The Invariance Principle and Income-Wealth Conservation Laws

Abstract

In the early part of the nineteenth century William Rowan Hamilton discovered a principle which can be generalized to encompass many areas of physics, engineering and applied mathematics. Hamilton’s principle roughly states that the evolution in time of a dynamic system takes place in such a manner that integral of the difference between the kinetic and potential energies for the system is stationary. If the “action” integral is free of the time variable, the sum of the kinetic and potential energies, the Hamiltonian, is constant—the conservation law of the total energy.

This chapter is a revised and expanded version of Sato (1985) and the author wishes to express his appreciation to Paul A. Samuelson, William A. Barnett, Hal R. Varian, Gilbert Suzawa, Takayuki Nôno, Fumitake Mimura, and Shigeru Maeda for their helpful comments on an earlier version of this chapter.

Appendix

Consider the variational integral

$$\displaystyle\begin{array}{rcl} & & J(x) =\int _{ a}^{b}L(t,x(t),\dot{x}(t))\,dt \\ & & (x(t) = ({x}^{1}(t),\ldots,{x}^{n}(t)),\quad \dot{x}(t) = (\dot{{x}}^{1}(t),\ldots,\dot{{x}}^{n}(t))).{}\end{array}$$

(8.62)

Also consider r-parameter transformations

$$\displaystyle\begin{array}{rcl} T: \overline{t} =\phi (t,x;\varepsilon ),\quad \varepsilon = {(\varepsilon }^{1},\ldots {,\varepsilon }^{n}),\quad {\overline{x}}^{i} {=\psi }^{i}(t,x;\varepsilon )\quad (i = 1,\ldots,n).& &{}\end{array}$$

(8.63)

where

$$\displaystyle\begin{array}{rcl} \phi (t,x;0)\!& =& \!t, \\ {\psi }^{i}(t,x;0)\!& =& \!{x}^{i}.{}\end{array}$$

(8.64)

In addition, it is assumed that T does not change the end points (a, α) and (b, β) where

$$\displaystyle\begin{array}{rcl} \phi (a,x(a);\varepsilon )\!& =& \!a,\quad \phi (b,x(b);\varepsilon ) = b, \\ {\psi }^{i}(a,x(a);\varepsilon )\!& =& \!{\alpha }^{i},\quad {\psi }^{i}(b,x(b);\varepsilon ) ={\beta }^{i}.{}\end{array}$$

(8.65)

Let X _s be the infinitesimal transformations for s = 1, …, r, then we write X _s as

$$\displaystyle\begin{array}{rcl} X_{s} =\tau _{s}(t,x) \frac{\partial } {\partial t} +\xi _{ s}^{i}(t,x) \frac{\partial } {\partial {x}^{i}} +\eta _{ s}^{i}(t,x,\dot{x}) \frac{\partial } {\partial \dot{{x}}^{i}}& &{}\end{array}$$

(8.66)

where

$$\displaystyle\begin{array}{rcl} \tau _{s}(t,x)\!& =& \! \frac{\partial \phi } {{\partial \varepsilon }^{s}}(t,x;0),\quad \xi _{s}^{i}(t,x) = \frac{{\partial \psi }^{i}} {{\partial \varepsilon }^{s}}(t,x;0), \\ \eta _{s}^{i}(t,x,\dot{x})\!& =& \!\frac{d\xi _{s}^{i}} {dt} - {x}^{i}\frac{d\tau _{s}} {dt} {}\end{array}$$

(8.67)

and (8.65) obeys

$$\displaystyle\begin{array}{rcl} \tau _{s}(a,x(a))\!& =& \!\tau _{s}(b,x(b)) = 0, \\ \xi _{s}^{i}(a,x(a))\!& =& \!\xi _{ s}^{i}(b,x(b)) = 0.{}\end{array}$$

(8.68)

If (8.62) is invariant under (8.63), we have

$$\displaystyle\begin{array}{rcl} \int _{a}^{b}X_{ s}(L\,dt) = 0.& &{}\end{array}$$

(8.69)

Lemma 8.1.

For any τ _s (t,x) and ξ _s ⁱ (t,x), we have

$$\displaystyle\begin{array}{rcl} N_{s} = (\xi _{s}^{i} -\dot{ {x}}^{i}\tau _{ s})E_{i} + \frac{d\Omega _{s}} {dt} & &{}\end{array}$$

(8.70)

where

$$\displaystyle\begin{array}{rcl} N_{s}\!& =& \!\frac{\partial L} {\partial t} \tau _{s} + \frac{\partial L} {\partial {x}^{i}}\xi _{s}^{i} + \frac{\partial L} {\partial \dot{{x}}^{i}}{\Bigl (\frac{d\xi _{s}^{i}} {dt} -\dot{ {x}}^{i}\frac{d\tau _{s}} {dt}\Bigr )} + L\frac{d\tau _{s}} {dt}, {}\\ E_{i}\!& =& \!\mbox{ Euler\textendash Lagrange equation} = \frac{\partial L} {\partial {x}^{i}} - \frac{d} {dt}{\Bigl ( \frac{\partial L} {\partial \dot{{x}}^{i}}\Bigr )}, {}\\ \frac{d\Omega } {dt} \!& =& \! \frac{d} {dt}{\Bigl [{\Bigl (L -\dot{ {x}}^{i} \frac{\partial L} {\partial \dot{{x}}^{i}}\Bigr )}\tau _{s} + \frac{\partial L} {\partial \dot{{x}}^{i}}\xi _{s}^{i}\Bigr ]}. {}\\ \end{array}$$

Proof.

By differentiating $\Omega _{s}$ with respect to t we have

$$\displaystyle\begin{array}{rcl} \frac{d\Omega _{s}} {dt} \!& =& \! \frac{d} {dt}{\Bigl [{\Bigl (L -\dot{ {x}}^{i} \frac{\partial L} {\partial \dot{{x}}^{i}}\Bigr )}\tau _{s} + \frac{\partial L} {\partial \dot{{x}}^{i}}\xi _{s}^{i}\Bigr ]} {}\\ \!& =& \!\frac{\partial L} {\partial t} \tau _{s} + \frac{\partial L} {\partial \dot{{x}}^{i}}{\Bigl (\frac{d\xi _{s}^{i}} {dt} -\dot{ {x}}^{i}\frac{d\tau _{s}} {dt}\Bigr )} + L\frac{d\tau _{s}} {dt} {}\\ & & \!+{x}^{i}\tau _{ s}{\Bigl [ \frac{\partial L} {\partial {x}^{i}} - \frac{d} {dt}{\Bigl ( \frac{\partial L} {\partial \dot{{x}}^{i}}\Bigr )}\Bigr ]} + \frac{d} {dt}{\Bigl ( \frac{\partial L} {\partial \dot{{x}}^{i}}\Bigr )}\xi _{s}^{i} {}\\ \!& =& \!N_{s} - (\xi _{s}^{i} -\dot{ {x}}^{i}\tau _{ s})E_{i}. {}\\ \end{array}$$

 □ 

When (8.62) is optimized, E _i vanishes and (8.70) reduces to

$$\displaystyle\begin{array}{rcl} \frac{d\Omega _{s}} {dt} = N_{s}.& &{}\end{array}$$

(8.71)

There are two cases: (1) when N _s  = 0 and N _s ≠0. The conservation law when N _s  = 0 is

$$\displaystyle\begin{array}{rcl} \frac{d\Omega _{s}} {dt} = 0\quad \mbox{ or}\quad \Omega _{s} = \mbox{ constant}.& &{}\end{array}$$

(8.72a)

When N _s ≠0 and $d\Phi /dt = N_{s}$, we have

$$\displaystyle\begin{array}{rcl} \frac{d(\Omega _{s} - \Phi _{s})} {dt} = 0\quad \mbox{ or}\quad \Omega _{s} - \Phi _{s} = \mbox{ constant}.& &{}\end{array}$$

(8.72b)

This is Noether’s invariance up to divergence.