All Questions
156
questions
1
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Energy change under point transformation
How do the energy and generalized momenta change under the following
coordinate
transformation $$q= f(Q,t).$$
The new momenta: $$P = \partial L / \partial \dot Q = \partial L / \partial \dot q\times ...
- 980
0
votes
1
answer
781
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Degrees of freedom for Constrained Motion
I'm starting to learn about Degrees of freedom, and the idea of 'constrained motion' seems strange to me, surely any particle with a predefined path is 'constrained' in its motion, We also had ...
- 263
-2
votes
1
answer
61
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Can we take any physical quantity as a generalized co-ordinate in Lagrangian function?
Lagrangian function is a function of generalized co-ordinates $q_1,q_2,....$ & possibly of time $t$.
i.e. $L=L(q_1,q_2,....;t)$
Consider a simple pendulum.
Can I take
$q_1$ = kinetic energy of ...
- 1,034
1
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1
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98
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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
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1
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212
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I'm studying analytical mechanics and it states that it always true that generalized coordinates times generalized forces have the dimension of energy
Since the terms $q$ "generalized coordinates" are not necessarily ‘lengths’, the quantities $Q$ "generalized forces" also do not necessarily have the dimension of a ‘force’. ...
- 11
3
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1
answer
148
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What does Thornton and Marion mean by "validity of Lagrange's equations"?
I am a bit confused about the 2nd statement below from Thornton and Marion 7.4:
It is important to realize that the validity of Lagrange's equations requires
the following two conditions:
The forces ...
- 243
2
votes
2
answers
644
views
Time dependence of generalized coordinates and virtual displacement
The Cartesian coordinates of particles are related to the generalized coordinates via a transformation (for the $x$ component of the $j$-th particle) as:
$$x_j = x_j(q_1, q_2, \ldots, q_N, t)$$
What I ...
- 1,648
3
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1
answer
325
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Intuition about non-invariance of the Hamiltonian in canonical transformation
Suppose $q={\{q_i\}}_{i=1}^n$ is the set of generalized coordinates of a dynamical system. $L(q,\dot q,t)$ is the Lagrangian of the system. Now we make coordinate transformation $Q_i=Q_i(q,t)$. Then ...
- 436
0
votes
2
answers
521
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Decomposing Lagrangian into CM and relative parts with presence of uniform gravitational field
Most problems concerning two-body motion (using Lagrangian methods) often only consider the motion of two particles subject to no external forces. However, the Lagrangian should be decomposable into ...
- 343
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482
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Central force motion and angular cyclic coordinates
(Goldstein 3rd edition pg 72)
After reducing two-body problem to one-body problem
We now restrict ourselves to conservative central forces, where the potential is $V(r)$ function of $r$ only, so that ...
- 1,270
1
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0
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93
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Representation of Holonomic Constraints by independent generalized coordinates
Say we have a system with N particles described by N position vectors: $\{\vec{r_{i}}\};$ $i=1,...N$
Say we have a holonomic constraint: $$f(\{\vec{r_{i}}\},t)=0 \tag{1}$$
Since we have one holonomic ...
- 409
0
votes
1
answer
67
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Dissipation function is homogeneous in $\dot{q}$
We have Rayleigh's dissipation function, defined as
$$\mathcal{F}=\frac{1}{2} \sum_{i}\left(k_{x} v_{i x}^{2}+k_{y} v_{i j}^{2}+k_{z} v_{i z}^{2}\right)$$
Also we have transformation equations to ...
- 1,270
1
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0
answers
27
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How to pick a constraint function for Lagrangian mechanics? [duplicate]
Motivating Example
Consider a system which consists of two masses $m_1$ and $m_2$ at positions $x_1$ and $x_2$ respectively joined together by a rigid rod of negligible mass and length $l$ . We have ...
- 1,590
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66
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Definition of Generalised Coordinates: Confusion with Notation
My analytical mechanics lecturer gave a definition for Generalised Coordinates today, it is as follows:
Generalised Coordinates: Let $(q_1,q_2)\in\Omega\in\mathbb{R}^2$ such that $x=x(q_1,q_2)$ and $...
0
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1
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579
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Independent generalized coordinates are dependent
(This is not about independence of $q$,$\dot q$)
A system has some holonomic constraints. Using them we can have a set of coordinates ${q_i}$. Since any values for these coordinates is possible we say ...
- 1,270