Principle of the least constraint

This article deals with the Gaussian principle of least constraint. The Le Chatelier principle can be found under the Lemma principle of the smallest constraint .

Principle of the smallest compulsion (also Gaussian principle of the smallest compulsion ) is a set of classical mechanics drawn up by Carl Friedrich Gauß in 1829 and supplemented by Philip Jourdain , according to which a mechanical system moves in such a way that the compulsion is minimized at all times . ${\displaystyle t}$

The compulsion is defined as:

${\displaystyle Z=\sum _{i}{\frac {1}{2m_{i}}}({\vec {F}}_{i}-m_{i}{\vec {a}}_{i})^{2}=\sum _{i}{\frac {m_{i}}{2}}\left({\frac {{\vec {F}}_{i}}{m_{i}}}-{\vec {a}}_{i}\right)^{2}}$

where the mass points i are added together with the given applied forces , the masses of the point particles and the accelerations . The individual point particles, from which the system is thought to be composed, are subject to additional constraints. The applied forces may explicitly depend on time, location and speed, but not on the acceleration. ${\displaystyle {\vec {F}}_{i}}$ ${\displaystyle m_{i}}$ ${\displaystyle {\vec {a}}_{i}}$

With the minimization of the constraint with regard to the accelerations, all movements compatible with the constraint conditions are in competition, in which the positions and the speeds currently match. Competition means that all possible movements are considered - even those that do not occur in reality because of the principle of the least bit of compulsion . ${\displaystyle t}$

The above equation shows the differences between the accelerations of the mass elements and the accelerations that they would experience as free masses under the action of the applied forces acting on them . The principle can be formulated as follows:

${\displaystyle Z={\mathsf {min}}}$

or.

${\displaystyle \delta Z=0}$,

with (only the acceleration is varied). ${\displaystyle \delta {\vec {r}}=0,\quad \delta {\vec {v}}=0}$

The principle of the smallest constraint is valid for very generally formulated constraints. The time, the places and the speeds can be included in this non-linearly. In this way, the principle of the smallest constraint is differentiated, for example, from d'Alembert's principle of virtual work, in which holonomic constraints are required in the simplest version. Cornelius Lanczos calls it an ingenious reinterpretation of d'Alembert's principle of mechanics by Carl Friedrich Gauß , who found a formulation of the mechanical principles that was closely related in the form of his method of least squares .

example

Figure 1: Two-mass pendulum

A pendulum with 2 point masses and a massless rigid rod is given (see Figure 1). The forces F ^e ₁ and F ^e ₂ are the applied forces with the amounts m ₁ g and m ₂ g. a _t1 and a _t2 are the tangential accelerations of the masses m ₁ and m ₂ , a _n1 and a _{n2 are} the associated normal accelerations . The compulsion is thus:

${\displaystyle \sum _{i=1}^{2}m_{i}\left[\left(a_{ti}+g\sin \phi \right)^{2}+\left(a_{ni}+g\cos \phi \right)^{2}\right]}$

When determining the minimum for the above expression, it should be noted that the variation of the normal accelerations vanishes because of the articulated suspension, while the following applies to the tangential accelerations:

${\displaystyle a_{ti}=r_{i}{\ddot {\phi }}}$

and

${\displaystyle \delta a_{ti}=r_{i}\delta {\ddot {\phi }}}$

Thus becomes

${\displaystyle \delta \sum _{i=1}^{2}m_{i}\left[\left(a_{ti}+g\sin \phi \right)^{2}+\left(a_{ni}+g\cos \phi \right)^{2}\right]=}$

${\displaystyle =2\sum _{i=1}^{2}m_{i}r_{i}\left(r_{i}{\ddot {\phi }}+g\sin \phi \right)\delta {\ddot {\phi }}=0}$

Because of the arbitrariness of , after reducing the factor 2, the equation of motion follows: ${\displaystyle \delta {\ddot {\phi }}}$

${\displaystyle \left(m_{1}r_{1}^{2}+m_{2}r_{2}^{2}\right){\ddot {\phi }}+\left(m_{1}r_{1}+m_{2}r_{2}\right)g\sin \phi =0}$

A formal interpretation

The following is an interpretation of the Gaussian principle for a general point mass system with constraints.

System description

${\displaystyle \left(k=1,\ldots ,n\right)}$Point masses with coordinates move under the influence of applied forces that depend on time, place and speed. ${\displaystyle x=\left(x_{1},\ldots ,x_{3n}\right)^{T}}$

The motion of the free system is given by the equation

${\displaystyle M\partial _{t}^{2}x(t)=F(t,x(t),\partial _{t}x(t))}$

( is the mass matrix), where the location is to be interpreted as a time-dependent function and the first and second time derivatives are. ${\displaystyle M}$ ${\displaystyle x(t)}$ ${\displaystyle \partial _{t}x,\partial _{t}^{2}x}$

In the system to be examined, however, there are additional constraints that are defined by the equation ${\displaystyle z\in \mathbb {N} }$ ${\displaystyle (0\leq z\leq 3n)}$

${\displaystyle G(t,x,\partial _{t}x)=0\in \mathbb {R} ^{z}}$

can be described with a vector-valued function . ${\displaystyle G\colon (t,x,\partial _{t}x)\mapsto G(t,x,\partial _{t}x)\in \mathbb {R} ^{z}}$

With the help of the Gaussian principle, the equation of motion of the system with constraints should be established, which takes the place of the equation of motion for the free system.

Interpretation of the Gaussian principle

The above-verbally formulated Gaussian principle is not only a programming problem, but rather a whole family by the time parameterized optimization problems, because the compulsion to at all times a minimum accept (this is one of the subtle differences of the Gaussian principle to the principle of stationary action in where the effect is a functional that depends on the entire movement ). ${\displaystyle t}$ ${\displaystyle S[x]}$ ${\displaystyle x}$

At each fixed point in time , all curves that are twice continuously differentiable in the curve parameter compete ${\displaystyle t}$ ${\displaystyle \tau }$

${\displaystyle \tau \mapsto x(t,\tau )\in \mathbb {R} ^{3n}}$

which the constraint

${\displaystyle G(\tau ,x(t,\tau ),\partial _{\tau }x(t,\tau ))=0}$

meet at the point by the same place ${\displaystyle \tau =t}$

${\displaystyle \left.x(t,\tau )\right|_{\tau =t}=x(t)}$

go and same speed

${\displaystyle \left.\partial _{\tau }x(t,\tau )\right|_{\tau =t}={\dot {x}}(t)}$

have around the mandatory minimum.

To set up an equation for the movement that minimizes the constraint , a method presented in the section “An aid from the analysis of real functions in a real variable” of the entry on the calculation of variations is used. ${\displaystyle x}$

From the set of all competing curves, any real parametric family is picked out, which is differentiable according to the family parameter . The curve for , well , is supposed to coincide with the physical movement . That means that at all times the constraint dependent on the flock parameter ${\displaystyle \{\tau \mapsto x(t,\tau ,\alpha )\}_{\alpha }}$ ${\displaystyle \alpha }$ ${\displaystyle \alpha =0}$ ${\displaystyle \tau \mapsto x(t,\tau ,0)}$ ${\displaystyle t\mapsto x(t)}$ ${\displaystyle t}$ ${\displaystyle \alpha }$

${\displaystyle Z(t,\alpha ):=\left.\left(\partial _{\tau }^{2}x(t,\tau ,\alpha )-M^{-1}F\right)^{T}M\left(\partial _{\tau }^{2}x(t,\tau ,\alpha )-M^{-1}F\right)\right|_{\tau =t}}$

${\displaystyle =\left.\left({\sqrt {M}}\left(\partial _{\tau }^{2}x(t,\tau ,\alpha )-M^{-1}F)\right)\right)^{2}\right|_{\tau =t}}$

assumes a minimum at this point (the second representation is essentially used for a clearer notation). If you keep the time , then it is only dependent on. A necessary condition for this function to take on a minimum is that the derivative of the constraint is equal to zero, i.e. ${\displaystyle \alpha =0}$ ${\displaystyle t}$ ${\displaystyle \alpha \mapsto Z(t,\alpha )}$ ${\displaystyle \alpha }$ ${\displaystyle \alpha =0}$ ${\displaystyle \alpha }$ ${\displaystyle \alpha =0}$

${\displaystyle \left.\partial _{\alpha }Z(t,\alpha )\right|_{\alpha =0}=0.}$

If one takes into account that this equation must apply to any family of curves selected according to the above conditions , one obtains the equation of motion for the system with the given constraints. ${\displaystyle \left\{x(t,\tau ,\alpha )\right\}_{\alpha }}$

This is elaborated on in the next section.

Transition to Jourdain's principle and Lagrangian representation

According to the procedure outlined above, the equations of motion are now set up in a form that is more accessible for calculation. The resulting system of equations is also interpreted as the Jourdain principle or principle of virtual performance .

First, the differentiation according to the last equation is carried out. ${\displaystyle \alpha }$

${\displaystyle \left.\partial _{\alpha }Z(t,\alpha )\right|_{\alpha =0}=2\left(\partial _{\tau }^{2}x(t,\tau ,0)-M^{-1}F(\,t,x(t,\tau ,\alpha ),\partial _{\tau }x(t,\tau ,\alpha )\,)\right)^{T}M\,\partial _{\alpha }\left.\partial _{\tau }^{2}x(t,\tau ,\alpha )\right|_{\alpha =0,\tau =t}}$

It was used here that many terms are due to the inner derivative and equal to zero. ${\displaystyle \partial _{\alpha }x(t,\tau ,\alpha )|_{\tau =t}=0}$ ${\displaystyle \partial _{\alpha }\partial _{\tau }x(t,\tau ,\alpha )|_{\tau =t}=0}$

To make it clear that the left side of the balance of forces stands for the free system in the brackets , the mass matrix is drawn into the brackets. ${\displaystyle M}$

${\displaystyle \left.\partial _{\alpha }Z(t,\alpha )\right|_{\alpha =0}=2\left(M\partial _{\tau }^{2}x(t,\tau ,\alpha )-F(\,t,x(t,\tau ,\alpha ),\partial _{\tau }x(t,\tau ,\alpha )\,)\right)^{T}\partial _{\alpha }\left.\partial _{\tau }^{2}x(t,\tau ,\alpha )\right|_{\alpha =0,\tau =t}}$

The variations in acceleration that are compatible with the constraints are obtained by deriving the constraint ${\displaystyle \partial _{\alpha }\left.\partial _{\tau }^{2}x(t,\tau ,\alpha )\right|_{\alpha =0,\tau =t}}$

${\displaystyle G(\tau ,x(t,\tau ,\alpha ),\partial _{\tau }x(t,\tau ,\alpha ))=0}$

after at the point and then after . ${\displaystyle \tau }$ ${\displaystyle \tau =t}$ ${\displaystyle \alpha }$

${\displaystyle 0=\left.\partial _{\tau }G(\tau ,x(t,\tau ,\alpha ),\partial _{\tau }x(t,\tau ,\alpha ))\right|_{\tau =t}}$

${\displaystyle =\left[\partial _{1}G+\partial _{2}G\partial _{\tau }x+\partial _{3}G\partial _{\tau }^{2}x\right]_{\tau =t}}$

For the sake of clarity, the arguments have been omitted here and the partial derivatives according to time (i = 1), location (i = 2) and speed (i = 3) are designated. In the subsequent differentiation , one takes advantage of the fact that the variations in position and speed are equal to zero, and the desired condition is obtained that is compatible with the constraints: ${\displaystyle \partial _{i}}$ ${\displaystyle \alpha }$ ${\displaystyle \tau =t}$ ${\displaystyle \partial _{\alpha }\partial _{\tau }^{2}x|_{\alpha =0,\tau =t}}$

${\displaystyle \left.\partial _{3}G(t,x(t,t,0),\partial _{\tau }x(t,\tau ,0)\right|_{\tau =t})\left.\partial _{\alpha }\partial _{\tau }^{2}x\right|_{\alpha =0,\tau =t}=0}$

If one introduces the symbol for the variation of the acceleration in the last equation and in the last equation and substitutes (correctly) and , one finally obtains the usual notation for the Jourdainian principle of virtual power from the Gaussian principle : ${\displaystyle \left.\partial _{\alpha }Z(t,\alpha )\right|_{\alpha =0}=0}$ ${\displaystyle \partial _{\alpha }\partial _{\tau }^{2}x|_{\alpha =0,\tau =t}}$ ${\displaystyle \delta v}$ ${\displaystyle x(t,\tau ,\alpha )|_{\alpha =0,\tau =t}=x(t)}$ ${\displaystyle \partial _{\tau }x(t,\tau ,\alpha )|_{\alpha =0,\tau =t}={\dot {x}}(t)}$

The physically distinguished motion runs precisely in such a way that at any point in time the equation ${\displaystyle t\mapsto x(t)}$ ${\displaystyle t}$

${\displaystyle \left(M{\ddot {x}}(t)-F(\,t,x(t),{\dot {x}}(t)\,)\right)^{T}\delta v=0}$

for everyone with ${\displaystyle \delta v\in \mathbb {R} ^{3n}}$

${\displaystyle \partial _{3}G(t,x(t),{\dot {x}}(t))\delta v=A\delta v=0}$

is satisfied.

This can be interpreted in such a way that at least in the directions in which the system can currently move freely, the system with constraints must also fulfill the equations of motion of the free system. ${\displaystyle \delta v}$

The sizes are called virtual speeds . ${\displaystyle \delta v}$

For a more effective calculation, the above system of equations can be converted into the Lagrangian representation ( Lagrangian equation of the first type ) as follows , which is also equivalent to the d'Alembert principle.

The second equation expresses that the set of all admissible is precisely the core of the matrix and the first equation says that this set lies in the orthogonal complement. So overall you get ${\displaystyle \delta v}$ ${\displaystyle A=\partial _{3}G(t,x(t),{\dot {x}}(t))}$ ${\displaystyle \left(M{\ddot {x}}(t)-F(\,t,x(t),{\dot {x}}(t)\,)\right)}$ ${\displaystyle \left(M{\ddot {x}}(t)-F(\,t,x(t),{\dot {x}}(t)\,)\right)\in \left(\operatorname {Kern} \,A\right)^{\perp }=\operatorname {Bild} \,A^{T}}$

Because it follows . So there is a (time-dependent) vector (the Lagrange multiplier) with which ${\displaystyle \,A{\vec {x}}=0}$ ${\displaystyle \,{\vec {y}}^{T}A{\vec {x}}={(A^{T}{\vec {y}})}^{T}{\vec {x}}=0}$ ${\displaystyle \lambda (t)\in \mathbb {R} ^{z}}$

${\displaystyle M{\ddot {x}}(t)-F(\,t,x(t),{\dot {x}}(t)\,)=A^{T}\lambda (t)}$

holds (Lagrangian equations of the first kind).

One interpretation for this is that any constraining forces can act perpendicular to the possible virtual speeds . ${\displaystyle \delta v}$ ${\displaystyle \,A^{T}\lambda (t)}$

Explicit derivation of the d'Alembert principle

Holonomic constraints , in which the velocities do not explicitly occur, can be included in the previous treatment by setting: ${\displaystyle \,C(t,x(t))=0}$

${\displaystyle G(t,x(t),{\dot {x}}(t)):={\frac {\mathrm {d} }{\mathrm {d} t}}C(t,x(t))=\partial _{1}C(t,x(t))+\partial _{2}C(t,x(t))\,{\dot {x}}(t)=0}$

From this view it is clear that the constraint for the location, which forces the system into a certain path, also limits the possible speeds. It results in Jourdain's principle:

${\displaystyle \partial _{3}G(t,x(t),{\dot {x}}(t))\delta v=\partial _{2}C(t,x(t))\delta v=0}$

Since then the speed changes in the constrained surfaces, it can be replaced by the virtual displacements and the usual form of the d'Alembert principle results: ${\displaystyle \,\delta v}$ ${\displaystyle \,\delta x}$

The physically distinguished movement proceeds in such a way that at any point in time the equation ${\displaystyle t\mapsto x(t)}$ ${\displaystyle t}$

${\displaystyle \left(M{\ddot {x}}(t)-F(t,x(t),{\dot {x}}(t))\right)\delta x=0}$

for everyone with ${\displaystyle \delta x\in \mathbb {R} ^{3n}}$

${\displaystyle \partial _{2}C(t,x(t))\delta x=0}$

is satisfied. The type 1 Lagrangian equations follow as above:

${\displaystyle M{\ddot {x}}(t)-F(t,x(t),{\dot {x}}(t))=\partial _{2}C(t,x(t))^{T}\,\lambda (t)}$

with . ${\displaystyle \lambda (t)\in \mathbb {R} ^{z}}$

literature

• Georg Hamel: Theoretical Mechanics. Springer-Verlag, Berlin 1949.
• Werner Schiehlen: Technical Dynamics. Teubner study books, Stuttgart 1986.
• Cornelius Lanczos: The Variational Principles of Mechanics. Dover.
• Gauss Werke Vol. 5, On a new general basic law of mechanics , Journal for Pure and Applied Mathematics Vol. 4, 1829

Web links

• Evans et al. a. to the principle of the smallest compulsion

References

1. ↑ Form based on the book by Cornelius Lanczos, where the prefactor 1/2 comes from the derivation from the d'Alembert principle, it can also be omitted with other authors
2. ^ The Variational Principles of Mechanics , Dover, p. 106