Gauss's principle of least constraint
Principle of least constraint (also Gaussian principle of least constraint ) is a by Carl Friedrich Gauss in 1829 be imputed and supplemented by Philip Jourdain sentence of classical mechanics, according to which a mechanical system moves so that the constraint is minimized at all times.
The constraint is defined as follows:
Being summed over the mass points i, with the given impressed forces, the masses of point particles and the accelerations. The individual point particles, from which the system thinks you are assembled, thereby subject to additional forced Bedingungenen. The active forces must explicitly on time, dependent on location and speed, but not by the acceleration.
By the minimization of coercion with respect to the accelerations are all compatible with the constraints movements to the competition, which currently match the positions and the velocities. Competition means that all possible movements are considered - even those that do not occur because of the principle of least constraint in reality.
In the above equation are the differences between the acceleration of the mass elements and the accelerations they would learn as free masses under the effect of acting on them active forces. The principle can thus be formulated as follows:
Or
With ( only the acceleration is varied ).
The principle of least constraint is valid for very generally formulated constraints. In this time, the positions and velocities can undergo non-linear. Thus, the principle of least constraint is bounded from for example from d' Alembert'schen principle of virtual work, which calls for the holonomic constraints in the simplest version. Cornelius Lanczos calls it a brilliant reinterpretation of d' Alembert's principle of mechanics by Carl Friedrich Gauss, who had thus a formulation of the mechanical principles found, which was closely related in the form of the method of least squares.
Example
Where is a pendulum with two point masses and massless rigid rod (see picture). The forces Fe1 and Fe2 are the active forces with the amounts and M1G M2G. at1 and at2 are the Tangentialbeschleunigungen of the masses m1 and m2, an1 and an2 its normal accelerations. The constraint is thus:
When determining the minimum for the above expression is to be noted that the variation of the normal acceleration disappears due to the articulated suspension, as is true for the Tangentialbeschleunigungen:
And
Thus,
Because of the arbitrariness of the following after reduction of the factor 2, the equation of motion:
A formal interpretation
In the following, an interpretation of the Gaussian principle for a general point mass system is given with constraints.
System Description
Point masses with coordinates move under the influence of load-independent forces, which depend on the time, place and speeds.
The movement of the free system is determined by the equation
Described (M is the mass matrix ), where now the place is to be interpreted as a time-dependent function and the first and second time derivative are.
In the system to be examined, additional constraints are given by the equation
Be described by a vector-valued function.
Using the Gaussian principle, the equation of motion of the system is to be installed with constraints that takes the place of the equation of motion for the free system.
Interpretation of the Gaussian principle
The verbally formulated above Gaussian principle is not only an optimization task is, but a whole family by the time of parametric optimization problems, because the constraint is at all times a minimum accept (which is one of the subtle differences of the Gaussian principle to the principle of stationary action, in which the effect of a dependent of the entire movement is functional ).
At any fixed time all compete twice continuously differentiable curves in the curve parameters
The constraint condition, the
Meet at the place through the same place
Go the same speed
To have the forced minimum.
For placing an equation for the forced -minimizing motion an imagined in " A tool from the Analysis of real functions in one real variable " of the entry to the calculus of variations method.
From the set of all curves of competing any real - parameter family is singled out, which is differentiable with respect to the family parameter. The curve for, therefore, is to just match the physically excellent movement. This means that at any time the dependent family parameter constraint
Assumes a minimum at the point (the second representation is essentially a clearer notation). If you hold the time fixed, as is only dependent. A necessary condition that this function takes at a minimum is that the derivative of the forced upon is at zero, so
Taking into account that this equation must hold for any chosen according to the above conditions of curves, one obtains the equation of motion for the system with the given constraints.
This is done in the next section.
Transition to jourdainschen principle and Lagrangian representation
According to the procedure outlined above, the equations of motion are now in a better position to access the form in one of the calculation. The resulting system of equations is interpreted as jourdainsches principle or principle of virtual power.
First one performs the differentiation with respect to the last remote equation further.
This was used that many terms are due to the internal dissipation and zero.
To illustrate that in the parenthesis is the left side of the force balance for the free system, the mass matrix will be drawn into the clip into it.
The compatible with the constraints of variations of acceleration is obtained by differentiating the constraint
According to the location and then.
Here, the arguments have been omitted for clarity and with the partial derivatives with respect to time (i = 1), and location (i = 2 ) and speed ( i = 3), respectively. In the subsequent differentiation with respect to one uses again that for the variations of the position and velocity are equal to zero and obtain the desired condition for the constraint is compatible:
If one introduces in the last equation and the last equation for the variation of the acceleration, the characters and substituted one ( correctly ) and we obtain finally from the Gaussian principle, the usual notation for the jourdainsche principle of virtual power:
The excellent physical movement is just so that at any time the equation
With for all
Is satisfied.
This can be interpreted to mean that at least in the directions in which the system can move freely currently, the system must also satisfy the equations of motion of the free system with constraints.
The values are referred to as virtual speed.
For a more effective computation can be the above system of equations in the Lagrangian representation as follows ( Lagrangegleichung 1.Art ) transform, which is also the d' Alembert principle equivalent.
The second equation is expressed, that the set of all allowable is just the core of the matrix and the first equation states that lies in the orthogonal complement of that set. Overall, we obtain therefore
Because follows. Thus there is a ( time-dependent ) vector ( Lagrange multiplier ), which
Applies ( Lagrange equations first kind ).
One interpretation of this is that perpendicular to the possible virtual velocities can act any constraining forces.
Explicit derivation of the d' Alembert principle
Holonomic constraints, where the speeds are not explicitly occur may be included in the treatment received by setting:
From the intuition is clear that the constraint also restricts the possible speeds for the place that forces the system into a particular path. This results in jourdainschen principle:
Since then carried the variation of velocities in the constraint surfaces, can be replaced by the virtual displacements and the result is the usual form of the d' Alembert Primzips:
The excellent physical movement such that at all times runs the equation
With for all
Is satisfied. The Lagrange equations first type follow as above:
With.