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 ...
1 vote
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 ...
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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 ...
• 45
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 ...
• 2,075
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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. ...
• 741
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 ...
• 9,947
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 ...
• 111
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 ...
• 1,052
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 ...
• 1,350
1 vote
163 views

• 323
1 vote