Roberto Pavani
- 297
- 124
- TL;DR
- Given a scalar field
## f(\mathbf{x})=\sum_a f_a(\mathbf{x}), ##
its Hessian satisfies
## H(\mathbf{x})=\sum_a H_a(\mathbf{x}). ##
Local observables are obtained from the eigendecomposition of the total Hessian,
## H\,\mathbf e_i=\lambda_i\,\mathbf e_i, ##
not by combining the eigenvalues of the individual contributions.
This seems like a trivial consequence of linear differential operators plus the spectral theorem, yet similar patterns appear in many areas of physics and applied
I recently encountered a situation where local observables are obtained from the eigendecomposition of a Hessian matrix.
A scalar field is written as a superposition of individual source contributions:
##
f(\mathbf{x}) = \sum_a f_a(\mathbf{x})
##
Therefore, by linearity of differentiation,
##
H(\mathbf{x})
=
\nabla\nabla f
=
\sum_a \nabla\nabla f_a
=
\sum_a H_a(\mathbf{x}).
##
So in a multi-source (multi-body) configuration the total Hessian is simply
##
H(\mathbf{x}) = \sum_a H_a(\mathbf{x}).
##
The observables are then obtained from the spectral decomposition of the total Hessian:
##
H(\mathbf{x})\,\mathbf e_i(\mathbf{x})
=
\lambda_i(\mathbf{x})\,\mathbf e_i(\mathbf{x}).
##
In other words,
##
\text{sources}
\rightarrow
f
\rightarrow
H
\rightarrow
\{\lambda_i,\mathbf e_i\}
\rightarrow
\text{observables}.
##
Importantly, I am not asking about summing eigenvalues. In general,
##
\lambda_i(A+B)
\neq
\lambda_i(A)+\lambda_i(B).
##
The point is that the underlying geometric object (the Hessian) superposes linearly, while the physically relevant quantities are extracted from the spectrum of the resulting operator.
This feels almost trivial (linearity of differentiation plus the spectral theorem), yet similar patterns seem to appear in continuum mechanics, elasticity, fluid dynamics, tidal tensors in GR, etc.
Is there a standard name for this viewpoint, principle, construction, or framework? Or is it considered so trivial that it is usually left unnamed, as a straightforward consequence of operator linearity and spectral theory?
A scalar field is written as a superposition of individual source contributions:
##
f(\mathbf{x}) = \sum_a f_a(\mathbf{x})
##
Therefore, by linearity of differentiation,
##
H(\mathbf{x})
=
\nabla\nabla f
=
\sum_a \nabla\nabla f_a
=
\sum_a H_a(\mathbf{x}).
##
So in a multi-source (multi-body) configuration the total Hessian is simply
##
H(\mathbf{x}) = \sum_a H_a(\mathbf{x}).
##
The observables are then obtained from the spectral decomposition of the total Hessian:
##
H(\mathbf{x})\,\mathbf e_i(\mathbf{x})
=
\lambda_i(\mathbf{x})\,\mathbf e_i(\mathbf{x}).
##
In other words,
##
\text{sources}
\rightarrow
f
\rightarrow
H
\rightarrow
\{\lambda_i,\mathbf e_i\}
\rightarrow
\text{observables}.
##
Importantly, I am not asking about summing eigenvalues. In general,
##
\lambda_i(A+B)
\neq
\lambda_i(A)+\lambda_i(B).
##
The point is that the underlying geometric object (the Hessian) superposes linearly, while the physically relevant quantities are extracted from the spectrum of the resulting operator.
This feels almost trivial (linearity of differentiation plus the spectral theorem), yet similar patterns seem to appear in continuum mechanics, elasticity, fluid dynamics, tidal tensors in GR, etc.
Is there a standard name for this viewpoint, principle, construction, or framework? Or is it considered so trivial that it is usually left unnamed, as a straightforward consequence of operator linearity and spectral theory?