Families of Planar Orbits in Polar Coordinates Compatible with Potentials
In light of the planar inverse problem of Newtonian Dynamics, we study the monoparametric family of regular orbits <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo&g...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-11-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/12/21/3435 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850197222209093632 |
|---|---|
| author | Thomas Kotoulas |
| author_facet | Thomas Kotoulas |
| author_sort | Thomas Kotoulas |
| collection | DOAJ |
| description | In light of the planar inverse problem of Newtonian Dynamics, we study the monoparametric family of regular orbits <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>r</mi><mo>,</mo><mi>θ</mi><mo>)</mo><mo>=</mo><mi>c</mi></mrow></semantics></math></inline-formula> in polar coordinates (where <i>c</i> is the parameter varying along the family of orbits), which are generated by planar potentials <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>=</mo><mi>V</mi><mo>(</mo><mi>r</mi><mo>,</mo><mi>θ</mi><mo>)</mo></mrow></semantics></math></inline-formula>. The corresponding family of orbits can be uniquely represented by the “<i>slope function</i>” <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>γ</mi><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mi>f</mi><mi>θ</mi></msub><msub><mi>f</mi><mi>r</mi></msub></mfrac></mstyle></mrow></semantics></math></inline-formula>. By using the basic partial differential equation of the planar inverse problem, which combines families of orbits and potentials, we apply a <i>new</i> methodology in order to find specific potentials, e.g., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>=</mo><mi>A</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>+</mo><mi>B</mi><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>=</mo><mi>H</mi><mo>(</mo><mi>γ</mi><mo>)</mo></mrow></semantics></math></inline-formula> and one-dimensional potentials, e.g., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>=</mo><mi>A</mi><mo>(</mo><mi>r</mi><mo>)</mo></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>=</mo><mi>G</mi><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></semantics></math></inline-formula>. In order to determine such potentials, differential conditions on the family of orbits <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>r</mi><mo>,</mo><mi>θ</mi><mo>)</mo></mrow></semantics></math></inline-formula> = <i>c</i> are imposed. If these conditions are fulfilled, then we can find a potential of the above form analytically. For the given families of curves, such as ellipses, parabolas, Bernoulli’s lemniscates, etc., we find potentials that produce them. We present suitable examples for all cases and refer to the case of straight lines. |
| format | Article |
| id | doaj-art-a468c1cfe72d4cfdaa08f6947aeae357 |
| institution | OA Journals |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-a468c1cfe72d4cfdaa08f6947aeae3572025-08-20T02:13:14ZengMDPI AGMathematics2227-73902024-11-011221343510.3390/math12213435Families of Planar Orbits in Polar Coordinates Compatible with PotentialsThomas Kotoulas0Department of Physics, Aristotle University of Thessaloniki, 541 24 Thessaloniki, GreeceIn light of the planar inverse problem of Newtonian Dynamics, we study the monoparametric family of regular orbits <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>r</mi><mo>,</mo><mi>θ</mi><mo>)</mo><mo>=</mo><mi>c</mi></mrow></semantics></math></inline-formula> in polar coordinates (where <i>c</i> is the parameter varying along the family of orbits), which are generated by planar potentials <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>=</mo><mi>V</mi><mo>(</mo><mi>r</mi><mo>,</mo><mi>θ</mi><mo>)</mo></mrow></semantics></math></inline-formula>. The corresponding family of orbits can be uniquely represented by the “<i>slope function</i>” <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>γ</mi><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><msub><mi>f</mi><mi>θ</mi></msub><msub><mi>f</mi><mi>r</mi></msub></mfrac></mstyle></mrow></semantics></math></inline-formula>. By using the basic partial differential equation of the planar inverse problem, which combines families of orbits and potentials, we apply a <i>new</i> methodology in order to find specific potentials, e.g., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>=</mo><mi>A</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>+</mo><mi>B</mi><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>=</mo><mi>H</mi><mo>(</mo><mi>γ</mi><mo>)</mo></mrow></semantics></math></inline-formula> and one-dimensional potentials, e.g., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>=</mo><mi>A</mi><mo>(</mo><mi>r</mi><mo>)</mo></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>=</mo><mi>G</mi><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></semantics></math></inline-formula>. In order to determine such potentials, differential conditions on the family of orbits <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>r</mi><mo>,</mo><mi>θ</mi><mo>)</mo></mrow></semantics></math></inline-formula> = <i>c</i> are imposed. If these conditions are fulfilled, then we can find a potential of the above form analytically. For the given families of curves, such as ellipses, parabolas, Bernoulli’s lemniscates, etc., we find potentials that produce them. We present suitable examples for all cases and refer to the case of straight lines.https://www.mdpi.com/2227-7390/12/21/3435classical mechanicsinverse problem of Newtonian dynamicsmonoparametric families of orbitsseparable potentials in polar coordinatesdynamical systemsintegrable systems |
| spellingShingle | Thomas Kotoulas Families of Planar Orbits in Polar Coordinates Compatible with Potentials Mathematics classical mechanics inverse problem of Newtonian dynamics monoparametric families of orbits separable potentials in polar coordinates dynamical systems integrable systems |
| title | Families of Planar Orbits in Polar Coordinates Compatible with Potentials |
| title_full | Families of Planar Orbits in Polar Coordinates Compatible with Potentials |
| title_fullStr | Families of Planar Orbits in Polar Coordinates Compatible with Potentials |
| title_full_unstemmed | Families of Planar Orbits in Polar Coordinates Compatible with Potentials |
| title_short | Families of Planar Orbits in Polar Coordinates Compatible with Potentials |
| title_sort | families of planar orbits in polar coordinates compatible with potentials |
| topic | classical mechanics inverse problem of Newtonian dynamics monoparametric families of orbits separable potentials in polar coordinates dynamical systems integrable systems |
| url | https://www.mdpi.com/2227-7390/12/21/3435 |
| work_keys_str_mv | AT thomaskotoulas familiesofplanarorbitsinpolarcoordinatescompatiblewithpotentials |