ABSTRACT
I deform the 2-D oscillator by adding a constant force f, and I then contract it by letting the spring constant k go to zero. Its deformed symmetry algebra contracts to the Heisenberg-Weyl algebra https://www.w3.org/1998/Math/MathML"> h ( 3 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq3171.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> instead of the Euclidean algebra https://www.w3.org/1998/Math/MathML"> e ( 2 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq3172.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . I then use the Levi-Cevita transformation to map the above system and its algebra onto 2-D Kepler system with a deformation by the 'SmorodinskyWinternitz' potential https://www.w3.org/1998/Math/MathML"> β r - 1 / 2 c o s φ / 2 - θ f https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq3173.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . I identify the symmetry of the deformed Kepler system https://www.w3.org/1998/Math/MathML"> L ( β ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq3174.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> as a Kac-Moody loop algebra. I show that https://www.w3.org/1998/Math/MathML"> L ( β ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq3175.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is smaller by two elements than the loop algebra https://www.w3.org/1998/Math/MathML"> L 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429076619/c00d8d0d-b3a0-431a-a3c2-e044ea15f7a5/content/eq3176.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> of the standard 2-D Kepler system. This difference is an interesting manifestation of the symmetry breaking.
