On the alpine ski with dry friction and air resistance. Some optimization problems for it

Aldo Bressan

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1999)

  • Volume: 10, Issue: 2, page 79-129
  • ISSN: 1120-6330

Abstract

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In the present work, divided in three parts, one considers a real skis-skier system, Σ R , descending along a straight-line l with constant dry friction; and one schematizes it by a holonomic system Σ = A ∪ U , having any number n ≥ 4 of degrees of freedom and subjected to (non-ideal) constraints, partly one-sided. Thus, e.g., jumps and also «steps made with sliding skis» can be schematized by Σ . Among the n Lagrangian coordinates for Σ two are the Cartesian coordinates ξ and η of its center of mass, C , relative to the downward axis that includes l and to the upward axis that is normal to l in a vertical plane; the others are to be regarded as controls in that their values can be determined by the skier. Four alternative laws of air resistance, A2.5,1 to A2.5,4, are considered for Σ . They have increasing simplicity and according to all of them the resultant of air resistance, m ⁢ R , is parallel to l and independent of Σ 's possible asymmetries with respect to the vertical plane trough l . Briefly, m ⁢ R is independent of the skier’s configuration C U with respect to the skis, these being always supposed to be parallel to l ; according to A2.5,3 m ⁢ R is a (possibly non-homogeneous) linear function of C 's velocity ξ ˙ ; and according to A2.5,4 m ⁢ R ≡ 0 . In Part 1, after some preliminaries, Σ 's dynamic equations are written in a suitable form by which, under the law A2.5,3, a control-free first integral can be deduced, notwithstanding controls can raise and lower C , which affects C 's velocity ξ because of dry friction. Given Σ 's initial conditions at t = 0 , this first integral is a relation between ξ , η , ξ ˙ , η ˙ and the present time t . In the case m ⁢ R ≡ 0 it can be integrated again. Thus ξ appears to be determined by η and t . The afore-mentioned results on Σ are simple; and here it is convenient to note that the present work does not aim at refined results; furthermore its Parts 2 and 3 are completely based on the afore-mentioned result valid for m ⁢ R ≡ 0 ; and they treat two problems on Σ that have a special interest for Σ R , these problems being useful in connection with races and hence with tourism. This occurs in that, by a suitable device, the conclusions of the above treatment can be used to obtain good informations on Σ R also in case m ⁢ R is for Σ R practically constant and large. At the end of Part 1, under an air resistance law more general than A2.5,4, one considers the possibility of rendering the (negative) work of dry friction in a given time interval 0 , T arbitrarily small by means of «steps made with sliding skis»; and one shows that this fact has a negligible influence on the length ξ ⁢ T - ξ ⁢ 0 of the ski-run’s stretch covered by C in 0 , T . This result reasonably holds for Σ R with a good approximation; thus it is «explained» why the above steps are not made in practice. In Part 2, where the identity m ⁢ R ≡ 0 is assumed, the following is considered: Problem 9.1. Given ξ ¯ > 0 and the initial conditions at t = 0 , how can one minimize the time t ¯ > 0 taken by C 's absciss ξ to cover the ski-run’s stretch 0 , ξ ¯ ? This Problem concerns alpine ski. On the other hand the few mathematical works on ski, that are known to the author but not related to his papers, deal with-ski jumps. Furthermore Problem 9.1 is different from all preceding ski problems treated by the author in that it involves dry friction, air resistance, and one-sided constraints. For the same reasons Σ cannot be regarded as a special case of some holonomic system to which the author has applied control theory. In conformity with this, instead of solving Problem 9.1 by this theory (Pontriagin’s maximum principle), it is convenient to preliminarily consider the following Problem 6.1. Given T > 0 and the initial conditions at t = 0 , how can one maximize the length ξ ⁢ T - ξ ⁢ 0 of the ski-run’s stretch 0 , ξ ¯ covered by C in the time T ? For this problem ∞ ∞ solutions are exhibited in C 1 ∩ P ⁢ C 2 , so that they are much more regular than the solutions (in L 1 ) assured by the most known existence theorem in control theory (if applicable). Lastly the solutions of the above two problems are shown to be the same when a certain relation holds between T and ξ ¯ . The optimal values of ξ ⁢ T and t ¯ for Problem 6.1 and Problem 9.1 respectively can be expressed by means of the data, independently of Σ 's optimal motions. Various properties of these are considered; and it is shown that, generally, the skier can affect the values of ξ ⁢ T and t ¯ very little. Part 3 treats Σ 's motions without jumps, i.e., the most common ones. E.g. some upper bounds for the afore-mentioned little influence of the skier are given. Furthermore for every t ∈ 0 , T one exhibits some conditions equivalent to the possibility of extending a jumpless motion of Σ in 0 , t to such a motion in 0 , T . One shows that this may be useful to implement the «skier U » as a robot, in order to compare jumpless motions of Σ R , with the corresponding dynamic motions of Σ considered in Part 2 where the identity m ⁢ R ≡ 0 is assumed. The afore-mentioned device allows us to perform such a comparison in an interesting real case where m ⁢ R keeps very near a possibly large constant value.

How to cite

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Bressan, Aldo. "On the alpine ski with dry friction and air resistance. Some optimization problems for it." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 10.2 (1999): 79-129. <http://eudml.org/doc/252414>.

@article{Bressan1999,
abstract = {In the present work, divided in three parts, one considers a real skis-skier system, \( \Sigma\_\{R\} \), descending along a straight-line \( l \) with constant dry friction; and one schematizes it by a holonomic system \( \Sigma = A \cup U \), having any number \( n \ge 4 \) of degrees of freedom and subjected to (non-ideal) constraints, partly one-sided. Thus, e.g., jumps and also «steps made with sliding skis» can be schematized by \( \Sigma \). Among the \( n \) Lagrangian coordinates for \( \Sigma \) two are the Cartesian coordinates \( \xi \) and \( \eta \) of its center of mass, \( C \), relative to the downward axis that includes \( l \) and to the upward axis that is normal to \( l \) in a vertical plane; the others are to be regarded as controls in that their values can be determined by the skier. Four alternative laws of air resistance, A2.5,1 to A2.5,4, are considered for \( \Sigma \). They have increasing simplicity and according to all of them the resultant of air resistance, \( m \mathcal\{R\} \), is parallel to \( l \) and independent of \( \Sigma \)'s possible asymmetries with respect to the vertical plane trough \( l \). Briefly, \( m \mathcal\{R\} \) is independent of the skier’s configuration \( \mathcal\{C\}\_\{U\} \) with respect to the skis, these being always supposed to be parallel to \( l \); according to A2.5,3 \( m \mathcal\{R\} \) is a (possibly non-homogeneous) linear function of \( C \)'s velocity \( \dot\{\xi\} \); and according to A2.5,4 \( m \mathcal\{R\} \equiv 0 \). In Part 1, after some preliminaries, \( \Sigma \)'s dynamic equations are written in a suitable form by which, under the law A2.5,3, a control-free first integral can be deduced, notwithstanding controls can raise and lower \( C \), which affects \( C \)'s velocity \( \xi \) because of dry friction. Given \( \Sigma \)'s initial conditions at \( t = 0 \), this first integral is a relation between \( \xi, \eta, \dot\{\xi\}, \dot\{\eta\} \) and the present time \( t \). In the case \( m \mathcal\{R\} \equiv 0 \) it can be integrated again. Thus \( \xi \) appears to be determined by \( \eta \) and \( t \). The afore-mentioned results on \( \Sigma \) are simple; and here it is convenient to note that the present work does not aim at refined results; furthermore its Parts 2 and 3 are completely based on the afore-mentioned result valid for \( m \mathcal\{R\} \equiv 0 \); and they treat two problems on \( \Sigma \) that have a special interest for \( \Sigma \_\{R\} \), these problems being useful in connection with races and hence with tourism. This occurs in that, by a suitable device, the conclusions of the above treatment can be used to obtain good informations on \( \Sigma \_\{R\} \) also in case \( m \mathcal\{R\} \) is for \( \Sigma \_\{R\} \) practically constant and large. At the end of Part 1, under an air resistance law more general than A2.5,4, one considers the possibility of rendering the (negative) work of dry friction in a given time interval \( \left[0, T \right] \) arbitrarily small by means of «steps made with sliding skis»; and one shows that this fact has a negligible influence on the length \( \xi(T ) - \xi(0) \) of the ski-run’s stretch covered by \( C \) in \( \left[0, T \right] \). This result reasonably holds for \( \Sigma \_\{R\} \) with a good approximation; thus it is «explained» why the above steps are not made in practice. In Part 2, where the identity \( m \mathcal\{R\} \equiv 0 \) is assumed, the following is considered: Problem 9.1. Given \( \bar\{\xi\} > 0 \) and the initial conditions at \( t = 0 \), how can one minimize the time \( \bar\{t\} (> 0) \) taken by \( C \)'s absciss \( \xi \) to cover the ski-run’s stretch \( \left[0, \bar\{\xi\} \right] \)? This Problem concerns alpine ski. On the other hand the few mathematical works on ski, that are known to the author but not related to his papers, deal with-ski jumps. Furthermore Problem 9.1 is different from all preceding ski problems treated by the author in that it involves dry friction, air resistance, and one-sided constraints. For the same reasons \( \Sigma \) cannot be regarded as a special case of some holonomic system to which the author has applied control theory. In conformity with this, instead of solving Problem 9.1 by this theory (Pontriagin’s maximum principle), it is convenient to preliminarily consider the following Problem 6.1. Given \( T > 0 \) and the initial conditions at \( t = 0 \), how can one maximize the length \( \xi(T ) - \xi(0) \) of the ski-run’s stretch \( \left[ 0, \bar\{\xi\} \right] \) covered by \( C \) in the time \( T \)? For this problem \( \infty^\{\infty\} \) solutions are exhibited in \( C^\{1\} \cap PC^\{2\} \), so that they are much more regular than the solutions (in \( L^\{1\} \)) assured by the most known existence theorem in control theory (if applicable). Lastly the solutions of the above two problems are shown to be the same when a certain relation holds between \( T \) and \( \bar\{\xi\} \). The optimal values of \( \xi(T ) \) and \( \bar\{t\} \) for Problem 6.1 and Problem 9.1 respectively can be expressed by means of the data, independently of \( \Sigma \)'s optimal motions. Various properties of these are considered; and it is shown that, generally, the skier can affect the values of \( \xi(T ) \) and \( \bar\{t\} \) very little. Part 3 treats \( \Sigma \)'s motions without jumps, i.e., the most common ones. E.g. some upper bounds for the afore-mentioned little influence of the skier are given. Furthermore for every \( t \in \left[ 0, T \right[ \) one exhibits some conditions equivalent to the possibility of extending a jumpless motion of \( \Sigma \) in \( \left[ 0, t \right] \) to such a motion in \( \left[ 0, T \right] \). One shows that this may be useful to implement the «skier \( U \)» as a robot, in order to compare jumpless motions of \( \Sigma\_\{R\} \), with the corresponding dynamic motions of \( \Sigma \) considered in Part 2 where the identity \( m \mathcal\{R\} \equiv 0 \) is assumed. The afore-mentioned device allows us to perform such a comparison in an interesting real case where \( m \mathcal\{R\} \) keeps very near a possibly large constant value.},
author = {Bressan, Aldo},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Ski; Holonomic systems; Optimization problems; Controls; alpin ski},
language = {eng},
month = {6},
number = {2},
pages = {79-129},
publisher = {Accademia Nazionale dei Lincei},
title = {On the alpine ski with dry friction and air resistance. Some optimization problems for it},
url = {http://eudml.org/doc/252414},
volume = {10},
year = {1999},
}

TY - JOUR
AU - Bressan, Aldo
TI - On the alpine ski with dry friction and air resistance. Some optimization problems for it
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1999/6//
PB - Accademia Nazionale dei Lincei
VL - 10
IS - 2
SP - 79
EP - 129
AB - In the present work, divided in three parts, one considers a real skis-skier system, \( \Sigma_{R} \), descending along a straight-line \( l \) with constant dry friction; and one schematizes it by a holonomic system \( \Sigma = A \cup U \), having any number \( n \ge 4 \) of degrees of freedom and subjected to (non-ideal) constraints, partly one-sided. Thus, e.g., jumps and also «steps made with sliding skis» can be schematized by \( \Sigma \). Among the \( n \) Lagrangian coordinates for \( \Sigma \) two are the Cartesian coordinates \( \xi \) and \( \eta \) of its center of mass, \( C \), relative to the downward axis that includes \( l \) and to the upward axis that is normal to \( l \) in a vertical plane; the others are to be regarded as controls in that their values can be determined by the skier. Four alternative laws of air resistance, A2.5,1 to A2.5,4, are considered for \( \Sigma \). They have increasing simplicity and according to all of them the resultant of air resistance, \( m \mathcal{R} \), is parallel to \( l \) and independent of \( \Sigma \)'s possible asymmetries with respect to the vertical plane trough \( l \). Briefly, \( m \mathcal{R} \) is independent of the skier’s configuration \( \mathcal{C}_{U} \) with respect to the skis, these being always supposed to be parallel to \( l \); according to A2.5,3 \( m \mathcal{R} \) is a (possibly non-homogeneous) linear function of \( C \)'s velocity \( \dot{\xi} \); and according to A2.5,4 \( m \mathcal{R} \equiv 0 \). In Part 1, after some preliminaries, \( \Sigma \)'s dynamic equations are written in a suitable form by which, under the law A2.5,3, a control-free first integral can be deduced, notwithstanding controls can raise and lower \( C \), which affects \( C \)'s velocity \( \xi \) because of dry friction. Given \( \Sigma \)'s initial conditions at \( t = 0 \), this first integral is a relation between \( \xi, \eta, \dot{\xi}, \dot{\eta} \) and the present time \( t \). In the case \( m \mathcal{R} \equiv 0 \) it can be integrated again. Thus \( \xi \) appears to be determined by \( \eta \) and \( t \). The afore-mentioned results on \( \Sigma \) are simple; and here it is convenient to note that the present work does not aim at refined results; furthermore its Parts 2 and 3 are completely based on the afore-mentioned result valid for \( m \mathcal{R} \equiv 0 \); and they treat two problems on \( \Sigma \) that have a special interest for \( \Sigma _{R} \), these problems being useful in connection with races and hence with tourism. This occurs in that, by a suitable device, the conclusions of the above treatment can be used to obtain good informations on \( \Sigma _{R} \) also in case \( m \mathcal{R} \) is for \( \Sigma _{R} \) practically constant and large. At the end of Part 1, under an air resistance law more general than A2.5,4, one considers the possibility of rendering the (negative) work of dry friction in a given time interval \( \left[0, T \right] \) arbitrarily small by means of «steps made with sliding skis»; and one shows that this fact has a negligible influence on the length \( \xi(T ) - \xi(0) \) of the ski-run’s stretch covered by \( C \) in \( \left[0, T \right] \). This result reasonably holds for \( \Sigma _{R} \) with a good approximation; thus it is «explained» why the above steps are not made in practice. In Part 2, where the identity \( m \mathcal{R} \equiv 0 \) is assumed, the following is considered: Problem 9.1. Given \( \bar{\xi} > 0 \) and the initial conditions at \( t = 0 \), how can one minimize the time \( \bar{t} (> 0) \) taken by \( C \)'s absciss \( \xi \) to cover the ski-run’s stretch \( \left[0, \bar{\xi} \right] \)? This Problem concerns alpine ski. On the other hand the few mathematical works on ski, that are known to the author but not related to his papers, deal with-ski jumps. Furthermore Problem 9.1 is different from all preceding ski problems treated by the author in that it involves dry friction, air resistance, and one-sided constraints. For the same reasons \( \Sigma \) cannot be regarded as a special case of some holonomic system to which the author has applied control theory. In conformity with this, instead of solving Problem 9.1 by this theory (Pontriagin’s maximum principle), it is convenient to preliminarily consider the following Problem 6.1. Given \( T > 0 \) and the initial conditions at \( t = 0 \), how can one maximize the length \( \xi(T ) - \xi(0) \) of the ski-run’s stretch \( \left[ 0, \bar{\xi} \right] \) covered by \( C \) in the time \( T \)? For this problem \( \infty^{\infty} \) solutions are exhibited in \( C^{1} \cap PC^{2} \), so that they are much more regular than the solutions (in \( L^{1} \)) assured by the most known existence theorem in control theory (if applicable). Lastly the solutions of the above two problems are shown to be the same when a certain relation holds between \( T \) and \( \bar{\xi} \). The optimal values of \( \xi(T ) \) and \( \bar{t} \) for Problem 6.1 and Problem 9.1 respectively can be expressed by means of the data, independently of \( \Sigma \)'s optimal motions. Various properties of these are considered; and it is shown that, generally, the skier can affect the values of \( \xi(T ) \) and \( \bar{t} \) very little. Part 3 treats \( \Sigma \)'s motions without jumps, i.e., the most common ones. E.g. some upper bounds for the afore-mentioned little influence of the skier are given. Furthermore for every \( t \in \left[ 0, T \right[ \) one exhibits some conditions equivalent to the possibility of extending a jumpless motion of \( \Sigma \) in \( \left[ 0, t \right] \) to such a motion in \( \left[ 0, T \right] \). One shows that this may be useful to implement the «skier \( U \)» as a robot, in order to compare jumpless motions of \( \Sigma_{R} \), with the corresponding dynamic motions of \( \Sigma \) considered in Part 2 where the identity \( m \mathcal{R} \equiv 0 \) is assumed. The afore-mentioned device allows us to perform such a comparison in an interesting real case where \( m \mathcal{R} \) keeps very near a possibly large constant value.
LA - eng
KW - Ski; Holonomic systems; Optimization problems; Controls; alpin ski
UR - http://eudml.org/doc/252414
ER -

References

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