All Questions
78
questions
2
votes
2
answers
3k
views
Is the potential term in a Lagrangian velocity-dependent?
I know that the Lagrangian of a system has to be dependent on the coordinate (as the type of potential in it is dependent on the coordinate) and on velocity and time (per KE and PE, respectively). ...
- 79
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 ...
1
vote
1
answer
258
views
Feynman Lecture Principle of Least Action: Glossed over Taylor expansion?
His initial one dimensional derivation of Newton's Second Law using the Principle of Least Action, I believe is fairly concise and easy to read. However, I did get hung up on his use of the Taylor ...
- 802
3
votes
4
answers
2k
views
Generalized force arising from a velocity-dependent potential
On slide 16 of this presentation it is stated without proof that given a velocity-dependent potential $U(q,\dot q, t)$, the associated generalized force is $$Q_j = - \frac{\partial U}{\partial q^j} + \...
- 1,826
2
votes
2
answers
178
views
Susskind & Hrabovsky: For any system $F_i=-\partial_{i}V$?
In the following $\left\{x\right\}$ means a configuration point in $3N\text{-dimensional}$ configuration space. Each $x_i$ represents one coordinate of one particle of the system of $N$ particles.
...
- 2,015
4
votes
1
answer
385
views
Does the negative sign in the Lagrangian $L=T-V$ relate to the $(+,-,-,-)$ Minkowski signature of relativity?
I've read many derivations of the Euler-Lagrange equation, but this is more of a physics-philosophical point.
Kinetic energy $T$ involves time derivatives, while potential involves spatial location. ...
- 568
2
votes
1
answer
125
views
Potential energy and conservation law
I'm preparing for my masters entrance exam on pure mathematics (thought some problems are devoted to classical/lagrangian mechanics). I would be grateful to clarify some basics regarding the ...
- 123
1
vote
0
answers
527
views
Particle in electromagnetic field Lagrangian
Given the two definitions of $\vec E$ and $\vec B$ by scalar potential $\phi$ and vector potential $\vec A$:
$$\vec B=\vec \nabla \times \vec A$$
$$\vec E=-\vec \nabla \phi -\frac 1 c\frac {\partial \...
- 141
1
vote
3
answers
320
views
Lagrangian of two balls in space
I know the Lagrangian of a free ball with mass $m$ is $$\mathcal{L}=\frac{1}{2}m\dot{\vec{x}}^2.$$
How can I produce a Lagrangian with two interacting balls with masses $m_1,m_2$ and radii $r_1,r_2$? ...
- 229
1
vote
1
answer
548
views
How can a Lagrangian containing no potential energy be used to describe motion of a pendulum?
I'm reading Leonard Susskind's Theoretical Minimum and I got confused with the task number 5 in the lecture 6. The task is given after considering a particle with no forces acting on it. Its position ...
- 15
0
votes
2
answers
360
views
Lagrangian & Newtonian Force
It is written in standard textbooks on classical mechanics that, advantage of Lagrangian equations is that nowhere do enter statement regarding (Newtonian) force.
e.g. To find Lagrangian $L = T - U$,...
- 1,034
0
votes
2
answers
797
views
What is the implication of using the Lagrangian to find the equation of motion?
Consider a particle executing 1D simple harmonic oscillation. To formulate Lagrangian, we use $V=(1/2)kx^2$ for potential energy, and via Euler-Lagrange's equation we formulate the equation of motion $...
- 53
2
votes
0
answers
386
views
Definition of effective potential energy in two-body central-force problems [duplicate]
The Lagrangian of the two-body problem in terms of polar coordinate $r$ and $\phi$ is
$$\mathcal{L}=\frac{1}{2} \mu (\dot{r}^2+r^2\dot{\phi}^2)-U(r)$$
And the Lagrange equation corresponding to $\phi$ ...
- 103
1
vote
1
answer
190
views
Why does mechanical equilibrium depend only on potential energy?
As far as I understand, for a system to be considered in equilibrium, the sum of the forces that is applied to it must be $0$:
$\vec F = 0$
which is
$\partial \frac{E_p}{\partial x}\bigg\rvert _{x=...
- 113
1
vote
0
answers
52
views
Why should the potential of a non-relativistic isolated system be velocity independent?
The lagrangian function of an non-relativistic isolated system of point masses is
$$L=\sum_i\frac{m_i}{2}\dot{\vec r}_i^2-V,$$
where the potential function $V$ represents all interactions.
If we ...
- 17.8k