: The (unique) extremal from the definition of HPF, δ γ = δ γ ( τ ) {\displaystyle \delta \gamma =\delta \gamma (\tau )} a vector field along γ , {\displaystyle \gamma ,} and γ ε = γ ε ( τ ; q ε , q 0 , t , t 0 ) {\displaystyle \gamma _{\varepsilon }=\gamma _{\varepsilon }(\tau ;\mathbf {q} _{\varepsilon },\mathbf {q} _{0},t,t_{0})}

8154-513: The (unique) extremal from the definition of the Hamilton's principal function ⁠ S {\displaystyle S} ⁠ . Call v = def γ ˙ ( τ ; t , t 0 , q , q 0 ) | τ = t {\displaystyle \mathbf {v} \,{\stackrel {\text{def}}{=}}\,{\dot {\gamma }}(\tau ;t,t_{0},\mathbf {q} ,\mathbf {q} _{0})|_{\tau =t}}

8305-1381: The (uniquely solvable for q ˙ ) {\displaystyle \mathbf {\dot {q}} )} equation p = ∂ L ( q , q ˙ , t ) ∂ q ˙ , {\textstyle \mathbf {p} ={\frac {\partial {\cal {L}}(\mathbf {q} ,\mathbf {\dot {q}} ,t)}{\partial \mathbf {\dot {q}} }},} obtain ∂ S ∂ t = L ( q , q ˙ , t ) − ∂ S ∂ q q ˙ = − H ( q , ∂ S ∂ q , t ) , {\displaystyle {\frac {\partial S}{\partial t}}={\cal {L}}(\mathbf {q} ,\mathbf {\dot {q}} ,t)-{\frac {\partial S}{\mathbf {\partial q} }}\mathbf {\dot {q}} =-H\left(\mathbf {q} ,{\frac {\partial S}{\partial \mathbf {q} }},t\right),} where q = ξ ( t ) {\displaystyle \mathbf {q} =\xi (t)} and q ˙ = ξ ˙ ( t ) . {\displaystyle \mathbf {\dot {q}} ={\dot {\xi }}(t).} Alternatively, as described below,

8456-423: The 19th century; the city's economy and population expanded dramatically, and it became the main railway hub and economic center of Germany. Additional suburbs soon developed and increased the area and population of Berlin. In 1861, neighboring suburbs including Wedding , Moabit and several others were incorporated into Berlin. In 1871, Berlin became capital of the newly founded German Empire . In 1881, it became

8607-502: The 20th century. 17% of Berlin's buildings are Gründerzeit or earlier and nearly 25% are of the 1920s and 1930s, when Berlin played a part in the origin of modern architecture . Devastated by the bombing of Berlin in World War II many of the buildings that had survived in both East and West were demolished during the postwar period. After the reunification, many important heritage structures have been reconstructed , including

8758-795: The Federal Republic of Germany was founded in West Germany and eventually included all of the American, British and French zones, excluding those three countries' zones in Berlin, while the Marxist–Leninist German Democratic Republic was proclaimed in East Germany . West Berlin officially remained an occupied city, but it politically was aligned with the Federal Republic of Germany despite West Berlin's geographic isolation. Airline service to West Berlin

8909-510: The German language are stylistic masterworks. Carathéodory also spoke and wrote English , Italian , Turkish , and the ancient languages without any effort. Such an impressive linguistic arsenal enabled him to communicate and exchange ideas directly with other mathematicians during his numerous travels, and greatly extended his fields of knowledge. Much more than that, Carathéodory was a treasured conversation partner for his fellow professors in

9060-458: The HJE can be useful in other problems of the calculus of variations and, more generally, in other branches of mathematics and physics , such as dynamical systems , symplectic geometry and quantum chaos . For example, the Hamilton–Jacobi equations can be used to determine the geodesics on a Riemannian manifold , an important variational problem in Riemannian geometry . However as

9211-423: The HJE leads directly to constants of motion . For example, the time t can be separated if the Hamiltonian does not depend on time explicitly. In that case, the time derivative ∂ S ∂ t {\displaystyle {\frac {\partial S}{\partial t}}} in the HJE must be a constant, usually denoted ( − E {\displaystyle -E} ), giving

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9362-555: The HJE, a generating function G 2 ( q , P , t ) {\displaystyle G_{2}(\mathbf {q} ,\mathbf {P} ,t)} is chosen in such a way that, it will make the new Hamiltonian K = 0 {\displaystyle K=0} . Hence, all its derivatives are also zero, and the transformed Hamilton's equations become trivial P ˙ = Q ˙ = 0 {\displaystyle {\dot {\mathbf {P} }}={\dot {\mathbf {Q} }}=0} so

9513-668: The HPF is known. Indeed, let a time instant t 0 {\displaystyle t_{0}} and a point q 0 {\displaystyle \mathbf {q} _{0}} in the configuration space be fixed. For every time instant t {\displaystyle t} and a point q , {\displaystyle \mathbf {q} ,} let γ = γ ( τ ; t , t 0 , q , q 0 ) {\displaystyle \gamma =\gamma (\tau ;t,t_{0},\mathbf {q} ,\mathbf {q} _{0})} be

9664-461: The Hamilton's principal function. Step 1. Let ξ = ξ ( t ) {\displaystyle \xi =\xi (t)} be a path in the configuration space, and δ ξ = δ ξ ( t ) {\displaystyle \delta \xi =\delta \xi (t)} a vector field along ξ {\displaystyle \xi } . (For each t , {\displaystyle t,}

9815-594: The Hamiltonian H = H ( q 1 , q 2 , … , q k − 1 , q k + 1 , … , q N ; p 1 , p 2 , … , p k − 1 , p k + 1 , … , p N ; ψ ; t ) . {\displaystyle H=H(q_{1},q_{2},\ldots ,q_{k-1},q_{k+1},\ldots ,q_{N};p_{1},p_{2},\ldots ,p_{k-1},p_{k+1},\ldots ,p_{N};\psi ;t).} In that case,

9966-401: The Hamilton–Jacobi equation is considered the "closest approach" of classical mechanics to quantum mechanics . The qualitative form of this connection is called Hamilton's optico-mechanical analogy . In mathematics, the Hamilton–Jacobi equation is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations . It can be understood as

10117-566: The Hamilton–Jacobi equation may be derived from Hamiltonian mechanics by treating S {\displaystyle S} as the generating function for a canonical transformation of the classical Hamiltonian H = H ( q 1 , q 2 , … , q N ; p 1 , p 2 , … , p N ; t ) . {\displaystyle H=H(q_{1},q_{2},\ldots ,q_{N};p_{1},p_{2},\ldots ,p_{N};t).} The conjugate momenta correspond to

10268-612: The Hamilton–Jacobi equation shows that the function ψ must be a constant (denoted here as Γ k {\displaystyle \Gamma _{k}} ), yielding a first-order ordinary differential equation for S k ( q k ) , {\displaystyle S_{k}(q_{k}),} ψ ( q k , d S k d q k ) = Γ k . {\displaystyle \psi \left(q_{k},{\frac {dS_{k}}{dq_{k}}}\right)=\Gamma _{k}.} In fortunate cases,

10419-702: The Huguenot immigration. Many other immigrants came from Bohemia , Poland , and Salzburg . Since 1618, the Margraviate of Brandenburg had been in personal union with the Duchy of Prussia . In 1701, the dual state formed the Kingdom of Prussia , as Frederick III, Elector of Brandenburg , crowned himself as king Frederick I in Prussia . Berlin became the capital of the new Kingdom, replacing Königsberg . This

10570-437: The Munich Department of Philosophy. The well-respected German philologist and professor of ancient languages, Kurt von Fritz , praised Carathéodory on the grounds that from him one could learn an endless amount about the old and new Greece, the old Greek language, and Hellenic mathematics. Von Fritz conducted numerous philosophical discussions with Carathéodory. The mathematician sent his son Stephanos and daughter Despina to

10721-500: The Museum, Athanasios Lipordezis (Αθανάσιος Λιπορδέζης), has noted that the museum provides a home for original manuscripts of the mathematician running to about 10,000 pages, including correspondence with the German mathematician Arthur Rosenthal for the algebraization of measure. At the showcase, visitors are also able to view the books " Gesammelte mathematische Schriften Band 1,2,3,4 ", "Mass und ihre Algebraiserung", " Reelle Functionen Band 1", " Zahlen/Punktionen Funktionen " , and

10872-780: The Museumsinsel U-Bahn station opened in 2021, which completed all new works on the U5. A partial opening by the end of 2020 of the Humboldt Forum museum, housed in the reconstructed Berlin Palace , was postponed until March 2021. On 16 September 2022, the opening of the eastern wing, the last section of the Humboldt Forum museum, meant the Humboldt Forum museum was finally completed. It became Germany's currently most expensive cultural project. The legal basis for

11023-458: The US support for West Berlin. Berlin was completely divided. Although it was possible for Westerners to pass to the other side through strictly controlled checkpoints, for most Easterners, travel to West Berlin or West Germany was prohibited by the government of East Germany. In 1971, a Four-Power Agreement guaranteed access to and from West Berlin by car or train through East Germany. In 1989, with

11174-595: The University of Munich named one of the largest lecture rooms in the mathematical institute the Constantin-Carathéodory Lecture Hall. In the town of Nea Vyssa, Caratheodory's ancestral home, a unique family museum is to be found. The museum is located in the central square of the town near to its church, and includes a number of Karatheodory's personal items, as well as letters he exchanged with Albert Einstein. More information

11325-623: The Western Allies extended the currency reform in the Western zones of Germany to the three western sectors of Berlin, the Soviet Union imposed the Berlin Blockade on the access routes to and from West Berlin, which lay entirely inside Soviet-controlled territory. The Berlin airlift , conducted by the three western Allies, overcame this blockade by supplying food and other supplies to the city from June 1948 to May 1949. In 1949,

11476-1610: The action S {\displaystyle {\cal {S}}} at the point ξ {\displaystyle \xi } in the direction δ ξ {\displaystyle \delta \xi } is given by the formula δ S δ ξ [ ξ , t 1 , t 0 ] = ∫ t 0 t 1 ( ∂ L ∂ q − d d t ∂ L ∂ q ˙ ) δ ξ d t + ∂ L ∂ q ˙ δ ξ | t 0 t 1 , {\displaystyle \delta {\cal {S}}_{\delta \xi }[\xi ,t_{1},t_{0}]=\int _{t_{0}}^{t_{1}}\left({\frac {\partial {\cal {L}}}{\partial \mathbf {q} }}-{\frac {d}{dt}}{\frac {\partial {\cal {L}}}{\partial \mathbf {\dot {q}} }}\right)\delta \xi \,dt+{\frac {\partial {\cal {L}}}{\partial \mathbf {\dot {q}} }}\,\delta \xi {\Biggl |}_{t_{0}}^{t_{1}},} where one should substitute q i = ξ i ( t ) {\displaystyle q^{i}=\xi ^{i}(t)} and q ˙ i = ξ ˙ i ( t ) {\displaystyle {\dot {q}}^{i}={\dot {\xi }}^{i}(t)} after calculating

11627-1022: The analytic geometry of conic sections . He also visited the Cheops pyramid and made measurements which he wrote up and published in 1901. He also published a book on Egypt in the same year which contained a wealth of information on the history and geography of the country. Carathéodory studied engineering in Belgium at the Royal Military Academy , where he was considered a charismatic and brilliant student. Carathéodory had about 20 doctoral students among these being Hans Rademacher , known for his work on analysis and number theory, and Paul Finsler known for his creation of Finsler space . Carathéodory had numerous contacts in Germany. They included such famous names as: Hermann Minkowski , David Hilbert , Felix Klein , Albert Einstein , Edmund Landau , Hermann Amandus Schwarz , and Lipót Fejér . During

11778-554: The area of Berlin from 66 to 883 km (25 to 341 sq mi). The population almost doubled, and Berlin had a population of around four million. During the Weimar era , Berlin underwent political unrest due to economic uncertainties but also became a renowned center of the Roaring Twenties . The metropolis experienced its heyday as a major world capital and was known for its leadership roles in science, technology, arts,

11929-572: The area of modern Berlin are dated around 60,000 BC. A deer mask, dated to 9,000 BC, is attributed to the Maglemosian culture . In 2,000 BC dense human settlements along the Spree and Havel rivers gave rise to the Lusatian culture . Starting around 500 BC Germanic tribes settled in a number of villages in the higher situated areas of today's Berlin. After the Semnones left around 200 AD,

12080-419: The area of present-day Berlin date from the late 12th century. Spandau is first mentioned in 1197 and Köpenick in 1209. 1237 is considered the founding date of the city. The two towns over time formed close economic and social ties, and profited from the staple right on the two important trade routes , one was known as Via Imperii , and the other trade route reached from Bruges to Novgorod . In 1307

12231-548: The calculus of variations and the Hamilton-Jacobi equation has been taken into the theory of optimal control and dynamic programming. Carathéodory's theorem in convex geometry states that if a point x {\displaystyle x} of R d {\displaystyle \mathbb {R} ^{d}} lies in the convex hull of a set P {\displaystyle P} , then x {\displaystyle x} can be written as

12382-541: The calculus of variations. In 1926 he gave a strict and general proof that no system of lenses and mirrors can avoid aberration , except for the trivial case of plane mirrors. In his later work he gave the theory of the Schmidt telescope . In his Geometrische Optik (1937), Carathéodory demonstrated the equivalence of Huygens' principle and Fermat's principle starting from the former using Cauchy's theory of characteristics. He argued that an important advantage of his approach

12533-459: The capital of all of Germany. Due to its geographic location and history, Berlin has been called "the heart of Europe". The economy of Berlin is based on high tech and the service sector , encompassing a diverse range of creative industries , startup companies , research facilities, and media corporations. Berlin serves as a continental hub for air and rail traffic and has a complex public transportation network . Tourism in Berlin makes

12684-567: The cities and villages in northeastern Germany bear Slavic languages -derived names. Typical Germanization for place name suffixes of Slavic origin are -ow, -itz, -vitz, -witz, -itzsch and -in , prefixes are Windisch and Wendisch . The name Berlin has its roots in the language of the West Slavs , and may be related to the Old Polabian stem berl-/birl- ("swamp"). Of Berlin's twelve boroughs , five bear

12835-752: The city a popular global destination. Significant industries include information technology, the healthcare industry , biomedical engineering , biotechnology , the automotive industry , and electronics . Berlin is home to several universities such as the Humboldt University of Berlin , Technische Universität Berlin , the Berlin University of the Arts and the Free University of Berlin . The Berlin Zoological Garden

12986-521: The city lost half of its population. Frederick William , known as the "Great Elector", who had succeeded his father George William as ruler in 1640, initiated a policy of promoting immigration and religious tolerance. With the Edict of Potsdam in 1685, Frederick William offered asylum to the French Huguenots . By 1700, approximately 30 percent of Berlin's residents were French, because of

13137-688: The city. The East Side Gallery is an open-air exhibition of art painted directly on the last existing portions of the Berlin Wall. It is the largest remaining evidence of the city's historical division. Hamilton%E2%80%93Jacobi equation In physics, the Hamilton–Jacobi equation , named after William Rowan Hamilton and Carl Gustav Jacob Jacobi , is an alternative formulation of classical mechanics , equivalent to other formulations such as Newton's laws of motion , Lagrangian mechanics and Hamiltonian mechanics . The Hamilton–Jacobi equation

13288-512: The coldest was January 1709 , with a mean temperature of −13.2 °C (8.2 °F). The wettest month on record was July 1907, with 230 millimeters (9.1 in) of rainfall, whereas the driest were October 1866, November 1902, October 1908 and September 1928, all with 1 millimeter (0.039 in) of rainfall. Berlin's history has left the city with a polycentric metropolitan area and an eclectic mix of architecture. The city's appearance today has been predominantly shaped by German history during

13439-400: The conditions γ | τ = t 0 = q 0 {\displaystyle \gamma |_{\tau =t_{0}}=\mathbf {q} _{0}} and γ ˙ | τ = t 0 = v 0 {\displaystyle {\dot {\gamma }}|_{\tau =t_{0}}=\mathbf {v} _{0}} has

13590-438: The convex combination of at most d + 1 {\displaystyle d+1} points in P {\displaystyle P} . Namely, there is a subset P ′ {\displaystyle P'} of P {\displaystyle P} consisting of d + 1 {\displaystyle d+1} or fewer points such that x {\displaystyle x} lies in

13741-421: The convex hull of P ′ {\displaystyle P'} . Equivalently, x {\displaystyle x} lies in an r {\displaystyle r} - simplex with vertices in P {\displaystyle P} , where r ≤ d {\displaystyle r\leq d} . The smallest r {\displaystyle r} that makes

13892-463: The country) to about 80,000 due to emigration between 1933 and 1939. After Kristallnacht in 1938, thousands of the city's Jews were imprisoned in the nearby Sachsenhausen concentration camp . Starting in early 1943, many were deported to ghettos like Łódź , and to concentration and extermination camps such as Auschwitz . Berlin hosted the 1936 Summer Olympics for which the Olympic stadium

14043-600: The difficult period of World War II, his close associates at the Bavarian Academy of Sciences were Perron and Tietze. Einstein, then a member of the Prussian Academy of Sciences in Berlin, was working on his general theory of relativity when he contacted Carathéodory for clarifications on the Hamilton-Jacobi equation and canonical transformations . He wanted to see a satisfactory derivation of

14194-593: The end of October, and flights to and from Tegel Airport ending in November. Due to the fall in passenger numbers resulting from the COVID-19 pandemic , plans were announced to close BER's Terminal 5, the former Schönefeld Airport , beginning in March 2021. The connecting link of U-Bahn line U5 from Alexanderplatz to Hauptbahnhof, along with the new stations Rotes Rathaus and Unter den Linden, opened on 4 December 2020,

14345-657: The end of the Cold War and pressure from the East German population, the Berlin Wall fell on 9 November and was subsequently mostly demolished. Today, the East Side Gallery preserves a large portion of the wall. On 3 October 1990, the two parts of Germany were reunified as the Federal Republic of Germany, and Berlin again became a reunified city. After the fall of the Berlin Wall, the city experienced significant urban development Archived 16 December 2023 at

14496-423: The existence of entropy, where temperature is the integration function. The Second Law of Thermodynamics was expressed via the following axiom: "In the neighbourhood of any initial state, there are states which cannot be approached arbitrarily close through adiabatic changes of state." In this connexion he coined the term adiabatic accessibility . Carathéodory's work in optics is closely related to his method in

14647-536: The expression for δ S δ ξ [ ξ , t ; t 0 ] {\displaystyle \delta {\cal {S}}_{\delta \xi }[\xi ,t;t_{0}]} from Step 1 and compare the result with the formula derived in Step 2. The fact that, for t > t 0 , {\displaystyle t>t_{0},} the vector field δ γ {\displaystyle \delta \gamma }

14798-608: The first N {\displaystyle N} of them denoted as α 1 , α 2 , … , α N {\displaystyle \alpha _{1},\,\alpha _{2},\dots ,\alpha _{N}} , and the last one coming from the integration of ∂ S ∂ t {\displaystyle {\frac {\partial S}{\partial t}}} . The relationship between p {\displaystyle \mathbf {p} } and q {\displaystyle \mathbf {q} } then describes

14949-452: The first derivatives of S {\displaystyle S} with respect to the generalized coordinates p k = ∂ S ∂ q k . {\displaystyle p_{k}={\frac {\partial S}{\partial q_{k}}}.} As a solution to the Hamilton–Jacobi equation, the principal function contains N + 1 {\displaystyle N+1} undetermined constants,

15100-499: The first elected mayor of a reunited Berlin. On 18 June 1994, soldiers from the United States, France and Britain marched in a parade which was part of the ceremonies to mark the withdrawal of allied occupation troops allowing a reunified Berlin (the last Russian troops departed on 31 August, while the final departure of Western Allies forces was on 8 September 1994). On 20 June 1991, the Bundestag (German Parliament) voted to move

15251-587: The former and the origins of the latter. Einstein told Carathéodory his derivation was "beautiful" and recommended its publication in the Annalen der Physik. Einstein employed the former in a 1917 paper titled Zum Quantensatz von Sommerfeld und Epstein (On the Quantum Theorem of Sommerfeld and Epstein). Carathéodory explained some fundamental details of the canonical transformations and referred Einstein to E.T. Whittaker's Analytical Dynamics . Einstein

15402-720: The formula for p i = p i ( q , t ) {\displaystyle p_{i}=p_{i}(\mathbf {q} ,t)} and the coordinate-based definition of the Hamiltonian H ( q , p , t ) = p q ˙ − L ( q , q ˙ , t ) , {\displaystyle H(\mathbf {q} ,\mathbf {p} ,t)=\mathbf {p} \mathbf {\dot {q}} -{\cal {L}}(\mathbf {q} ,\mathbf {\dot {q}} ,t),} with q ˙ ( p , q , t ) {\displaystyle \mathbf {\dot {q}} (\mathbf {p} ,\mathbf {q} ,t)} satisfying

15553-556: The function S {\displaystyle S} can be separated completely into N {\displaystyle N} functions S m ( q m ) , {\displaystyle S_{m}(q_{m}),} S = S 1 ( q 1 ) + S 2 ( q 2 ) + ⋯ + S N ( q N ) − E t . {\displaystyle S=S_{1}(q_{1})+S_{2}(q_{2})+\cdots +S_{N}(q_{N})-Et.} In such

15704-573: The function S can be partitioned into two functions, one that depends only on q k and another that depends only on the remaining generalized coordinates S = S k ( q k ) + S rem ( q 1 , … , q k − 1 , q k + 1 , … , q N , t ) . {\displaystyle S=S_{k}(q_{k})+S_{\text{rem}}(q_{1},\ldots ,q_{k-1},q_{k+1},\ldots ,q_{N},t).} Substitution of these formulae into

15855-1392: The generating function equal to Hamilton's principal function, plus an arbitrary constant A {\displaystyle A} : G 2 ( q , α , t ) = S ( q , t ) + A , {\displaystyle G_{2}(\mathbf {q} ,{\boldsymbol {\alpha }},t)=S(\mathbf {q} ,t)+A,} the HJE automatically arises p = ∂ G 2 ∂ q = ∂ S ∂ q → H ( q , p , t ) + ∂ G 2 ∂ t = 0 → H ( q , ∂ S ∂ q , t ) + ∂ S ∂ t = 0. {\displaystyle \mathbf {p} ={\frac {\partial G_{2}}{\partial \mathbf {q} }}={\frac {\partial S}{\partial \mathbf {q} }}\,\rightarrow \,H(\mathbf {q} ,\mathbf {p} ,t)+{\partial G_{2} \over \partial t}=0\,\rightarrow \,H\left(\mathbf {q} ,{\frac {\partial S}{\partial \mathbf {q} }},t\right)+{\partial S \over \partial t}=0.} When solved for S ( q , α , t ) {\displaystyle S(\mathbf {q} ,{\boldsymbol {\alpha }},t)} , these also give us

16006-577: The humanities, city planning, film, higher education, government, and industries. Albert Einstein rose to public prominence during his years in Berlin, being awarded the Nobel Prize for Physics in 1921. In 1933, Adolf Hitler and the Nazi Party came to power . Hitler was inspired by the architecture he had experienced in Vienna , and he wished for a German Empire with a capital city that had

16157-534: The invitation of the Greek Prime Minister Eleftherios Venizelos , he submitted a plan on 20 October 1919 for the creation of a new university at Smyrna in Asia Minor, to be named Ionian University of Smyrna . In 1920 Carathéodory was appointed dean of the university and took a major part in establishing the institution, touring Europe to buy books and equipment. The university however never actually admitted students, due to

16308-518: The largest and most popular parks in Berlin, located in the centre of the city. The Fernsehturm (TV tower) at Alexanderplatz in Mitte is among the tallest structures in the European Union at 368 m (1,207 ft). Built in 1969, it is visible throughout most of the central districts of Berlin. The city can be viewed from its 204-meter-high (669 ft) observation floor. Starting here,

16459-399: The last statement valid for each x {\displaystyle x} in the convex hull of P is defined as the Carathéodory's number of P {\displaystyle P} . Depending on the properties of P {\displaystyle P} , upper bounds lower than the one provided by Carathéodory's theorem can be obtained. He is credited with the authorship of

16610-726: The low plateaus on both sides of the Spree Valley. Large parts of the boroughs Reinickendorf and Pankow lie on the Barnim Plateau, while most of the boroughs of Charlottenburg-Wilmersdorf , Steglitz-Zehlendorf , Tempelhof-Schöneberg , and Neukölln lie on the Teltow Plateau. The borough of Spandau lies partly within the Berlin Glacial Valley and partly on the Nauen Plain, which stretches to

16761-921: The lowest is the Spektesee in Spandau, at 28.1 meters (92 ft) elevation. Berlin has an oceanic climate ( Köppen : Cfb ) bordering on a humid continental climate ( Dfb ). This type of climate features mild to very warm summer temperatures and cold, though not very severe, winters. Annual precipitation is modest. Frosts are common in winter, and there are larger temperature differences between seasons than typical for many oceanic climates . Summers are warm and sometimes humid with average high temperatures of 22–25 °C (72–77 °F) and lows of 12–14 °C (54–57 °F). Winters are cold with average high temperatures of 3 °C (37 °F) and lows of −2 to 0 °C (28 to 32 °F). Spring and autumn are generally chilly to mild. Berlin's built-up area creates

16912-428: The new generalized coordinates Q {\displaystyle \mathbf {Q} } are typically denoted as β 1 , β 2 , … , β N {\displaystyle \beta _{1},\,\beta _{2},\dots ,\beta _{N}} , so Q m = β m {\displaystyle Q_{m}=\beta _{m}} . Setting

17063-529: The new generalized coordinates and momenta are constants of motion . As they are constants, in this context the new generalized momenta P {\displaystyle \mathbf {P} } are usually denoted α 1 , α 2 , … , α N {\displaystyle \alpha _{1},\,\alpha _{2},\dots ,\alpha _{N}} , i.e. P m = α m {\displaystyle P_{m}=\alpha _{m}} and

17214-600: The new variables P , Q {\displaystyle \mathbf {P} ,\,\mathbf {Q} } and new Hamiltonian K {\displaystyle K} have the same form: P ˙ = − ∂ K ∂ Q , Q ˙ = + ∂ K ∂ P . {\displaystyle {\dot {\mathbf {P} }}=-{\partial K \over \partial \mathbf {Q} },\quad {\dot {\mathbf {Q} }}=+{\partial K \over \partial \mathbf {P} }.} To derive

17365-511: The orbit in phase space in terms of these constants of motion . Furthermore, the quantities β k = ∂ S ∂ α k , k = 1 , 2 , … , N {\displaystyle \beta _{k}={\frac {\partial S}{\partial \alpha _{k}}},\quad k=1,2,\ldots ,N} are also constants of motion, and these equations can be inverted to find q {\displaystyle \mathbf {q} } as

17516-413: The original generalized coordinates q {\displaystyle \mathbf {q} } as a function of the constants α , β , {\displaystyle {\boldsymbol {\alpha }},\,{\boldsymbol {\beta }},} and t {\displaystyle t} , thus solving the original problem. When the problem allows additive separation of variables ,

17667-507: The partial derivatives on the right-hand side. (This formula follows from the definition of Gateaux derivative via integration by parts). Assume that ξ {\displaystyle \xi } is an extremal. Since ξ {\displaystyle \xi } now satisfies the Euler–Lagrange equations, the integral term vanishes. If ξ {\displaystyle \xi } 's starting point q 0 {\displaystyle \mathbf {q} _{0}}

17818-474: The potential energy is additively separable in each coordinate, where the potential energy term for each coordinate is multiplied by the coordinate-dependent factor in the corresponding momentum term of the Hamiltonian (the Staeckel conditions ). For illustration, several examples in orthogonal coordinates are worked in the next sections. In spherical coordinates the Hamiltonian of a free particle moving in

17969-420: The proof below assumes the configuration space to be an open subset of R n , {\displaystyle \mathbb {R} ^{n},} the underlying technique applies equally to arbitrary spaces . In the context of this proof, the calligraphic letter S {\displaystyle {\cal {S}}} denotes the action functional, and the italic S {\displaystyle S}

18120-534: The quantities p i ( q , q ˙ , t ) = ∂ L / ∂ q ˙ i . {\textstyle p_{i}(\mathbf {q} ,\mathbf {\dot {q}} ,t)=\partial {\cal {L}}/\partial {\dot {q}}^{i}.} This section shows that the dependency of p i {\displaystyle p_{i}} on q ˙ {\displaystyle \mathbf {\dot {q}} } disappears, once

18271-663: The relations p = ∂ G 2 ∂ q , Q = ∂ G 2 ∂ P , K ( Q , P , t ) = H ( q , p , t ) + ∂ G 2 ∂ t {\displaystyle \mathbf {p} ={\partial G_{2} \over \partial \mathbf {q} },\quad \mathbf {Q} ={\partial G_{2} \over \partial \mathbf {P} },\quad K(\mathbf {Q} ,\mathbf {P} ,t)=H(\mathbf {q} ,\mathbf {p} ,t)+{\partial G_{2} \over \partial t}} and Hamilton's equations in terms of

18422-531: The seat of the German capital from Bonn to Berlin, which was completed in 1999, during the chancellorship of Gerhard Schröder . Berlin's 2001 administrative reform merged several boroughs, reducing their number from 23 to 12. In 2006, the FIFA World Cup Final was held in Berlin. Construction of the "Berlin Wall Trail" (Berliner Mauerweg) began in 2002 and was completed in 2006. In

18573-552: The second law, alongside the statements of Clausius , and of Kelvin and Planck . Carathéodory's version attracted the attention of some of the top physicists of the time, including Max Planck, Max Born, and Arnold Sommerfeld. According to Bailyn's survey of thermodynamics, Carathéodory's approach is called "mechanical," rather than "thermodynamic." Max Born acclaimed this "first axiomatically rigid foundation of thermodynamics" and he expressed his enthusiasm in his letters to Einstein. However, Max Planck had some misgivings in that while he

18724-874: The separated solution S = W ( q 1 , q 2 , … , q N ) − E t {\displaystyle S=W(q_{1},q_{2},\ldots ,q_{N})-Et} where the time-independent function W ( q ) {\displaystyle W(\mathbf {q} )} is sometimes called the abbreviated action or Hamilton's characteristic function and sometimes written S 0 {\displaystyle S_{0}} (see action principle names ). The reduced Hamilton–Jacobi equation can then be written H ( q , ∂ S ∂ q ) = E . {\displaystyle H\left(\mathbf {q} ,{\frac {\partial S}{\partial \mathbf {q} }}\right)=E.} To illustrate separability for other variables,

18875-578: The system Lagrangian   L   {\displaystyle \ {\mathcal {L}}\ } by an indefinite integral of the form used in the principle of least action :   S = ∫ L   d ⁡ t +   s o m e   c o n s t a n t   {\displaystyle \ S=\int {\mathcal {L}}\ \operatorname {d} t+~{\mathsf {some\ constant}}~} Geometrical surfaces of constant action are perpendicular to system trajectories, creating

19026-480: The theory from sets to Boolean algebras . Thermodynamics had been a subject dear to Carathéodory since his time in Belgium. In 1909, he published a pioneering work "Investigations on the Foundations of Thermodynamics" in which he formulated the second law of thermodynamics axiomatically, that is, without the use of Carnot engines and refrigerators and only by mathematical reasoning. This is yet another version of

19177-448: The theory of functions of multiple complex variables. In his investigations on this subject he sought analogs of classical results from the single-variable case. He proved that a ball in C 2 {\displaystyle \mathbb {C} ^{2}} is not holomorphically equivalent to the bidisc. He is credited with the Carathéodory extension theorem which is fundamental to modern measure theory. Later Carathéodory extended

19328-459: The time t {\displaystyle t} . The generalized momenta do not appear, except as derivatives of S {\displaystyle S} , the classical action . For comparison, in the equivalent Euler–Lagrange equations of motion of Lagrangian mechanics , the conjugate momenta also do not appear; however, those equations are a system of N {\displaystyle N} , generally second-order equations for

19479-490: The time evolution of the generalized coordinates. Similarly, Hamilton's equations of motion are another system of 2 N first-order equations for the time evolution of the generalized coordinates and their conjugate momenta p 1 , p 2 , … , p N {\displaystyle p_{1},\,p_{2},\dots ,p_{N}} . Since the HJE is an equivalent expression of an integral minimization problem such as Hamilton's principle ,

19630-676: The time, Athens was the only major educational centre in the wider area and had limited capacity to sufficiently satisfy the growing educational needs of the eastern part of the Aegean Sea and the Balkans . Carathéodory, who was a professor at the University of Berlin at the time, proposed the establishment of a new university - the difficulties regarding the establishment of a Greek university in Constantinople led him to consider three other cities: Thessaloniki , Chios and Smyrna . At

19781-783: The town citizens against the building culminated in 1448, in the "Berlin Indignation" ("Berliner Unwille"). Officially, the Berlin-Cölln palace became permanent residence of the Brandenburg electors of the Hohenzollerns from 1486, when John Cicero came to power. Berlin-Cölln, however, had to give up its status as a free Hanseatic League city. In 1539, the electors and the city officially became Lutheran . The Thirty Years' War between 1618 and 1648 devastated Berlin. One third of its houses were damaged or destroyed, and

19932-474: The two towns formed an alliance with a common external policy, their internal administrations still being separated. Members of the Hohenzollern family ruled in Berlin until 1918, first as electors of Brandenburg, then as kings of Prussia , and eventually as German emperors . In 1443, Frederick II Irontooth started the construction of a new royal palace in the twin city Berlin-Cölln. The protests of

20083-666: The useful equations Q = β = ∂ S ∂ α , {\displaystyle \mathbf {Q} ={\boldsymbol {\beta }}={\partial S \over \partial {\boldsymbol {\alpha }}},} or written in components for clarity Q m = β m = ∂ S ( q , α , t ) ∂ α m . {\displaystyle Q_{m}=\beta _{m}={\frac {\partial S(\mathbf {q} ,{\boldsymbol {\alpha }},t)}{\partial \alpha _{m}}}.} Ideally, these N equations can be inverted to find

20234-455: The vast Northern European Plain which stretches all the way from northern France to western Russia. The Berliner Urstromtal (an ice age glacial valley ), between the low Barnim Plateau to the north and the Teltow plateau to the south, was formed by meltwater flowing from ice sheets at the end of the last Weichselian glaciation . The Spree follows this valley now. In Spandau, a borough in

20385-552: The vector δ ξ ( t ) {\displaystyle \delta \xi (t)} is called perturbation , infinitesimal variation or virtual displacement of the mechanical system at the point ξ ( t ) {\displaystyle \xi (t)} ). Recall that the variation δ S δ ξ [ γ , t 1 , t 0 ] {\displaystyle \delta {\cal {S}}_{\delta \xi }[\gamma ,t_{1},t_{0}]} of

20536-633: The velocity at ⁠ τ = t {\displaystyle \tau =t} ⁠ . Then ∂ S ∂ q i = ∂ L ∂ q ˙ i | q ˙ = v , i = 1 , … , n . {\displaystyle {\frac {\partial S}{\partial q^{i}}}=\left.{\frac {\partial {\cal {L}}}{\partial {\dot {q}}^{i}}}\right|_{\mathbf {\dot {q}} =\mathbf {v} }\!\!\!\!\!\!\!,\quad i=1,\ldots ,n.} While

20687-693: The west of Berlin, the Spree empties into the river Havel , which flows from north to south through western Berlin. The course of the Havel is more like a chain of lakes, the largest being the Tegeler See and the Großer Wannsee . A series of lakes also feeds into the upper Spree, which flows through the Großer Müggelsee in eastern Berlin. Substantial parts of present-day Berlin extend onto

20838-590: The west of Berlin. Since 2015, the Arkenberge hills in Pankow at 122 meters (400 ft) elevation, have been the highest point in Berlin. Through the disposal of construction debris they surpassed Teufelsberg (120.1 m or 394 ft), which itself was made up of rubble from the ruins of the Second World War. The Müggelberge at 114.7 meters (376 ft) elevation is the highest natural point and

20989-841: The École d'Application from 1893 to 1896. In 1897 a war broke out between the Ottoman Empire and Greece. This put Carathéodory in a difficult position since he sided with the Greeks, yet his father served the government of the Ottoman Empire. Since he was a trained engineer he was offered a job in the British colonial service. This job took him to Egypt where he worked on the construction of the Assiut dam until April 1900. During periods when construction work had to stop due to floods, he studied mathematics from some textbooks he had with him, such as Jordan's Cours d'Analyse and Salmon's text on

21140-477: Was a successful attempt to centralise the capital in the very far-flung state, and it was the first time the city began to grow. In 1709, Berlin merged with the four cities of Cölln, Friedrichswerder, Friedrichstadt and Dorotheenstadt under the name Berlin, "Haupt- und Residenzstadt Berlin". In 1740, Frederick II, known as Frederick the Great (1740–1786), came to power. Under the rule of Frederick II, Berlin became

21291-411: Was an unsuccessful attempt of unifying the states of Berlin and Brandenburg. Both share a common history, dialect and culture and in 2020, there are over 225,000 residents of Brandenburg that commute to Berlin. The fusion had the near-unanimous support by a broad coalition of both state governments, political parties, media, business associations, trade unions and churches. Though Berlin voted in favor by

21442-767: Was appointed professor of mathematics at the University of Munich, and held this position until retirement in 1938. He later worked at the Bavarian Academy of Sciences until his death in 1950. The new Greek university in the broader area of the Southeast Mediterranean region, as originally envisioned by Carathéodory, finally materialised with the establishment of the Aristotle University of Thessaloniki in 1925. Carathéodory excelled at languages, much like many members of his family. Greek and French were his first languages, and he mastered German with such perfection, that his writings composed in

21593-487: Was built. During World War II , Berlin was the location of multiple Nazi prisons, forced labour camps, 17 subcamps of the Sachsenhausen concentration camp for men and women, including teenagers, of various nationalities, including Polish, Jewish, French, Belgian, Czechoslovak, Russian, Ukrainian, Romani, Dutch, Greek, Norwegian, Spanish, Luxembourgish, German, Austrian, Italian, Yugoslavian, Bulgarian, Hungarian,

21744-1203: Was chosen arbitrarily completes the proof. Given the Hamiltonian H ( q , p , t ) {\displaystyle H(\mathbf {q} ,\mathbf {p} ,t)} of a mechanical system, the Hamilton–Jacobi equation is a first-order, non-linear partial differential equation for the Hamilton's principal function S {\displaystyle S} , − ∂ S ∂ t = H ( q , ∂ S ∂ q , t ) . {\displaystyle -{\frac {\partial S}{\partial t}}=H\left(\mathbf {q} ,{\frac {\partial S}{\partial \mathbf {q} }},t\right).} For an extremal ξ = ξ ( t ; t 0 , q 0 , v 0 ) , {\displaystyle \xi =\xi (t;t_{0},\mathbf {q} _{0},\mathbf {v} _{0}),} where v 0 = ξ ˙ | t = t 0 {\displaystyle \mathbf {v} _{0}={\dot {\xi }}|_{t=t_{0}}}

21895-598: Was designated the capital of the Margraviate of Brandenburg (1417–1701), Kingdom of Prussia (1701–1918), German Empire (1871–1918), Weimar Republic (1919–1933), and Nazi Germany (1933–1945). Berlin served as a scientific, artistic, and philosophical hub during the Age of Enlightenment , Neoclassicism , and the German revolutions of 1848–1849 . During the Gründerzeit , an industrialization-induced economic boom triggered

22046-650: Was directly involved with the reorganization of Greek universities. An especially close friend and colleague in Athens was Nicolaos Kritikos who had attended his lectures at Göttingen, later going with him to Smyrna, then becoming professor at Athens Polytechnic. Kritikos and Carathéodory helped the Greek topologist Christos Papakyriakopoulos take a doctorate in topology at Athens University in 1943 under very difficult circumstances. While teaching at Athens University, Carathéodory had Evangelos Stamatis as an undergraduate student, who subsequently achieved considerable distinction as

22197-490: Was granted only to American, British and French airlines. The founding of the two German states increased Cold War tensions. West Berlin was surrounded by East German territory, and East Germany proclaimed the Eastern part as its capital, a move the western powers did not recognize. East Berlin included most of the city's historic center. The West German government established itself in Bonn . In 1961, East Germany began to build

22348-405: Was impressed by Carathéodory's mathematical prowess, he did not accept that this was a fundamental formulation, given the statistical nature of the second law. In his theory he simplified the basic concepts, for instance heat is not an essential concept but a derived one. He formulated the axiomatic principle of irreversibility in thermodynamics stating that inaccessibility of states is related to

22499-541: Was on leave. Then in 1875 they went to Brussels when Stephanos was appointed there as Ottoman Ambassador. In Brussels, Constantin's younger sister Julia was born. The year 1879 was a tragic one for the family since Constantin's paternal grandfather died in that year, but much more tragically, Constantin's mother Despina died of pneumonia in Cannes . Constantin's maternal grandmother took on the task of bringing up Constantin and Julia in his father's home in Belgium. They employed

22650-655: Was that it covers the integral invariants of Henri Poincaré and Élie Cartan and completes the Malus law . He explained that in his investigations in optics, Pierre de Fermat conceived a minimum principle similar to that enunciated by Hero of Alexandria to study reflection. During the Second World War Carathéodory edited two volumes of Euler 's Complete Works dealing with the Calculus of Variations which were submitted for publication in 1946. At

22801-470: Was trying to solve the problem of "closed time-lines" or the geodesics corresponding to the closed trajectory of light and free particles in a static universe, which he introduced in 1917. Landau and Schwarz stimulated his interest in the study of complex analysis. While in Germany, Carathéodory retained numerous links with the Greek academic world, details of which can be found in Georgiadou's book. He

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