Difference between revisions of "D2. Interaction forces between particles"
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==D2.2 Classification of interaction forces== | ==D2.2 Classification of interaction forces== | ||
If the universe consisted only of dimensionless particles with no connecting elements between them, the list of possible interactions would be very short: in the mechanics, one could only speak of the force of gravitational attraction, which occurs | If the universe consisted only of dimensionless particles with no connecting elements between them, the list of possible interactions would be very short: in the mechanics, one could only speak of the force of gravitational attraction, which occurs “at a distance” (<math>\rho > 0</math>). Talking about contact interaction between particles (<math>\rho = 0</math>) is not possible: if <math>\mathbf{P}</math> and <math>\mathbf{Q}</math> “come into contact”, since the location of each of them is defined by a single point, the direction of the interaction is not determined. On the other hand, two particles cannot occupy the same point in space. <br> | ||
Revision as of 19:09, 6 November 2024
[math]\displaystyle{ \newcommand{\uvec}{\overline{\textbf{u}}} \newcommand{\vvec}{\overline{\textbf{v}}} \newcommand{\evec}{\overline{\textbf{e}}} \newcommand{\Omegavec}{\overline{\mathbf{\Omega}}} \newcommand{\velang}[2]{\Omegavec^{\textrm{#1}}_{\textrm{#2}}} \newcommand{\Alfavec}{\overline{\mathbf{\alpha}}} \newcommand{\accang}[2]{\Alfavec^{\textrm{#1}}_{\textrm{#2}}} \newcommand{\cs}{\textrm{c}} \newcommand{\ds}{\textrm{d}} \newcommand{\ms}{\textrm{m}} \newcommand{\ts}{\textrm{t}} \newcommand{\us}{\textrm{u}} \newcommand{\vs}{\textrm{v}} \newcommand{\Fs}{\textrm{F}} \newcommand{\Rs}{\textrm{R}} \newcommand{\Ts}{\textrm{T}} \newcommand{\Ls}{\textrm{L}} \newcommand{\Ns}{\textrm{N}} \newcommand{\Bs}{\textrm{B}} \newcommand{\es}{\textrm{e}} \newcommand{\fs}{\textrm{f}} \newcommand{\is}{\textrm{i}} \newcommand{\js}{\textrm{j}} \newcommand{\rs}{\textrm{r}} \newcommand{\Os}{\textbf{O}} \newcommand{\Gs}{\textbf{G}} \newcommand{\Cbf}{\textbf{C}} \newcommand{\Or}{\Os_\Rs} \newcommand{\Qs}{\textbf{Q}} \newcommand{\Cs}{\textbf{C}} \newcommand{\Ps}{\textbf{P}} \newcommand{\Ss}{\textbf{S}} \newcommand{\deg}{^\textsf{o}} \newcommand{\xs}{\textsf{x}} \newcommand{\ys}{\textsf{y}} \newcommand{\zs}{\textsf{z}} \newcommand{\dert}[2]{\left.\frac{\ds{#1}}{\ds\ts}\right]_{\textrm{#2}}} \newcommand{\ddert}[2]{\left.\frac{\ds^2{#1}}{\ds\ts^2}\right]_{\textrm{#2}}} \newcommand{\vec}[1]{\overline{#1}} \newcommand{\vecbf}[1]{\overline{\textbf{#1}}} \newcommand{\vecdot}[1]{\overline{\dot{#1}}} \newcommand{\OQvec}{\vec{\Os\Qs}} \newcommand{\QPvec}{\vec{\Qs\Ps}} \newcommand{\OPvec}{\vec{\Os\textbf{P}}} \newcommand{\OCvec}{\vec{\Os\Cs}} \newcommand{\OGvec}{\vec{\Os\Gs}} \newcommand{\abs}[1]{\left|{#1}\right|} \newcommand{\braq}[2]{\left\{{#1}\right\}_{\textrm{#2}}} \newcommand{\vector}[3]{ \begin{Bmatrix} {#1}\\ {#2}\\ {#3} \end{Bmatrix}} \newcommand{\vecdosd}[2]{ \begin{Bmatrix} {#1}\\ {#2} \end{Bmatrix}} \newcommand{\vel}[2]{\vvec_{\textrm{#2}} (\textbf{#1})} \newcommand{\acc}[2]{\vecbf{a}_{\textrm{#2}} (\textbf{#1})} \newcommand{\accs}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{s}} (\textbf{#1})} \newcommand{\accn}[2]{\vecbf{a}_{\textrm{#2}}^{\textrm{n}} (\textbf{#1})} \newcommand{\velo}[1]{\vvec_{\textrm{#1}}} \newcommand{\accso}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{s}}} \newcommand{\accno}[1]{\vecbf{a}_{\textrm{#1}}^{\textrm{n}}} \newcommand{\re}[2]{\Re_{\textrm{#2}}(\textbf{#1})} \newcommand{\psio}{\dot{\psi}_0} \newcommand{\Pll}{\textbf{P}_\textrm{lliure}} \newcommand{\agal}[1]{\vecbf{a}_{\textrm{Gal}} (#1)} \newcommand{\angal}[1]{\vecbf{a}_{\textrm{NGal}} (#1)} \newcommand{\vgal}[1]{\vecbf{v}_{\textrm{Gal}} (#1)} }[/math]
Newton's second law can be used to predict the acceleration of a particle P when all the interaction forces exerted on P by all the other particles (Q) are known (all [math]\displaystyle{ \overline{\mathbf{F}}_{\mathrm{Q} \rightarrow \mathrm{P}} }[/math]), or to calculate the forces needed to guarantee a predetermined movement ([math]\displaystyle{ \overline{\mathbf{O}_\Rs\mathbf{P}}(\ts) }[/math]).
This section deals with the interaction forces between particles, whether they can be formulated or not. Formulating an interaction is finding a mathematical expression that allows calculating its value at each time instant given the mechanical state of the particles and the constants associated with the type of interaction phenomenon. When a force cannot be formulated, it is an unknown of the dynamic problem.
D2.1 Kinematic dependence of interaction forces
According to the Principle of Determinacy, the forces of interaction between two particles P and Q at each time instant can only depend on the position and velocity of the particles at that time instant:
[math]\displaystyle{ \ms_\mathrm{p} \overline{\mathbf{a}}_{\mathrm{R}}(\mathbf{P}, \mathrm{t})=\overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}}\left(\overline{\mathbf{O}_\mathrm{Gal} \mathbf{P}}(\mathrm{t}), \overline{\mathbf{O}_\mathrm{Gal} \mathbf{Q}}(\mathrm{t}), \overline{\mathbf{v}}_\mathrm{Gal}(\mathbf{P}, \mathrm{t}), \overline{\mathbf{v}}_\mathrm{Gal}(\mathbf{Q}, \mathrm{t})\right)=-\mathrm{m}_{\mathbf{Q}} \overline{\mathbf{a}}_{\mathrm{R}}(\mathbf{Q}, \mathrm{t}) . }[/math]
Galileo's Principle of Relativity (equivalence of Galilean reference frames for the formulation of dynamics) imposes restrictions on the type of dependence of [math]\displaystyle{ \overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}} }[/math] on positions and velocities:
El Principi de Relativitat de Galileu (equivalència de les referències galileanes per a laformulació de la dinàmica) imposa restriccions al tipus de dependència de [math]\displaystyle{ \overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}} }[/math] en posicions i velocitats:
- La homogeneitat de l’espai en referències galileanes impedeix que [math]\displaystyle{ \overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}} }[/math] depengui en les posicions [math]\displaystyle{ \overline{\mathbf{O}_\mathrm{Gal} \mathbf{P}} }[/math] i [math]\displaystyle{ \overline{\mathbf{O}_\mathrm{Gal} \mathbf{Q}} }[/math] per separat, però permet la dependència en la seva diferència [math]\displaystyle{ \overline{\mathbf{P}\Qs}(=\overline{\mathbf{O}_\mathrm{Gal} \mathbf{Q}}-\overline{\mathbf{O}_\mathrm{Gal} \mathbf{P}}) }[/math]
- La isotropia de l'espai en referències galileanes impedeix que [math]\displaystyle{ \overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}} }[/math] depengui de les velocitats de [math]\displaystyle{ \mathbf{P} }[/math] i [math]\displaystyle{ \mathbf{Q} }[/math] per separat (doncs [math]\displaystyle{ \overline{\mathbf{v}}_{\text {Gal1 }}(\mathbf{P}, \mathbf{Q}) \neq \overline{\mathbf{v}}_{\text {Gal2 }}(\mathbf{P}, \mathbf{Q}) }[/math]), però permet que depengui de la seva diferència: [math]\displaystyle{ \overline{\mathbf{v}}_{\text {Gal }}(\mathbf{P})-\overline{\mathbf{v}}_{\text {Gal }}(\mathbf{Q}) \equiv \Delta \overline{\mathbf{v}}_{\forall\mathrm{Gal}} }[/math]. El vector [math]\displaystyle{ \Delta \overline{\mathbf{v}}_{\forall \mathrm{Gal}} }[/math] és el mateix independentment de quina sigui la referència Gal (per això s'ha posat el subíndex ' [math]\displaystyle{ \forall }[/math]Gal', que vol dir “per a tota referència Gal”). En general, [math]\displaystyle{ \Delta \overline{\mathbf{v}}_{\forall\text {Gal }} }[/math] té una component parallela a [math]\displaystyle{ \overline{\mathbf{P Q}} }[/math], i una de perpendicular a [math]\displaystyle{ \overline{\mathbf{P Q}} }[/math] : [math]\displaystyle{ \Delta \overline{\mathbf{v}}_{\forall\mathrm{Gal}}=\Delta \overline{\mathbf{v}}_{\| \overline{\mathbf{P Q}}}+\Delta \overline{\mathbf{v}}_{\perp \overline{\mathbf{P Q}}} \equiv \Delta \overline{\mathbf{v}}_\rho+\Delta \overline{\mathbf{v}}_{\mathrm{n}} }[/math]. Però la isotropia de l'espai tampoc no permet que [math]\displaystyle{ \overline{\mathbf{F}}_{\mathbf{Q} \rightarrow \mathbf{P}} }[/math] depengui d'una direcció que no sigui [math]\displaystyle{ \overline{\mathbf{P Q}} }[/math](com s'ha vist quan s'ha presentat la Tercera llei de Newton). Per tant, s'accepta la dependència en el vector [math]\displaystyle{ \Delta \overline{\mathbf{v}}_\rho }[/math] però només en el valor de [math]\displaystyle{ \left|\Delta \overline{\mathbf{v}}_{\mathrm{n}}\right| }[/math](tot i que usualment aquesta última dependència no apareix).
D2.2 Classification of interaction forces
If the universe consisted only of dimensionless particles with no connecting elements between them, the list of possible interactions would be very short: in the mechanics, one could only speak of the force of gravitational attraction, which occurs “at a distance” ([math]\displaystyle{ \rho \gt 0 }[/math]). Talking about contact interaction between particles ([math]\displaystyle{ \rho = 0 }[/math]) is not possible: if [math]\displaystyle{ \mathbf{P} }[/math] and [math]\displaystyle{ \mathbf{Q} }[/math] “come into contact”, since the location of each of them is defined by a single point, the direction of the interaction is not determined. On the other hand, two particles cannot occupy the same point in space.
D2.3 Gravitational attraction
D2.4 Interaction through springs
D2.5 Interaction through dampers
D2.6 Interaction through actuators
D2.7 Constraint interactions
D2.8 Friction
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