Linked Questions

3 votes
2 answers
141 views

How to prove that $ \delta \frac{dq_i}{dt} = \frac{d \delta q_i}{dt} $? [duplicate]

During the proof of least action principle my prof used the equation $ \delta \frac{dx}{dt} = \frac{d \delta x}{dt} $. We were not proved this equality. I was curious to know why this is true so I ...
QuantumOscillator's user avatar
1 vote
0 answers
119 views

When and why is $\frac{d}{dt}\delta q^{i}=\delta \frac{dq^{i}}{dt}$ true? [duplicate]

Apparently my question is different from Lagrangian Mechanics - Commutativity Rule $\frac{d}{dt}δq=δ\frac{dq}{dt}$. I hadn't noticed because the answer given in the comments to this question was ...
Steven Thomas Hatton's user avatar
0 votes
0 answers
63 views

Deriving Klein-Gordon equation from Euler-Lagrangian equation: Taking partial derivative inside [duplicate]

Lagrangian for Klein-Gordon equation is given by $$L=\frac{1}{2}\partial_\mu \phi \partial^\mu\phi - m^2\phi^2/2.$$ To derive Klein-Gordon Equation I have to Compute derivative in Euler-Lagrange ...
Vivek's user avatar
  • 45
144 votes
7 answers
17k views

Calculus of variations -- how does it make sense to vary the position and the velocity independently?

In the calculus of variations, particularly Lagrangian mechanics, people often say we vary the position and the velocity independently. But velocity is the derivative of position, so how can you treat ...
grizzly adam's user avatar
  • 2,075
40 votes
7 answers
10k 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. ...
Chin Yeh's user avatar
  • 741
28 votes
6 answers
8k views

Noether Theorem and Energy conservation in classical mechanics

I have a problem deriving the conservation of energy from time translation invariance. The invariance of the Lagrangian under infinitesimal time displacements $t \rightarrow t' = t + \epsilon$ can be ...
jak's user avatar
  • 9,947
11 votes
1 answer
3k views

Hamilton's principle with nonholonomic constraints in Goldstein

I am studying from Goldstein's Classical Mechanics, 3rd intl' edition, 2013. In section 2.4, he discussed Hamilton's principle with nonholonomic constraints. The constraints can be written in the form ...
radi's user avatar
  • 111
2 votes
3 answers
482 views

Hamilton's Principle - achieving Hamilton equations

Consider the action function: $$\mathcal{S}(t)=\int_{t_1}^{t_2}\mathcal{L}(q_i,\dot{q_i},t) dt$$ where $\mathcal{L}$ is the Lagrangian of the system. The Hamiltonian is defined by the following ...
Élio Pereira's user avatar
0 votes
1 answer
439 views

Least action principle : is $ \delta \frac{dx}{dt} = \frac{d \delta x}{dt} $ always true?

(Just some recalls) We have an action on which we want to apply Least action principle. $$ S=\int_{t_i}^{t_f} L(q,\dot{q},t)dt$$ We assume that $t \mapsto q(t)$ is the function that will extremise ...
StarBucK's user avatar
  • 1,350
1 vote
1 answer
163 views

Why does the path modification ($\delta$) commute with the time derivative in the derivation of the Euler-Lagrange equation?

When deriving the Euler-Lagrange equation my notes bring the $\delta$ into the action integral, which is fine, which gives $$\delta S=\int_{t_0}^{t_1}dt \frac{\partial \mathcal L}{\partial q_i}\delta ...
Charlie's user avatar
  • 6,955
1 vote
1 answer
143 views

Converse of the Lagrangian form-invariance

The form-invariance of the Lagrange equations implies the existence of a function $\ A( q_k, t)$ so that $\ \begin{equation} L' (q_k, v_k, t) -L(q_k, v_k, t) = \frac d {dt} A( q_k, t) \end{equation}...
carllacan's user avatar
  • 570
1 vote
1 answer
136 views

Variational Principle for Relativistic Action

I'm going through p. 27 in Landau & Lifshitz Classical Field Theory (vol 2), and I'm confused as to why only the contravariant part of the proper time is varied? They start with $$\delta S=-mc\...
Redcrazyguy's user avatar
1 vote
1 answer
120 views

Reasoning behind $\delta \dot q = \frac{d}{dt} \delta q$ in deriving E-L equations [duplicate]

Consider a Lagrangian $L(q, \dot{q}, t)$ for a single particle. The variation of the Lagrangian is given by: $$\delta L= \frac{\partial L}{\partial q}\delta q + \frac{\partial L}{\partial \dot q}\...
zack1123581321's user avatar
2 votes
0 answers
122 views

Interchanging of variation and integration operator for holonomic systems

Meirovitch says in his "Principles and Techniques of Vibrations" (1997) on p.85: In the case of holonomic systems, the variation and integration processes are interchangeable (...) which means ...
Max Herrmann's user avatar
1 vote
0 answers
45 views

Why $\delta (\partial_{\mu} \phi_{i}) = \partial_{\mu} (\delta \phi_{i})$ on field theory?

While one are using Hamilton Principle on density Lagrangian, he may find this $$\delta (\partial_{\mu} \phi_{i}) = \partial_{\mu} (\delta \phi_{i})$$ Why this statement are true?
evmiz's user avatar
  • 43