All Questions
Tagged with classical-mechanics coordinate-systems
417
questions
1
vote
1
answer
51
views
Invertibility between generalized and actual coordinates
Chapter $1$, page $13$ of Classical Mechanics by Goldstein ($2^{nd}$ edition), he states the following after defining a transformation equation:
"It is always assumed that one can transform back ...
0
votes
2
answers
78
views
Generalized momentum
I am studying Hamiltonian Mechanics and I was questioning about some laws of conservation:
in an isolate system, the Lagrangian $\mathcal{L}=\mathcal{L}(q,\dot q)$ is a function of the generalized ...
- 59
1
vote
1
answer
52
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 ...
- 7,388
0
votes
0
answers
14
views
Position and displacement vector in Arc coordinate system
In Arc coordinate system the position of the particle is given by the length of the path(which is pre-determined and may also be curved) that it has travelled so how can we write it's position vector ...
- 51
1
vote
1
answer
68
views
Generating function condition not satisfied?
We want to find a generating function $S(q_i,P_i,t)$ such that we get the best possible canonical transformations. So it must satisfy the Hamilton-Jacobi equation:
$$H(q_i,\frac{\partial S}{\partial ...
- 267
1
vote
3
answers
148
views
Why $q,p,Q,P$ are Independent Variables when Using Generating Functions?
In Hamiltonian formalism, specifically generating functions, why do the variables $q, p, Q, P$ are treated as independent when finding the equations that arise from the generating function?
I ...
- 150
0
votes
1
answer
66
views
Regarding Poission structure of Hamiltonian phase space
Why exactly do we need $$ \{q^i,p_j\}=\delta^i_j,$$ where $\delta^i_j$ is Kronecker delta and $\{\cdot,\cdot\}$ is the Poisson bracket? What happens to the phase space structure if these fundamental ...
0
votes
2
answers
74
views
Energy in different coordinates in central force motion
With reference to central force, we see that K.E has 2 terms in 2D cartesian cordinate but just 1 term in polar coordinates and potential energy has 1 term in cartesian but 2 terms in polar.
Basically ...
- 367
4
votes
3
answers
145
views
Analyzing uniform circular motion with Lagrangian mechanics
Consider swinging a ball around a center via uniform circular motion. The centripetal acceleration is provided by the tension of a rope. Now, is this force a constraint force? If it is, since it is ...
- 143
1
vote
2
answers
211
views
Question about canonical transformation and generating functions
In Goldsteins' Mechanics, page 371 (relevant part appears below), it follows from what he states in the first yellow part that the equations of transformation:
$$Q = Q(q, p,t), \quad P = P(q, p,t)\tag{...
- 150
3
votes
0
answers
62
views
Canonical Transformations and Poisson Brackets - Sufficient and Necessary Condition for Canonical Transformation [duplicate]
I am currently taking analytical mechanics,
My professor directed us in the lecture to Hand and Finch, problem 6.9, to prove ourselves (as he didn't have time) that the equation $$[Q,P]_{Q,P} = [Q(q,p)...
- 150
0
votes
1
answer
74
views
Derivation of lagrange equation in classical mechanics
I'm currently working on classical mechanics and I am stuck in a part of the derivation of the lagrange equation with generalized coordinates. I just cant figure it out and don't know if it's just ...
- 41
0
votes
1
answer
44
views
How to choses coordinate systems and change between them? [closed]
I understand that when choosing a system for the problem that interests me I need to consider all the things that effect what I want to calculate and try to pick the thing that fits my interests the ...
- 33
1
vote
1
answer
49
views
How do physicists determine where to place the world or inertial frame when describing the equation of motion of an object?
For example, I have a pendulum as shown in the diagram above. I would like to write down its equation of motion. To do this, I must define a world frame (or inertial frame, or origin).
But this is ...
- 320
1
vote
3
answers
102
views
Can (extended) canonical transformation involve change of time?
A map from $(q,p)$ to $(Q,P)$ is called an extended canonical transformation if it satisfies
$$
\lambda(pdq-H(q,p,t)dt)-(PdQ-K(Q,P,t)dt)=dF
$$
Here, to include the change of $t$, let us use
$$
\lambda(...
- 145