All Questions
56
questions
148
votes
8
answers
18k
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 ...
- 2,155
57
votes
7
answers
9k
views
Why isn't the Euler-Lagrange equation trivial?
The Euler-Lagrange equation gives the equations of motion of a system with Lagrangian $L$. Let $q^\alpha$ represent the generalized coordinates of a configuration manifold, $t$ represent time. The ...
- 1,826
25
votes
2
answers
2k
views
Lagrangian Mechanics - Commutativity Rule $\frac{d}{dt}\delta q=\delta \frac{dq}{dt} $
I am reading about Lagrangian mechanics.
At some point the difference between the temporal derivative of a variation and variation of the temporal derivative is discussed.
The fact that the two are ...
- 281
7
votes
3
answers
1k
views
In equation (3) from lecture 7 in Leonard Susskind’s ‘Classical Mechanics’, should the derivatives be partial?
Here are the equations. ($V$ represents a potential function and $p$ represents momentum.)
$$V(q_1,q_2) = V(aq_1 - bq_2)$$
$$\dot{p}_1 = -aV'(aq_1 - bq_2)$$
$$\dot{p}_2 = +bV'(aq_1 - bq_2)$$
Should ...
4
votes
4
answers
260
views
Variation of a function
I'm studying calculus of variations and Lagrangian mechanics and i don't understand something about the variational operator
Let's say for example that i got a Lagrangian $L [x(t), \dot{x}(t), t] $ ...
- 309
4
votes
2
answers
1k
views
Trouble with Landau & Lifshitz's expansion of the Lagrangian with respect to $\epsilon$ and $v$ [duplicate]
Hello I have a quick question on what I have been reading in Landau & Lifshitz's book on classical mechanics. I am in the very beginning of the book and I am having trouble with his derivation on ...
- 43
4
votes
1
answer
2k
views
How do total time derivatives of partial derivatives of functions work?
Say im trying to prove $\frac{\partial \dot{T}}{\partial \dot{q}^i} - 2\frac{\partial {T}}{\partial {q^i}} = - \frac{\partial {V}}{\partial {q^i}}$ from the Lagrangian equation: $L = T - V$, and the ...
- 75
3
votes
2
answers
148
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 ...
2
votes
6
answers
239
views
Lagrangian - How can we differentiate with respect to time if $v$ not a function of time?
In the Lagrangian itself, we know that $v$ and $q$ don't depend on $t$ (i.e - they are not functions of $t$ - i.e., $L(q,v,t)$ is a state function.)
Imagine $L = \frac{1}{2}mv^2 - mgq$
Euler-Lagrange ...
- 525
2
votes
1
answer
615
views
Proof that the Euler-Lagrange equations hold in any set of coordinates if they hold in one
This is a question about a specific proof presented in the book Introduction to Classical Mechanics by David Morin. I have highlighted the relevant portion in the picture below.
In the remark, he ...
anon
2
votes
2
answers
188
views
Take derivative to a cross product of two vectors with respect to the position vector [closed]
I'm doing classical mechanics about Lagrange formulation and confused about something about vector differentiation.The Lagrangian is given:
$$\mathcal{L}=\frac{m}{2}(\dot{\vec{R}}+\vec{\Omega} \times \...
- 103
2
votes
1
answer
225
views
Is Goldstein's matrix formalism to Hamiltonian mechanics necessary? [closed]
I am trying to see whether the matrix formalism of the Hamiltonian formalism (used in Goldstein's textbook) is truly necessary to solve problem in this framework.
It appears so based on the problem I'...
- 1,031
2
votes
1
answer
100
views
Confusion regarding the time derivative term in Lagrange's equation
I am solving a pendulum attached to a cart problem. Without going into unnecessary details, the generalised coordinates are chosen to be $x$ and $\theta$. The kinetic energy of the system contains a ...
- 192
2
votes
1
answer
3k
views
Lagrange equations in a conservative system, understanding $\nabla_i$
For a system of multiple particles with conservative forces: $\mathbf{F}_i = - \nabla_i V$, with $V \equiv V(\mathbf{r}_1,\dots,\mathbf{r}_N)$ the potential in function of the position of the $N$ ...
- 265
2
votes
4
answers
1k
views
The definition of the hamiltonian in lagrangian mechanics
So going through the "Analytical Mechanics by Hand and Finch". In section 1.10 of the book, the Hamiltonian $H$ is defined as: $$H = \sum_k{\dot{q_k}\frac{\partial L}{\partial \dot{q_k}} -L}.\tag{1.65}...
user63248