# The Analytical Mechanics of Consumption

The Analytical Mechanics of Consumption: In Mechanical and Economic Systems

AuthorHutters, Coenraad (TU Delft Mechanical, Maritime and Materials Engineering; TU Delft Delft Center for Systems and Control)

Delft University of Technology

Programme Date2019-04-24

AbstractThe Utility Lagrangian and the Surplus Hamiltonian in economic engineering do not depend on consumption.Two theories are proposed to include the effect of consumption in the Utility Lagrangian and the Surplus Hamiltonian. The second of these two theories resolves the dissipation obstacle in port-Hamiltonian systems as an additional result.The first theory includes consumption as a fractional-order derivative in the Fractional Utility Lagrangian, following an action principle for dissipative systems proposed in the literature.The principle of maximal utility from economic engineering results in a fractional Euler-Lagrange equation that relates a change in price to the accrued benefit less the accrued depreciation due to consumption.A Legendre transform of the Fractional Utility Lagrangian results in a Fractional Surplus Hamiltonian the reveals the effect of consumption on surplus.A drawback of this theory is that control formalisms of port-Hamiltonian systems theory cannot be applied to the Fractional Surplus Hamiltonian, since it is not canonical.The second theory includes consumption in the Surplus Hamiltonian with complex state variables and ---in general--- dissipation in the Hamiltonian formalism.The theory of Complex Hamiltonians is developed to model damped harmonic oscillators as canonical Hamiltonian systems.The Complex Hamilton's equations result in the equations of motion of a damped harmonic oscillator and are equivalent to the canonical Poisson bracket between the complex state and the Complex Hamiltonian.Applying control formalisms from port-Hamiltonian systems theory to the Complex Hamiltonian bypasses the dissipation obstacle that in real-valued port-Hamiltonian systems stymies the control of dissipative systems.Utilizing the analogies from economic engineering results in a Complex Surplus Hamiltonian.Evaluating the Complex Surplus Hamiltonian shows that the marginal propensity to consume is the economic analog of the damping ratio.Control formalisms from port-Hamiltonian systems can be applied to the Complex Surplus Hamiltonian.As an additional result, it is shown that the fractional derivative can be used as a storage variable for heat generated by frictional dissipation; this results in an expression for dissipated energy of the same form as the familiar expressions for kinetic and potential energy.

SubjectEconomic modelling

Analytical Mechanics

complex geometry

Hamiltonian mechanics

Lagrangian Mechanics

Fractional Calculus

Port-Hamiltonian

Control

Complex Hamiltonian

Utility Maximization

Surplus

http://resolver.tudelft.nl/uuid:258e378f-c55d-4024-bcb3-c1f7c88391c2

Bibliographical noteThis thesis is part of the Economic Engineering group at DCSC.

Part of collectionStudent theses

Document typemaster thesis

Rights© 2019 Coenraad Hutters