# What is Constraints: Definition and 217 Discussions

The theory of constraints (TOC) is a management paradigm that views any manageable system as being limited in achieving more of its goals by a very small number of constraints. There is always at least one constraint, and TOC uses a focusing process to identify the constraint and restructure the rest of the organization around it. TOC adopts the common idiom "a chain is no stronger than its weakest link". This means that processes, organizations, etc., are vulnerable because the weakest person or part can always damage or break them or at least adversely affect the outcome.

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1. ### How to calculate the degrees of freedom of a simple multi-body system?

I intuitively understand that it has 2 degrees of freedom (rolling without slipping - RWS), but I struggle to formalize this according to the rules of the art: I obtain: 10 - 3 (ground) - (rolling without slipping constraint) = 2 how to precisely calculate the RWS constraint? what formula and...
2. ### How should I calculate the stationary value of ## S[y] ##?

Consider the functional ## S[y]=\int_{1}^{2}x^2y^2dx ## stationary subject to the two constraints ## \int_{1}^{2}xydx=1 ## and ## \int_{1}^{2}x^2ydx=2 ##. Then the auxiliary functional is ## \overline{S}[y]=\int_{1}^{2}(x^2y^2+\lambda_{1}xy+\lambda_{2}x^2y)dx, y(1)=y(2)=0 ## where ## \lambda_{1}...
3. ### Holonomic constraints and non-holonomic system

The solution is given. What makes this solution unique is that there is a net turn for the disk. The note of the solution mentions this system is non-holonomic. My question is that are there other non-holonomic examples. What makes this particular set up non-holonomic? Thanks!
4. ### I New Constraints On The Higgs Boson Width

The New Measurement The ATLAS experiment at the Large Hadron Collider (LHC) has experimentally limited the "width" of the Standard Model Higgs boson with a rest mass of about 125 GeV to 4.5 + 3.3 - 2.5 MeV, with a maximum value of 10.5 MeV at a 95% confidence level. In the Standard Model, the...
5. ### I Shankar on constraints and free parameters for a particle in a box

On page 160 in Shankar, he discusses how we get quantized energy levels of bound states - specifically for the particle in a box. We have three regions in space; region I from ## \ - \infty, -L/2 ##, region II from ## \ -L/2, L/2 ##, and region III from ## \ L/2, \infty ##. For the...
6. ### Degrees of freedom and holonomic constraints

Hi, I am not quite sure if I have solved task 2a and 2b correctly. For task 2a I would say, because of the constraints, that the system has only 1 degree of freedom. Since the vectors must always have an angle of Pi/4 to each other, this would mean that if one vector moves up, the other must...
7. ### I Constraints of a mechanical system

I'm studying theoretical mechanics and I kind of find the notion of a "mechanical system" very slippery, especially when it comes to constraints. Take an example : I know that when a system consists of N particles and p constraints, it has 3N-p degrees of freedom; this is the definition. Then I...
8. ### I Degrees of freedom and constraints

In case of P holonomic constraints and N particles, I have 3N-P degrees of freedom and I have to look for 3N-P generalized coordinates if I want them to vary independently, but what about non-holonomic constraints? I know if I have N particles and P non-holonomic constraints, I still need 3N...
9. ### I Understanding Holonomic Constraints: Common Questions and Answers

I've got a couple of questions concerning holonomic constraint equations: 1- Suppose I've got k holonomic constraint equations for n particles, how can I be sure those are all the ones there are and I didn't miss any? I mean, in a given situation, I can be pretty sure that I've got all, but is...

11. ### Moving in a straight line with multiple constraints

Hi All, I am trying to solve what I think should be a simple problem, but I must be missing something because I am struggle to solve it. The situation is shown below: So to summarise: A body starts at Point A It moves in a straight line to Point B, covering a distance of 10m The time taken...
12. ### Investigating Lagrangians and Constraints for Tension Calculation

I had used the same constraint as the solution manual says. So my two Lagrangian would be ##L_1=\frac{1}{2}m_A\dot{x_A}^2+\frac{1}{2}m_B\dot{x_B}^2+\frac{1}{2}m_C\dot{x_C}^2+m_Cgx_C+T(x_A+x_B+2x_C-c)## whereas c is just a constant. Of course, I have to write my Lagrangian using constraints...
13. ### I Constraints on Lorentz Velocity Transformation

1. The 2nd line on the 3rd page of your notes, you have x=ct and x'=ct', thus ux=dx/dt and ux'= dx'/dt' =c according to Einstein's assumptiuon. 2. But near the end of the last page, you wrote dx'/dt' = (ux -v)/(1-vux/c2) . Compare with 1. This equation can be valid only for ux=c and...
14. ### Constraints on a hoop rolling on a cylinder

My question is about the contraint we need to use to solve this problem. The answer to the question use the following constraint: $$(r+R)\theta = r\phi$$ Where $\theta$ is angle from the radius of the fixed cylinder to, say, the vertical axis. And $\phi$ is the angle that the rolling cylinder...
15. ### Classical Exploring Gauss Principle of Less Constraints & Gibbs-Apell Equations

In the usual literature about analytical mechanics, I find very little about the Gauss principle of less constraints and the Gibbs-Apell equations. I think the only treatment I've seen on Gauss is given In Lanczos's The variational principle of mechanics". So, I'm looking for introductory and...
16. ### Find the minimum perimeter of a triangle with these constraints

My attempt: $$|\vec{BA} -\lambda \vec {BC}| \geq 2|\vec {BC}|$$ $$|\vec {BA}|^2 -2 \lambda (\vec {BA} \cdot \vec {BC}) +\lambda ^{2} |\vec {BC}|^2 \geq 4|\vec {BC}|^2$$ $$|\vec {BA}|^2 -2 \lambda |\vec {BA}| \cos \theta +\lambda ^{2} \geq 4$$ Am I even on the right track? Thanks
17. ### MHB Static or dynamic integrity constraints

Hey! 😊 Classify the following statements according to static or dynamic integrity constraints, based on the database that is to be created in the thread https://mathhelpboards.com/threads/creating-tables-integrity-constraints.29157/. 1. A customer's discount may not exceed 50 percent. 2. The...
18. ### MHB Creating tables - integrity constraints

Hey! 😊 Create the following tables: 1. Kundenstamm KNr (Primary Key) Name (at most 15 characters) Stadt (at most 10 characters) Land (at most 10 characters) Saldo (Type FLOAT) Rabatt (Type FLOAT) 2. Produkte PNr (greater than 1 and Primary Key) Bez (not NULL, at most 10 characters and...
19. ### A Non holonomic constraints in classical mechanics textbook

I want to learn about the non holonomic case in lagrangian and Hamiltonian mechanics. I've seen that many people say that Goldstein 3rd ed is wrong there. Where should I go to learn it. My mathematics level is at the level Goldstein uses. Please help
20. ### I Virtual displacement is not consistent with constraints

Goldstein 3rd ed says "First consider holonomic constraints. When we derive Lagrange's equation from either Hamilton's or D'Alembert's principle, the holonomic constraint appear in the last step when the variations in the ##q_i## were considered independent of each other. However, the virtual...
21. ### A Equilibrium, entropy, probability - Release of constraints

Hi everyone, I have a fundamental question to the first part of Swendsen's Intro to StatMech and Thermodynamics (book). Suppose we have two isolated systems of volumes ##V_1## and ##V_2##. We distribute ##N## ideal gas particles across the two systems with total energy ##E##. Suppose we bring...
22. ### I Reducing NxN Matrix to 2x2 w/ Physical Constraints

Gonna preface by saying I never thought linear algebra would be a class I would regret not taking so much... but in short the goal is to reduce an arbitrary symmetric NxN system using a set of auxiliary constraint relationships, e.g. for a 3x3 \begin{bmatrix} V_1\\ V_2\\ V_3\\ \end{bmatrix} =...
23. ### Strategy for scheduling with preference constraints

A computer program that was supposed to handle this was bought and tried, and it made a mess of it, so the idea is that perhaps it would be easier to give a strategy to someone (computer savvy, but an amateur, not a professional ) to program (not requiring too much power of the computer on which...
24. ### Classical Looking for a book on non-holonomic constraints with worked examples

A lot of the notes online about non-holonomic constraints are mathematically/theoretically heavy with no real worked examples. I feel like worked examples are a good place to start; it gives me an (incomplete) overview that helps me see the “forest from the trees”. If I can see the “forest from...
25. ### Maximize the function with constraints

I tried parametrizing the domain using spherical coordinates, with theta and phi. I also tried the method of lagrange multipliers, but the substitutions don't easily result in an easy solution. It requires solving five equations for five variables, and no easy way to isolate variables. I think...
26. ### Constraints in Rotation Matrix

In Rigid body rotation, we need only 3 parameters to make a body rotate in any orientation. So to define a rotation matrix in 3d space we only need 3 parameters and we must have 6 constraint equation (6+3=9 no of elements in rotation matrix) My doubt is if orthogonality conditions...
27. ### I Constraints on BSM from EDM measurements

Hello! I see that the experiments looking for electric dipole moment (EDM) of fundamental particles (especially for the electron) are able to place constraints on new BSM particles with masses of around 10 TeV or even more, in a model independent way i.e. we just need to measure a non-zero...
28. ### I Metric Tensor: Symmetry & Other Constraints

Aside from being symmetric, are there any other mathematical constraints on the metric?
29. ### Equations of motion of a system with non holonomic constraints

Hello, I have a system with 2 degrees of freedom with 2 non-holonomic constrains that can be expressed by:##A_1 dq_1 +Cdq_3 + Ddq_4 = 0## ##A_2 dq_1 + Bdq_2 = 0##Being ##q_1, q_2, q_3## and ##q_4## four generalized coordinates that can describe the movement of the system. And ##A_1, A_2, B...
30. ### Multiple constraints: Connecting rods for high performance engines

Summary:: How to combine two formulas to find the material index Attached is the problem I am having trouble understanding. I have been able to do the first two combinations by transposing for A in the mass equation then subsituting into the stress equation. The next combinations (in red box)...
31. ### A Extending Hamilton's principle to systems with constraints

I'm working my way through Goldstein's Classical Mechanics and have followed the arguments until section 2.4 (Extending Hamilton's Principle to Systems with Constraints). In the second paragraph, Goldstein states that "When we derive Lagrange's equations from either Hamilton's or D'Alembert's...
32. ### Lagrangian equations with other kinds of constraints

When a constraint is expressed as F(x)=0, I am quite comfortable in putting such constraints into the Lagrangian. Just add the function with an undetermined multiplier, then treat the multiplier as an additional coordinate, and proceed as before. ##L = T - V + \lambda F ## For example, you...
33. ### Biomechanical constraints on vagile autotrophs

On Earth, there are no vagile autotrophs. Microalgae are planctic, but lack ability for complex, active movement. Bigger plants have propargules that need to move - but only do so passively, not actively. Many animals are sessile, with vagile larvae - but not autotrophs. Very few, like corals...
34. ### I Galilean Invariance and constraints on Forces.

Let's say we have a system of two point particles that can interact with each other by forces that are position and velocity dependent. The forces might or might not be derivable from a generalized potential. Assuming Isotropy of space and homogeneity of space and time, what are the constraints...
35. ### I Applying a constraint in the calculus of variations

I have an analytical function F of the discrete variables ni, which are natural numbers. I also know that the sum of all ni is constant and equal to N. N also appears explicitly in F, but F is not a function of N. F exists in a coordinate system given by the ni only. Should I carry out the...
36. ### Constraint relation in a pulley spring system

Homework Statement How to apply constraints in the system to get a relationship between the displacements of block of mass m and pulley of mass M.? Homework Equations ∑T.a= 0 The Attempt at a Solution Assuming tension in both strings to be T . -T × a1 ( for the block) + 2T × a2 ( for the...
37. ### I Variation principle - holonomic constraints

in order to extend hamilton's principle to include holonomic constraints, out lecturer did the following : when we are under constraints, we cannot consider the variations of the coordinates as independent of each other. we know that the constraint equations are fa = 0. we can multiply each...
38. ### I GW170817: Limits on D>4 Spacetime Dimensions

I found this article about a paper on https://phys.org/news/2018-09-gravitational-dose-reality-extra-dimensions.html which I find interesting in his own respect, but especially for the fact that the GW experiments can actually reveal insights and that good old GR/SR is again at least supported...
39. ### Why do we need the Lagrangian formulation of Mechanics?

These images have been taken from Goldstein, Classical Mechanics. Why do we need Lagrangian formulation of mechanics when we already have Newtonian formulation of mechanics? Newtonian formulation of mechanics demands us to solve the equation of motion given by equation 1. 19. for this we need...
40. ### B Known properties and constraints of dark matter

I'm wondering what properties of Dark Matter are known for certain. Such as: How much does it collide with itself, and with ordinary matter? What does ##\sigma/m\le 7cm^2/g## mean (and why is the estimate rising in more recent studies)? How does it compare with ordinary matter, e.g. liquid...
41. ### Disabling Constraints in SQL server?

Hi, I want to practice queries that require having repeated records. This is not allowed by default since it violates uniqueness constraint. Do I just use an alter table drop unique constraint for this?
42. ### A Pontryagin minimum principle with control constraints

Hi, I am trying to solve a control problem where I have to minimize the fuel consumption of a vehicle: $$J=\int_{0}^{T} L(x(t), u(t),t) + g(x(T),T)dt$$ ##L(u(t),v(t))=\sum\limits_{i,j=0}^{2} K_{i,j} u(t)^i v(t)^j ## is convex (quadratic) and the term ##g(x(T),T)## is to have a constraint in the...
43. ### I FRW metric derivation: constraints from isotropic and homoge

I don't understand the reasoning for any of the three constraints imposed. why would ##dtdx^i## terms indicate a preferred direction? what if there was identical terms for each ##x^i## would there still be a specified or preferred direction? (or is it that in this case we could rename ##t## to...
44. ### What are the constraints on research progress?

While thinking of my previous thread here: https://www.physicsforums.com/threads/how-is-new-math-researched.935167/ I've come up with another general question on research. My airbnb guest is obviously brilliant and goes to a big and generally respected university of Georgia Tech. I got to...
45. ### Interpret constraints on scattering amplitude

Homework Statement For the following theory: ##\mathcal{L}=\frac{1}{2}[(\partial \phi)^2-m^2\phi^2+(\partial\Phi)^2-M^2\Phi^2]+g\phi^2 \Phi^2## Compute s-channel amplitude for process ##\phi\phi \rightarrow \phi\phi##. Interpret result for ##M>2m##. Homework Equations Scattering amplitude...
46. ### Integer programming model (alternating constraints)

Homework Statement Formulate as a mixed integer programming problem but do not solve. Maximize ##x_1 + x_2## subject to ##2x_1 + 3x_2 \le 12## or {##3x_1 + 4x_2 \le 24## and ##-x_1 + x_2 \ge 1##} ##x_1, x_2 \ge 0## Homework EquationsThe Attempt at a Solution if the first constraint is met, we...
47. ### Knapsack problem constraints help

Homework Statement 8 (a) (The Knapsack Problem) A backpacker's knapsack has a volume of V in.^3 and can hold up to W lb of gear. The backpacker has a choice of ##n## items to carry in it, with the ##i##th item requiring ##a_i## in.^3 of space, weighing ##w_i## lb, and providing ##c_i## units of...
48. ### Three Pulley - two masses system.

Homework Statement The given system is released from rest. Assuming no friction, mass-less pulleys and ideal strings; calculate the accelerations of the pulleys. Homework Equations Constraint equations. F=ma The Attempt at a Solution Taking the above assumptions; And applying constraints...
49. ### I Values of Lagrange multipliers when adding new constraints

Say we have a Lagrange function with one multiplier a times a constrain. I minimize and solve the system to find a. I now add another constrain to the same system multiplied by the constant b. Is the value of a the same or can it change?
50. ### A Falsifications and constraints due to GW measurements

This thread is to serve as - a collection of theories that have been falsified by and/or have had new constrained placed on them by the ongoing gravitational wave measurements. - a place to discuss the further constraining/falsifying of still existing models using GW data. I'll start by posting...