Symplectic Geometry Lecture 9 - Harvard …math.harvard.edu/archive/139_spring_05/sympgeom9.pdf ·...

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Symplectic Geometry Lecture 9 Hamilton’s principle Legendre Transformations Thermodynamics Generating functions

Transcript of Symplectic Geometry Lecture 9 - Harvard …math.harvard.edu/archive/139_spring_05/sympgeom9.pdf ·...

Page 1: Symplectic Geometry Lecture 9 - Harvard …math.harvard.edu/archive/139_spring_05/sympgeom9.pdf · Symplectic Geometry Lecture 9 Hamilton’s principle Legendre Transformations Thermodynamics

Symplectic GeometryLecture 9Hamilton’s principle

Legendre TransformationsThermodynamics

Generating functions

Page 2: Symplectic Geometry Lecture 9 - Harvard …math.harvard.edu/archive/139_spring_05/sympgeom9.pdf · Symplectic Geometry Lecture 9 Hamilton’s principle Legendre Transformations Thermodynamics
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Hamilton’s principle

Page 4: Symplectic Geometry Lecture 9 - Harvard …math.harvard.edu/archive/139_spring_05/sympgeom9.pdf · Symplectic Geometry Lecture 9 Hamilton’s principle Legendre Transformations Thermodynamics
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Page 9: Symplectic Geometry Lecture 9 - Harvard …math.harvard.edu/archive/139_spring_05/sympgeom9.pdf · Symplectic Geometry Lecture 9 Hamilton’s principle Legendre Transformations Thermodynamics
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Systems and states.

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Equilibrium states.

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Diathermal contact.

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Thermal equilibrium.

Page 24: Symplectic Geometry Lecture 9 - Harvard …math.harvard.edu/archive/139_spring_05/sympgeom9.pdf · Symplectic Geometry Lecture 9 Hamilton’s principle Legendre Transformations Thermodynamics

The zero-th law of thermodynamics.

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Configurational variables.

Page 26: Symplectic Geometry Lecture 9 - Harvard …math.harvard.edu/archive/139_spring_05/sympgeom9.pdf · Symplectic Geometry Lecture 9 Hamilton’s principle Legendre Transformations Thermodynamics

Adiabatic interactions.

Page 27: Symplectic Geometry Lecture 9 - Harvard …math.harvard.edu/archive/139_spring_05/sympgeom9.pdf · Symplectic Geometry Lecture 9 Hamilton’s principle Legendre Transformations Thermodynamics

The first law.

Page 28: Symplectic Geometry Lecture 9 - Harvard …math.harvard.edu/archive/139_spring_05/sympgeom9.pdf · Symplectic Geometry Lecture 9 Hamilton’s principle Legendre Transformations Thermodynamics

Heat

Page 29: Symplectic Geometry Lecture 9 - Harvard …math.harvard.edu/archive/139_spring_05/sympgeom9.pdf · Symplectic Geometry Lecture 9 Hamilton’s principle Legendre Transformations Thermodynamics

Reversible paths.

Page 30: Symplectic Geometry Lecture 9 - Harvard …math.harvard.edu/archive/139_spring_05/sympgeom9.pdf · Symplectic Geometry Lecture 9 Hamilton’s principle Legendre Transformations Thermodynamics

Pressure.

Page 31: Symplectic Geometry Lecture 9 - Harvard …math.harvard.edu/archive/139_spring_05/sympgeom9.pdf · Symplectic Geometry Lecture 9 Hamilton’s principle Legendre Transformations Thermodynamics

The heat form.

Page 32: Symplectic Geometry Lecture 9 - Harvard …math.harvard.edu/archive/139_spring_05/sympgeom9.pdf · Symplectic Geometry Lecture 9 Hamilton’s principle Legendre Transformations Thermodynamics

The second law.

Page 33: Symplectic Geometry Lecture 9 - Harvard …math.harvard.edu/archive/139_spring_05/sympgeom9.pdf · Symplectic Geometry Lecture 9 Hamilton’s principle Legendre Transformations Thermodynamics

Entropy.

Page 34: Symplectic Geometry Lecture 9 - Harvard …math.harvard.edu/archive/139_spring_05/sympgeom9.pdf · Symplectic Geometry Lecture 9 Hamilton’s principle Legendre Transformations Thermodynamics

The combined laws.

Page 35: Symplectic Geometry Lecture 9 - Harvard …math.harvard.edu/archive/139_spring_05/sympgeom9.pdf · Symplectic Geometry Lecture 9 Hamilton’s principle Legendre Transformations Thermodynamics

The gas constant.

Page 36: Symplectic Geometry Lecture 9 - Harvard …math.harvard.edu/archive/139_spring_05/sympgeom9.pdf · Symplectic Geometry Lecture 9 Hamilton’s principle Legendre Transformations Thermodynamics

The combined laws and symplectic geometry.

Page 37: Symplectic Geometry Lecture 9 - Harvard …math.harvard.edu/archive/139_spring_05/sympgeom9.pdf · Symplectic Geometry Lecture 9 Hamilton’s principle Legendre Transformations Thermodynamics

The entropy as a generating function.

Page 38: Symplectic Geometry Lecture 9 - Harvard …math.harvard.edu/archive/139_spring_05/sympgeom9.pdf · Symplectic Geometry Lecture 9 Hamilton’s principle Legendre Transformations Thermodynamics

Planck’s function as a generating function.

Page 39: Symplectic Geometry Lecture 9 - Harvard …math.harvard.edu/archive/139_spring_05/sympgeom9.pdf · Symplectic Geometry Lecture 9 Hamilton’s principle Legendre Transformations Thermodynamics

Massieu’s function as a generating function.

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The Massieu function and the partition function.

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The graph of a symplectomorphism.

Page 43: Symplectic Geometry Lecture 9 - Harvard …math.harvard.edu/archive/139_spring_05/sympgeom9.pdf · Symplectic Geometry Lecture 9 Hamilton’s principle Legendre Transformations Thermodynamics
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Problems with the definition of composition.

Page 46: Symplectic Geometry Lecture 9 - Harvard …math.harvard.edu/archive/139_spring_05/sympgeom9.pdf · Symplectic Geometry Lecture 9 Hamilton’s principle Legendre Transformations Thermodynamics

The cotangent case.

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