The Two-Thirds Power Law Derived from a Higher-Derivative Action
The two-thirds power law is a link between angular speed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula> and curvature <inline-formula>...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-11-01
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| Series: | Physics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2624-8174/6/4/77 |
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| Summary: | The two-thirds power law is a link between angular speed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula> and curvature <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>κ</mi></semantics></math></inline-formula> observed in voluntary human movements: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula> is proportional to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>κ</mi><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></semantics></math></inline-formula>. Squared jerk is known to be a Lagrangian leading to the latter law. However, it leads to unbounded movements and is therefore incompatible with quasi-periodic dynamics, such as the movement of the tip of a pen drawing ellipses. To solve this drawback, we give a class of higher-derivative Lagrangians that allow for both quasi-periodic and unbounded movements, and at the same time lead to the two-thirds power law. The current study extends this framework and investigates a wider class of Lagrangians admitting generalised conservation laws. |
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| ISSN: | 2624-8174 |