Chapter

The action functional and the harmonic oscillator

Jean-Michel Bismut

in Hypoelliptic Laplacian and Orbital Integrals (AM-177)

Published by Princeton University Press

Published in print August 2011 | ISBN: 9780691151298
Published online October 2017 | e-ISBN: 9781400840571
The action functional and the harmonic oscillator

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This chapter solves explicitly certain natural variational problems associated with a scalar hypoelliptic Laplacian, in the case where the underlying Riemannian manifold is an Euclidean vector space. It depends on the parameter b > 0. The behavior of the minimum values as well as of the minimizing trajectories is studied when b → 0 and when b → +∞. Finally, certain heat kernels are computed in terms of the minimum value of the action. The aforementioned variational problem has already been considered previously as a warm up to the more general problem on Riemannian manifolds, in order to study in detail the small time asymptotics of the hypoelliptic heat kernel.

Keywords: action functional; harmonic oscillator; variational problems; scalar hypoelliptic Laplacian; Euclidean vector space; heat kernels; Pontryagin maximum principle; orbital integrals

Chapter.  5707 words. 

Subjects: Geometry

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