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On the Hamilton–Jacobi Equation for a Harmonic Oscillator

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Abstract

The topic of this note is the classical Hamilton–Jacobi equation

$$S_t+\frac{1}{2}|\nabla S|^2+\frac{1}{2} \sum_{j=1}^n \omega_j^2 x_j^2=0\quad{\rm in}\,G_T=(0,T)\times\mathbb{R}^n.$$

In complete generality, a description of the superdifferential \({\partial^+_x S(t,x)}\) of the viscosity solution of the initial-value problem for this equation is furnished in terms of a convex hull construction and rotations. For background, the basic existence and uniqueness properties of the viscosity solution S are recalled.

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  1. Department of Mathematics, Luleå University of Technology, 971 87, Luleå, Sweden

    Thomas Strömberg

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  1. Thomas Strömberg
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Correspondence to Thomas Strömberg.

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Strömberg, T. On the Hamilton–Jacobi Equation for a Harmonic Oscillator. Results. Math. 57, 195–204 (2010). https://doi.org/10.1007/s00025-010-0017-5

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  • DOI: https://doi.org/10.1007/s00025-010-0017-5

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