What is Constraint forces: Definition and 18 Discussions
In computational chemistry, a constraint algorithm is a method for satisfying the Newtonian motion of a rigid body which consists of mass points. A restraint algorithm is used to ensure that the distance between mass points is maintained. The general steps involved are: (i) choose novel unconstrained coordinates (internal coordinates), (ii) introduce explicit constraint forces, (iii) minimize constraint forces implicitly by the technique of Lagrange multipliers or projection methods.
Constraint algorithms are often applied to molecular dynamics simulations. Although such simulations are sometimes performed using internal coordinates that automatically satisfy the bond-length, bond-angle and torsion-angle constraints, simulations may also be performed using explicit or implicit constraint forces for these three constraints. However, explicit constraint forces give rise to inefficiency; more computational power is required to get a trajectory of a given length. Therefore, internal coordinates and implicit-force constraint solvers are generally preferred.
Constraint algorithms achieve computational efficiency by neglecting motion along some degrees of freedom. For instance, in atomistic molecular dynamics, typically the length of covalent bonds to hydrogen are constrained; however, constraint algorithms should not be used if vibrations along these degrees of freedom are important for the phenomenon being studied.
Goldstein 2ed pg 36
So in the case of holonomic constraints we can move back and forth between Hamiltons principle and Lagrange equations given as ##\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{q}_{j}}\right)-\frac{\partial L}{\partial q_{j}}=0##
But the Lagrange equations were...
Consider the following setup
where the bead can glide along the rod without friction, and the rod rotates with a constant angular velocity ##\omega##, and we want to find the constraint force using Lagrange multipliers.
I chose the generalized coordinates ##q=\{r,\varphi\}## and the...
My question is about the general relationship between the constraint functions and the constraint forces, but I found it easier to explain my problem over the example of a double pendulum:
Consider a double pendulum with the generalized coordinates ##q=\{l_1,\theta_1,l_2,\theta_2\}##,:
The...
Suppose I'm considering a system of N particles that are constrained in their possible motions and so there are less that 3N generalized coordinates. Suppose now I perform a virtual displacement on one particle, which due to some constraints might force some other particles to more virtually...
Found a question on another website, I have the exact same question. Please help me
Goldstein says :
I do not understand how (2.34) shows that the virtual work done by forces of constraint is zero. How does the fact that "the same Hamilton's principle holds for both holonomic and...
From what I understand, constraint forces do no work because they are perpendicular to the allowed virtual displacements of the system. However, if you consider an unbalanced Atwood machine, in which both masses are accelerating in opposite directions, you'll find that the tension force of the...
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...
Homework Statement
Under the action of force P the constant acceleration of B is 6 m/s^2 up the incline as in figure. For the instant when B's velocity is 3 m/s up incline, what is the velocity of point C? How do I solve this using constraints?
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Homework Equations
For a pulley system ΣT.v...
Homework Statement
The situation is that of Goldstein's problem 1.21 (or 1.19 in some editions):
"Two mass points of mass m1 and m2 are connected by a string passing through
a hole in a smooth table so that m1 rests on the table and m2 hangs suspended.
Assume m2 moves only in a vertical line."...
I am now reading Lagrange's equations part in Taylor's Classical Mechanics text.
It says:
When a system of interest involves constraint forces, F_cstr, and all the nonconstraint forces are derivable from a potential energy(U), then the Lagrangian for the system L is L = T - U, where U is the...
According to d'Alembert's Principle, the virtual work done by constraint forces must be zero.
I have a few things needing to be clarified. First, as we know from friction, d'Alembert's Principle is not always true (friction usually does work, and is not normal to the constraint surface). On the...
On a rigid body we usually use the formula δL=F*δP to calculate virtual work. My problem is about the force. This kind of force exists only before the contact. If I imagine a movement δP of the constrained body outside ,in the free space, I will have δL≥0 but as soon as P moves the force F...
Suppose you are trying the solve the equation of motion of say a particle constrained to move on a surface f(x\vec{},t)=0. The equation of motion is:
mx\ddot{} = F\vec{} + N\vec{}, where F is an known external force and N is the unknown constraint force.
Now, when you assume that N always...
the fundamental basis of the lagrangian formulation is the fact that the virtual displacement are perpendicular to the constraint forces
so how does one define constraint forces?
is it necessary for the virtual displacement consistent with the given constraints be perpendicular to the...