All Questions
14
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
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
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
3k
views
Charge, velocity-dependent potentials and Lagrangian
Given an electric charge $q$ of mass $m$ moving at a velocity ${\bf v}$ in a region containing both electric field ${\bf E}(t,x,y,z)$ and magnetic field ${\bf B}(t,x,y,z)$ (${\bf B}$ and ${\bf E}$ are ...
- 1,133
2
votes
3
answers
1k
views
Effective potential of a two-body system
This question is with regards to a two body system consisting of two masses that interact via a conservative central force.
In polar coordinates, the Lagrangian can be written as
$$\frac{1}{2}M\dot{...
- 581
4
votes
2
answers
1k
views
Internal potential energy and relative distance of the particle
Today, I read a line in Goldstein Classical mechanics and got confused about one line.
To satisfy the strong law of action and reaction, $V_{ij}$ can be a function only of the distance between the ...
- 269
4
votes
1
answer
7k
views
What is an effective potential in classical mechanics?
What is an effective potential in classical mechanics? I have read the wikipedia article and David Tong's lectures notes, but I didn't understand how an effective potential simplifies a situation or ...
- 3,455
3
votes
1
answer
9k
views
Lagrangian Equations of Motion, Conservative Forces
I'm new to this topic so please bear with me. Here on wikipedia we have the Lagrangian equations of motion:
$$ \frac{d}{dt}\left(\frac{\partial T}{\partial \dot{q}}\right) - \frac{\partial T}{\...
- 33
3
votes
1
answer
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
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 $$Q_j = - \frac{\partial U}{\partial q^j} + \...
- 1,826
3
votes
1
answer
2k
views
Non-conservative system and velocity dependent potentials
I'm studying Lagrangian mechanics, but I'm a little bit upset because when dealing with Lagrange's equations, we mostly consider conservative systems. If the system is non conservative they are very ...
- 751
2
votes
2
answers
3k
views
Is the potential term in a Lagrangian velocity-dependent?
I know that the Lagrangian of a system has to be dependent on the coordinate (as the type of potential in it is dependent on the coordinate) and on velocity and time (per KE and PE, respectively). ...
- 79
1
vote
0
answers
527
views
Particle in electromagnetic field Lagrangian
Given the two definitions of $\vec E$ and $\vec B$ by scalar potential $\phi$ and vector potential $\vec A$:
$$\vec B=\vec \nabla \times \vec A$$
$$\vec E=-\vec \nabla \phi -\frac 1 c\frac {\partial \...
- 141
1
vote
1
answer
1k
views
Force and energy relation: in case of time dependent force
The equivalent problems are also found in Marion problem 7-22, and other formal classical mechanics textbook. Here what i want to know why instructor solution and some websites gives this kinds of ...
- 3,622