All Questions
78
questions
41
votes
7
answers
11k
views
Is there a proof from the first principle that the Lagrangian $L = T - V$?
Is there a proof from the first principle that for the Lagrangian $L$,
$$L = T\text{(kinetic energy)} - V\text{(potential energy)}$$
in classical mechanics? Assume that Cartesian coordinates are used. ...
- 761
7
votes
3
answers
1k
views
In equation (3) from lecture 7 in Leonard Susskind’s ‘Classical Mechanics’, should the derivatives be partial?
Here are the equations. ($V$ represents a potential function and $p$ represents momentum.)
$$V(q_1,q_2) = V(aq_1 - bq_2)$$
$$\dot{p}_1 = -aV'(aq_1 - bq_2)$$
$$\dot{p}_2 = +bV'(aq_1 - bq_2)$$
Should ...
7
votes
2
answers
2k
views
Example in motivation for Lagrangian formalism
I started reading Quantum Field Theory for the Gifted Amateur by Lancaster & Blundell, and I have a conceptual question regarding their motivation of the Lagrangian formalism. They start by ...
- 203
4
votes
1
answer
7k
views
What is an effective potential in classical mechanics?
What is an effective potential in classical mechanics? I have read the wikipedia article and David Tong's lectures notes, but I didn't understand how an effective potential simplifies a situation or ...
- 3,455
4
votes
2
answers
1k
views
The "stationary potential energy" condition for static equilibrium in mechanical systems
I've often read that, for a mechanical system which can be described by $n$ generalized coordinates $q_1,...,q_n$, a point $\mathbf{Q}=(Q_1,...,Q_n)$ is a point of equilibrium if and only if the ...
- 4,654
4
votes
2
answers
167
views
Classical Mechanics: Relation between general velocity and general potential function for velocity-dependent potential
How is the general force derived from the general potential for a velocity-dependent potential $U = U(q_j,\dot{q_j})$?
$$Q_j=-\frac{\partial U}{\partial q_j}+ \frac{\mathrm{d}}{\mathrm{dt}}(\frac{\...
- 301
4
votes
1
answer
385
views
Does the negative sign in the Lagrangian $L=T-V$ relate to the $(+,-,-,-)$ Minkowski signature of relativity?
I've read many derivations of the Euler-Lagrange equation, but this is more of a physics-philosophical point.
Kinetic energy $T$ involves time derivatives, while potential involves spatial location. ...
- 568
4
votes
2
answers
1k
views
Internal potential energy and relative distance of the particle
Today, I read a line in Goldstein Classical mechanics and got confused about one line.
To satisfy the strong law of action and reaction, $V_{ij}$ can be a function only of the distance between the ...
- 269
4
votes
1
answer
945
views
speed of sound and the potential energy of an ideal gas; Goldstein derivation
I am looking the derivation of the speed of sound in Goldstein's Classical Mechanics (sec. 11-3, pp. 356-358, 1st ed). In order to write down the Lagrangian, he needs the kinetic and potential ...
- 163
4
votes
0
answers
178
views
Deriving the Lagrangian of a set of interacting particles only from symmetry
In section 5 of Landau and Lifshitz's Mechanics book, they show that the Lagrangian of a free particle must be proportional to its velocity squared, $\mathcal{L} = \alpha\mathbf{v}^2$ using only ...
- 2,157
3
votes
1
answer
9k
views
Lagrangian Equations of Motion, Conservative Forces
I'm new to this topic so please bear with me. Here on wikipedia we have the Lagrangian equations of motion:
$$ \frac{d}{dt}\left(\frac{\partial T}{\partial \dot{q}}\right) - \frac{\partial T}{\...
- 33
3
votes
1
answer
323
views
Why can't $U=b\mathbf{v} \cdot \mathbf{r}$ be considered a potential for the drag force ${\bf F}=-b {\bf v}$?
Consider the function $U(\mathbf{r},\dot{\mathbf{r}})=b\mathbf{r} \cdot \mathbf{v}=bx\dot{x}+by\dot{y}+bz\dot{z}$, where $b$ is the drag coefficient. Why can't this be considered as a potential energy ...
- 1,092
3
votes
1
answer
2k
views
Why is the potential independent of the generalized velocity?
In Goldstein, Classical Mechanics, Chap. 1.4 we derive Lagrange's equations from D'Alembert's Principle. My question is regarding the last part of the derivation, specifically the part where he ...
- 149
3
votes
1
answer
537
views
Doubt in the expression of Lagrangian of a system [duplicate]
There is a problem given in Goldstein's Classical Mechanics Chapter-1 as
20. A particle of mass $\,m\,$ moves in one dimension such that it has the Lagrangian
\begin{equation}
L\boldsymbol{=}\...
- 436
3
votes
2
answers
566
views
Deriving effective potential energy from the Lagrangian of a two-body system [duplicate]
I'm having some issues understanding how the effective potential energy of a two-body system is derived from the Lagrangian of the system. Specifically my issue is with one step...
Suppose we are ...
