Linked Questions

12
votes
1answer
4k views

How do non-conservative forces affect Lagrange equations?

If we have a system and we know all the degrees of freedom, we can find the Lagrangian of the dynamical system. What happens if we apply some non-conservative forces in the system? I mean how to deal ...
3
votes
3answers
943 views

Hamilton's principle and virtual work by constraint forces

I have a question about the following pages(pg 47 and 48) from Goldstein's "Classical Mechanics" I do not understand how (2.34) shows that the virtual work done by forces of constraint is zero. How ...
3
votes
1answer
370 views

Holonomic constraints and degrees of freedom

Wikipedia and other sources define holonomic constraints as a function $$ f(\vec{r}_1, \ldots, \vec{r}_N, t) \equiv 0, $$ and says the number of degrees of freedom in a system is reduced by the ...
2
votes
1answer
268 views

Can the Lagrange multiplier method be used with non-holonomic constraints?

The confusion for me comes from page 46 of Goldstein, where he says "However, it has been proven that no such varied path can be constructed unless [the differential equations of constraint] are ...
2
votes
2answers
161 views

Conversion of non-holonomic constraints to holonomic

In the case of a disc rolling without slipping, we have a constraint $\dot{x}=a\dot{\theta}$ where $a$ is the radius of the disc. Note that I have considered $x$ and $\theta$ as the generalized ...
1
vote
2answers
203 views

When the constraints are not holonomic, why is it not possible to find such $q_i$s that $\delta q_i$s are independent of each other?

In the book of Classical Mechanics by Goldstein, at page 20, it is given that However, I cannoot understand from what has been presented so far that when the constraints are not holonomics, why is it ...
1
vote
1answer
209 views

Non-holonomic constraints, degree of freedom and generalized coordinates

If a system has $N$ coordinates and $M$ number of holonomic constraints then number of degree of freedom $=N-M$ and generalized coordinates $=N-M$ too. But if there are $k$ non-holonomic constraints ...
2
votes
2answers
126 views

Finding the value of the holonomic constraint forces

So let's say I have a Lagrangian augmented with some holonomic constraints. $$L' = L + \sum_i \lambda_i(t) f_i(q,t).\tag{i}$$ The solutions is the system of differential equations: $$\frac{\partial ...
4
votes
1answer
95 views

How to find an underlying holonomic constraints from a differential constraint?

I have been reading through "The Variational Principles of Mechanics" by Lanczos (if anyone is familiar with this text), and I am currently reading through the section discussing holonomic constraints....
1
vote
2answers
66 views

Lagrange's Equation for Dependent Generalized Coordinates

In learning about Lagrange's Equations, I had always used generalized coordinates that are independent from each other. However, in this post it was mentioned that generalized coordinates can be ...
0
votes
0answers
107 views

Integrable semi-holonomic constraints: more than holonomic constraints?

I have a Lagrangian $L(q,\dot q, t)$ that defines a variational problem, along with a constraint that takes the form $$ df = 0 \tag{1}$$ with $f(q,t)$. I take it is what is called a "semi-holonomic" ...
1
vote
1answer
58 views

On the use of Lagrange multipliers in deriving the Lagrange eqn. in classical mechanics

Can one derive the Lagrange eqn based on the methods of Lagrange multipliers? That is, we need to minimize the action with respect to the trajectory keeping the net energy of the body in motion ...