A Ramsey-Theory-Based Approach to the Dynamics of Systems of Material Points
We propose a Ramsey-theory-based approach for the analysis of the behavior of isolated mechanical systems containing interacting particles. The total momentum of the system in the frame of the center of masses is zero. The mechanical system is described by a Ramsey-theory-based, bi-colored, complete...
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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: | Dynamics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2673-8716/4/4/43 |
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| Summary: | We propose a Ramsey-theory-based approach for the analysis of the behavior of isolated mechanical systems containing interacting particles. The total momentum of the system in the frame of the center of masses is zero. The mechanical system is described by a Ramsey-theory-based, bi-colored, complete graph. Vectors of momenta of the particles <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo>→</mo></mover></mrow><mrow><mi>i</mi></mrow></msub><mo> </mo></mrow></semantics></math></inline-formula> serve as the vertices of the graph. We start from the graph representing the system in the frame of the center of masses, where the momenta of the particles in this system are <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo>→</mo></mover></mrow><mrow><mi>c</mi><mi>m</mi><mi>i</mi></mrow></msub><mo>.</mo></mrow></semantics></math></inline-formula> If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>p</mi></mrow><mo>→</mo></mover></mrow><mrow><mi>c</mi><mi>m</mi><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>·</mo><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo>→</mo></mover></mrow><mrow><mi>c</mi><mi>m</mi><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≥</mo><mn>0</mn></mrow></semantics></math></inline-formula> is true, the vectors of momenta of the particles numbered <i>i</i> and <i>j</i> are connected with a red link; if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>p</mi></mrow><mo>→</mo></mover></mrow><mrow><mi>c</mi><mi>m</mi><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>·</mo><msub><mrow><mover accent="true"><mrow><mi>p</mi></mrow><mo>→</mo></mover></mrow><mrow><mi>c</mi><mi>m</mi><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula> takes place, the vectors of momenta are connected with a green link. Thus, the complete, bi-colored graph emerges. Considering an isolated system built of six interacting particles, according to the Ramsey theorem, the graph inevitably comprises at least one monochromatic triangle. The coloring procedure is invariant relative to the rotations/translations of frames; thus, the graph representing the system contains at least one monochromatic triangle in any of the frames emerging from the rotation/translation of the original frame. This gives rise to a novel kind of mechanical invariant. Similar coloring is introduced for the angular momenta of the particles. However, the coloring procedure is sensitive to Galilean/Lorenz transformations. Extensions of the suggested approach are discussed. |
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| ISSN: | 2673-8716 |