All Questions
44
questions
12
votes
2
answers
2k
views
Can the Lagrangian be written as a function of ONLY time?
The lagrangian is always phrased as $L(t,q,\dot{q})$.
If you magically knew the equations $q(t)$ and $\dot{q}(t)$, could the Lagrangian ever be written only as a function of time?
Take freefall for ...
- 484
8
votes
1
answer
2k
views
If the Lagrangian depends explicitly on time then the Hamiltonian is not conserved?
Why is the Hamiltonian not conserved when the Lagrangian has an explicit time dependence? What I mean is that it is very obvious to argue that if the Lagrangian has no an explicit time dependence $L=L(...
- 227
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
5
votes
2
answers
914
views
In a four mass six spring vibration, how is the kinetic energy represented
This is from Hobson, Riley, Bence Mathematical Methods, p 322. A spring system is described as follows (they are floating in air like molecules):
The equilibrium positions of four equal masses M of a ...
4
votes
1
answer
576
views
Sufficient conditions for the energy to be not conserved?
I'm almost embarrased to ask this question because I thought I was by now very confident with classical mechanics.
Someone has stated that given a mechanical system with a Lagrangian $L$ s.t. $\frac{...
- 909
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
1
answer
1k
views
General Form for Kinetic Energy Given Velocity Independent Potential such that $\mathcal{H}=E$
Suppose the potential energy is independent of $\dot{q},$ i.e $\frac{\partial V}{\partial\dot{q}}=0$. What is the most general form of the kinetic energy such that the Hamiltonian is the total energy? ...
- 32
3
votes
1
answer
1k
views
Why is total kinetic energy always equal to the sum of rotational and translational kinetic energies?
My derivation is as follows.
The total KE, $T_r$ for a rigid object purely rotating about an axis with angular velocity $\bf{ω}$ and with the $i$th particle rotating with velocity $ \textbf{v}_{(rot)...
user86425
3
votes
4
answers
660
views
Misunderstanding in deriving Newton’s law from Euler-Lagrange equation
When deriving Newton’s law from Euler-Lagrange equation for a particle, the Lagrangian is defined as the kinetic energy minus the potential energy, but the problem is that the kinetic energy is ...
- 31
3
votes
1
answer
113
views
Gram-Schmidt Orthogonalisation for scalars
I'm reading Chapter 11 (Normal Modes) of Classical Mechanics (5th ed.) by Berkshire and Kibble and came across this on pg. 253:
The kinetic energy in terms in terms of the generalised coordinates is ...
- 1,047
2
votes
1
answer
156
views
Differentiation of the on-shell action with respect to time
From the on-shell action, we derive the following two:
$\frac{\partial S}{\partial t_1} = H(t_1)$,
$\frac{\partial S}{\partial t_2} = -H(t_2)$,
where $H = vp - L$ is the energy function.
I have two ...
- 525
2
votes
1
answer
1k
views
Condition that the Lagrangian energy function $h\equiv\sum_i\frac{\partial L}{\partial\dot q_i}\dot q_i-L$ would be same as the mechanical energy $E$
I'm studying Classical Mechanics by Goldstein. I solved a problem but I have a question.
Pro 2.18
A point mass is constrained to move on a massless hoop of radius a fixed in a vertical plane ...
- 216
2
votes
1
answer
333
views
Lagrangian under time transformation
Given a Lagrangian $$L(q,\dot{q},t)=\sum_{ij}a_{ij}(q)\dot{q}_i\dot{q}_j-V(q_1,q_2,\cdots,q_f)$$show that under a time transformation $t=\lambda T$ ($\lambda$ = constant), the invariance of $\int_1^...
- 592
2
votes
1
answer
145
views
Is it possible to derive the equations of motion from the energy of a system alone, without knowing canonical coordinates or the Lagrangian?
Is it possible to derive the equations of motion from the energy of a system alone, without knowing canonical coordinates or the Lagrangian?
I'm confused about which parts of the fundamental ...
- 21
2
votes
1
answer
1k
views
Difference between the energy and the Hamiltonian in a specific example
The problem is the following:
Consider a particle of mass $m$ confined in a long and thin hollow pipe, which rotates in the $xy$ plane with constant angular velocity $\omega$. The rotation axis ...
- 1,045