Fundamental Physical How to Select an . . . Equations Can Be - PowerPoint PPT Presentation

Introduction Newton’s Physics: . . . First Step: Selecting a . . . Resulting . . . Fundamental Physical How to Select an . . . Equations Can Be Derived Resulting Model This Model Leads to . . . By Applying Fuzzy Beyond the Simplest . . . Beyond Netwon’s . . . Methodology to Informal Home Page Physical Ideas Title Page ◭◭ ◮◮ Eric Gutierrez and Vladik Kreinovich ◭ ◮ Department of Computer Science Page 1 of 16 University of Texas at El Paso 500 W. University Go Back El Paso, TX 79968, USA Full Screen ejgutierrez@miners.utep.edu vladik@utep.edu Close Quit

Introduction Newton’s Physics: . . . 1. Introduction First Step: Selecting a . . . • Fuzzy methodology has been invented to transform: Resulting . . . How to Select an . . . – expert ideas – formulated in terms of words from Resulting Model natural language, This Model Leads to . . . – into precise rules and formulas, rules and formulas Beyond the Simplest . . . understandable by a computer. Beyond Netwon’s . . . • Fuzzy methodology: main success is intelligent (fuzzy) Home Page control. Title Page • We show that the same fuzzy methodology can also ◭◭ ◮◮ lead to the exact fundamental equations of physics. ◭ ◮ • This fact provides an additional justification for the Page 2 of 16 fuzzy methodology. Go Back Full Screen Close Quit

Introduction Newton’s Physics: . . . 2. Newton’s Physics: Informal Description First Step: Selecting a . . . • A body usually tries to go to the points x where its Resulting . . . potential energy V ( x ) is the smallest. How to Select an . . . Resulting Model • For example, a moving rock on the mountain tries to This Model Leads to . . . go down. Beyond the Simplest . . . • The sum of the potential energy V ( x ) and the kinetic Beyond Netwon’s . . . energy K is preserved: Home Page 3 � 2 K = 1 � dx i � 2 · m · . Title Page dt i =1 ◭◭ ◮◮ • Thus, when the body minimizes its potential energy, it ◭ ◮ thus tries to maximize its kinetic energy. Page 3 of 16 • We will show that when we apply the fuzzy techniques Go Back to this informal description, we get Newton’s equations m · d 2 x i dt 2 = − ∂V Full Screen . ∂x i Close Quit

Introduction Newton’s Physics: . . . 3. First Step: Selecting a Membership Function First Step: Selecting a . . . • The body tries to get to the areas where the potential Resulting . . . energy V ( x ) is small. How to Select an . . . Resulting Model • We need to select the corresponding membership func- This Model Leads to . . . tion µ ( V ). Beyond the Simplest . . . • For example, we can poll several ( n ) experts and if Beyond Netwon’s . . . n ( V ) of them consider V small, take µ ( V ) = n ( V ) . Home Page n Title Page • In physics, we only know relative potential energy – relative to some level. ◭◭ ◮◮ • If we change that level by V 0 , we replace V by V + V 0 . ◭ ◮ • So, values V and V + V 0 represent the same value of Page 4 of 16 the potential energy – but for different levels. Go Back • A seemingly natural formalization: µ ( V ) = µ ( V + V 0 ). Full Screen • Problem: we get useless µ ( V ) = const. Close Quit

Introduction Newton’s Physics: . . . 4. Re-Analyzing the Polling Method First Step: Selecting a . . . • In the poll, the more people we ask, the more accurate Resulting . . . is the resulting opinion. How to Select an . . . Resulting Model • Thus, to improve the accuracy of the poll, we add m This Model Leads to . . . folks to the original n top experts. Beyond the Simplest . . . • These m extra folks may be too intimidated by the Beyond Netwon’s . . . original experts. Home Page • With the new experts mute, we still have the same Title Page number n ( V ) of experts who say “yes”. ◭◭ ◮◮ • As a result, instead of the original value µ ( V ) = n ( V ) , ◭ ◮ n we get µ ′ ( V ) = n ( V ) n Page 5 of 16 n + m = c · µ ( V ), where c = n + m. Go Back • These two membership functions µ ( V ) and µ ′ ( V ) = Full Screen c · µ ( V ) represent the same expert opinion. Close Quit

Introduction Newton’s Physics: . . . 5. Resulting Formalization of the Physical Intu- First Step: Selecting a . . . ition Resulting . . . • How to describe that potential energy is small? How to Select an . . . Resulting Model • Idea: value V and V + V 0 are equivalent – they differ This Model Leads to . . . by a starting level for measuring potential energy. Beyond the Simplest . . . • Conclusion: membership functions µ ( V ) and µ ( V + V 0 ) Beyond Netwon’s . . . should be equivalent. Home Page • We know: membership functions µ ( V ) and µ ′ ( V ) are Title Page equivalent if µ ′ ( V ) = c · µ ( V ). ◭◭ ◮◮ • Hence: for every V 0 , there is a value c ( V 0 ) for which ◭ ◮ µ ( V + V 0 ) = c ( V 0 ) · µ ( V ) . Page 6 of 16 • It is known that the only monotonic solution to this Go Back equation is µ ( V ) = a · exp( − k · V ) . Full Screen • So we will use this membership function to describe that the potential energy is small. Close Quit

Introduction Newton’s Physics: . . . 6. Resulting Formalization of the Physical Intu- First Step: Selecting a . . . ition (cont-d) Resulting . . . • Reminder: we use µ ( V ) = a · exp( − k · V ) to describe How to Select an . . . that potential energy is small. Resulting Model This Model Leads to . . . • How to describe that kinetic energy is large? Beyond the Simplest . . . • Idea: K is large if − K is small. Beyond Netwon’s . . . • Resulting membership function: Home Page µ ( K ) = exp( − k · ( − K )) = exp( k · K ) . Title Page ◭◭ ◮◮ • We want to describe the intuition that ◭ ◮ – the potential energy is small and Page 7 of 16 – that the kinetic energy is large and – that the same is true at different moments of time. Go Back Full Screen • According to fuzzy methodology, we must therefore se- lect an appropriate “and”-operation (t-norm) f & ( a, b ). Close Quit

Introduction Newton’s Physics: . . . 7. How to Select an Appropriate t-Norm First Step: Selecting a . . . • In principle, if we have two completely independent Resulting . . . systems, we can consider them as a single system. How to Select an . . . Resulting Model • Since these systems do not interact with each other, This Model Leads to . . . the total energy E is simply equal to E 1 + E 2 . Beyond the Simplest . . . • We can estimate the smallness of the total energy in Beyond Netwon’s . . . two different ways: Home Page – we can state that the total energy E = E 1 + E 2 is Title Page small: certainty µ ( E 1 + E 2 ), or ◭◭ ◮◮ – we can state that both E 1 and E 2 are small: ◭ ◮ f & ( µ ( E 1 ) , µ ( E 2 )) . Page 8 of 16 • It is reasonable to require that these two estimates co- Go Back incide: µ ( E 1 + E 2 ) = f & ( µ ( E 1 ) , µ ( E 2 )) . Full Screen • This requirement enables us to uniquely determine the corresponding t-norm: f & ( a 1 , a 2 ) = a 1 · a 2 . Close Quit

Introduction Newton’s Physics: . . . 8. Resulting Model First Step: Selecting a . . . • Idea: at all moments of time t 1 , . . . , t N , the potential Resulting . . . energy V is small, and the kinetic energy K is large. How to Select an . . . Resulting Model • Small is exp( − k · V ), large is exp( k · K ), “and” is prod- This Model Leads to . . . uct, thus the degree µ ( x ( t )) is Beyond the Simplest . . . N N � � µ ( x ( t )) = exp( − k · V ( t i )) · exp( k · K ( t i )) . Beyond Netwon’s . . . Home Page i =1 i =1 N Title Page def � • So, µ ( x ( t )) = exp( − k · S ), w/ S = ( V ( t i ) − K ( t i )) . ◭◭ ◮◮ i =1 � • In the limit t i +1 − t i → 0, S → ( V ( t ) − K ( t )) dt . ◭ ◮ • The most reasonable trajectory is the one for which Page 9 of 16 � µ ( x ( t )) → max, i.e., S = L dt → min, where Go Back 3 � 2 � dx i = V ( t ) − K ( t ) = V ( t ) − 1 def � Full Screen L 2 · m · . dt i =1 Close Quit

Introduction Newton’s Physics: . . . 9. This Model Leads to Newton’s Equations First Step: Selecting a . . . � • Reminder: S = L dt → min, where Resulting . . . How to Select an . . . 3 � 2 � dx i = V ( t ) − K ( t ) = V ( t ) − 1 def � L 2 · m · . Resulting Model dt i =1 This Model Leads to . . . • Most physical laws are now formulated in terms of the Beyond the Simplest . . . � Principle of Least Action S = L dt → min. Beyond Netwon’s . . . Home Page • E.g., for the above L , we get Newtonian physics. Title Page • So, fuzzy indeed implies Newton’s equations . ◭◭ ◮◮ • Newton’s physics: only one trajectory, with S → min. ◭ ◮ • With the fuzzy approach, we also get the degree Page 10 of 16 exp( − k · S ) w/which other trajectories are reasonable. Go Back • In quantum physics, each non-Newtonian trajectory is possible with “amplitude” exp( − k · S ) (for complex k ). Full Screen • This makes the above derivation even more interesting. Close Quit