All Questions
30
questions with no upvoted or accepted answers
3
votes
0
answers
130
views
What is the geometric interpretation of a general 'state space' in classical mechanics?
Let $\pmb{q}\in\mathbb{R}^n$ be some n generalized coordinates for the system (say, a double pendulum). Then the 'state space' is often examined using either the 'Lagrangian variables', $(\pmb{q},\dot{...
- 477
2
votes
1
answer
122
views
Independence of generalized coordinates in the derivation of Lagrange equations from d'Alembert's Principle
I am confused by this remark in the derivation of Lagrange equations from d'Alembert's principle in Goldstein:
I am not comfortable that I understand why, at this late stage of the derivation, they ...
- 31
2
votes
2
answers
108
views
How to explain the independence of coordinates from physics aspect and mathmetics aspect?
When I was studying Classical Mechanics, particularly Lagrangian formulation and Hamiltonian formulation. I always wondering how to understand the meaning of independence of parameters used of ...
- 529
2
votes
0
answers
117
views
Choosing coordinates to solve problems using Lagrangian mechanics
I am trying to obtain the equations of motion using the euler-lagrange equation.
First, let $x$ be the distance of disc R from the wall. Let $y_p$ and $y_q$ be the distance of disc P and disc Q from ...
- 21
2
votes
1
answer
245
views
How to calculate generalized force $Q_\phi$ with d'Alembert's principle?
The related post was found here Lagrangian formalism application on a particle falling system with air resistance and also Wikipedia's definition on generalized force. Essential
$$\frac{d}{dt}\frac{\...
- 3,256
1
vote
1
answer
56
views
Confusing Goldstein Statement about Magnitude of the Lagrangian
On page 345 of Goldstein's Classical Mechanics 3rd Ed., he writes:
...the Hamiltonian is dependent both in magnitude and in functional form upon the initial choice of generalized coordinates. For the ...
- 7,398
1
vote
1
answer
72
views
Requirement of Holonomic Constraints for Deriving Lagrange Equations
While deriving the Lagrange equations from d'Alembert's principle, we get from $$\displaystyle\sum_i(m\ddot x_i-F_i)\delta x_i=0\tag{1}$$ to $$\displaystyle\sum_k (\frac {\partial\mathcal L}{\partial\ ...
- 193
1
vote
1
answer
255
views
Canonical equations of motion
The Hamiltonian is obtained as the Legendre transform of the Lagrangian:
\begin{equation}
H(q,p,t)=\sum_i p_i \dot{q_i} - L(q,\dot{q},t)\tag{1}
\end{equation}
If the Hamiltonian is expressed in ...
- 860
1
vote
0
answers
44
views
Is there always a transformation between canonical variables?
Let us suppose that a for a given monogenic and holonomic system we can construct two different collection of canonical variables $\{\underline{q}, \underline{p}\}$ and $\{\underline{Q}, \underline{P}\...
- 402
1
vote
1
answer
98
views
Why is this the requirement for invertibility within the context of canonical transformations in mechanics?
I'm reading "Analytical Mechanics", by Hand and Finch. In page 210, there's the following statement, regarding the generating function $F$ for some lagrangian such that $L'=\lambda L-\frac{\...
- 1,775
1
vote
0
answers
93
views
Representation of Holonomic Constraints by independent generalized coordinates
Say we have a system with N particles described by N position vectors: $\{\vec{r_{i}}\};$ $i=1,...N$
Say we have a holonomic constraint: $$f(\{\vec{r_{i}}\},t)=0 \tag{1}$$
Since we have one holonomic ...
- 409
1
vote
2
answers
304
views
About virtual displacement
Thornton Marion
The varied path represented by $\delta y$ can be thought of physically as a virtual displacement from the actual path consistent with all the forces and constraints (see Figure above)....
- 1,270
1
vote
1
answer
101
views
What is meant by equivalent directions of space?
My Mechanics professor assigned me this question but I needed a bit of clarification on a given condition, the question is as follows:
Compute the Lagrangian for a free particle in an $(n+m)$-...
1
vote
2
answers
327
views
How do I check if a transformation is a point transformation?
In Lagrangian mechanics, I came across the notion of a point transformation which leaves the Lagrangian invariant. Normally it is denoted as follows.
$$Q = Q(q,t).$$
Now, unlike in the case of a ...
- 593
0
votes
1
answer
95
views
Virtual work of constraints in Hamilton‘s principle
Goldstein 2ed pg 36
So in the case of holonomic constraints we can move back and forth between Hamilton's principle and Lagrange equations given as $$\frac{d}{d t}\left(\frac{\partial L}{\partial \...
- 1,270