All Questions
Tagged with calculus lagrangian-formalism
29 questions
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Why are independent variables treated differently in kinetic energy calculations across problems?
In two different problems involving Lagrangian mechanics, I am confused about how independent variables are treated in the kinetic energy calculations. Specifically, in one case, an independent ...
- 101
6
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3
answers
927
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Conservation of energy in a mechanical system with a discontinuous potential function
I've started reading Landau-Lifshitz Mechanics, and I'm having trouble with the problem at the end of section 7.
A particle of mass $m$ moving with velocity $\mathbf{v}_1$ leaves a half-space in ...
- 63
4
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1
answer
68
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Need help understanding lagrangian applied to mass on spring problem [closed]
I am interested in learning physics, and I was trying to learn about the Lagrangian method for classical mechanics through https://scholar.harvard.edu/files/david-morin/files/cmchap6.pdf, but I ...
- 43
6
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3
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492
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How to linearise on Lagrangian level?
Consider a Lagrangian density
$$\mathcal{L}(\phi, \nabla \phi) = \frac{1}{2} \, g^{\mu \nu} \, \partial_{\mu} \phi \; \partial_{\nu} \phi + V(\phi) \tag{1}$$
The equation of motion (EOM), i.e. the ...
- 695
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3
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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 ...
1
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2
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230
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On Landaus&Lifshitz's derivation of the lagrangian of a free particle [duplicate]
I'm reading the first pages of Landaus&Lifshitz's Mechanics tome. I'm looking for some clarification on the derivation of the Lagrange function for the mechanical system composed of a single free ...
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1
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Derivation of lagrange equation in classical mechanics
I'm currently working on classical mechanics and I am stuck in a part of the derivation of the lagrange equation with generalized coordinates. I just cant figure it out and don't know if it's just ...
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Why is the regular Taylor expansion so similar to the construction of the Hamiltonian from a Lagrangian/Legendre transform?
An example of a first order Taylor expansion of a function with two variables is given by:
$$f\left(x,y\right)=f\left(x_0,y_0\right)+\left(x-x_0\right)\frac{\partial f}{\partial x}+\left(y-y_0\right)\...
- 2,180
1
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2
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Lagrangian total time derivative - continues second-order differential
In the lagrangian, adding total time derivative doesn't change equation of motion.
$$L' = L + \frac{d}{dt}f(q,t).$$
After playing with it, I realize that this is only true if the $f(q,t)$ function has ...
- 535
1
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What does it mean to differentiate a scalar with respect to a vector?
I am reading the special relativity lecture notes that I got from a professor of mine. It says that the Lagrangian is
$$L = \frac{1}{2}m|\dot{\boldsymbol{x}}|^2 - V(\boldsymbol{x}) \tag{1}$$
The notes ...
- 1,254
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1
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48
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Lagrangian for 2 inertial frames where only Speed is different by small amount
In Landau & Liftshitz’s book p.5, they go ahead and writes down lagrangians for 2 different inertial frames. They say that Lagrangian is a function of $v^2$.
So in one frame, we got $L(v^2)$.
In ...
- 535
1
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0
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61
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From Lagrangian to Lagrangian density in discrete mass-springs to continuum wave limit [duplicate]
I am currently reading these Weigand's QFT lecture notes. But already in the beginning I dont understand one step. Given is the energy $E$ of a string of length $L$ of $N$ connected particles that ...
- 131
1
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1
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Deriving smoothing kernels
I'm watching a video on smoothed particle hydrodynamics it just blindly claims that these smoothing kernels are pretty good.
$$W(r-r_b,h)\equiv\dfrac{315}{64\pi h^9}\left(h^2-|r-r_b|^2\right)^3$$
$$\...
- 114
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1
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Lagrange Multipliers
In this Lagrangian (from the paper: https://arxiv.org/abs/1302.0192 - page 4), $\eta, \mu, \nu, \& \lambda$ are lagrange multipliers.
My question is: why do they include $\nu$ and $\lambda$ ...
- 241
1
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1
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Interpretation of Variation Notes
I would like an explanation to how this Lagragian partial derivative was taken (eq. 3). This probably is more suited for the math Stack Exchange, however this is for a physics course which is why I am ...
- 15
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1
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The use of $x_\varepsilon (t) = x(t) + \varepsilon (t)$ and $x_\varepsilon (t) = x(t) + \varepsilon \eta (t)$ in proving Hamilton's principle
The following Wikipedia page uses $x_\varepsilon (t) = x(t) + \varepsilon (t)$ in the proof.
https://en.wikipedia.org/wiki/Hamilton%27s_principle#Mathematical_formulation
But in my mechanics book (by ...
- 485
2
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2
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639
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Is this a Taylor approximation? How is it done? [closed]
I know what a Taylor series involves but you have to know the function; here $\mathcal{L}^*$ is just a function depending on $(v+w)^2$, any kind of function could be inside. How can the below ...
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3
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Showing the equivalence between the chain rule's Leibniz and Lagrange Notations
This may seem more math related but this question crossed my mind as I was reading the derivation of the Euler-Lagrange Equation.
In math, we were introduced to the Lagrange notation of the derivative ...
- 97
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1
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1k
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How does the small angle approximation work for cosine?
In newtonian mechanics equation of motion of a simple pendulum:
$$\ddot{\theta}=\frac{g}{l}\sin\theta$$
And then I approximated for small angles $\sin\theta\simeq\theta$ that yields the equation of ...
- 2,029
1
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1
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555
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Lorentz Invariance of the Euler-Lagrange equation for fields
Given an Lorentz invariant Lagrangian density $L$ of a Lorentz invariant scalar field $\phi$,
How does one show that the following term in the Euler-Lagrange equation is invariant under Lorentz ...
- 651
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Partial Integration and the Levi-Civita Symbol
I'm currently working through the book Heisenberg's Quantum Mechanics (Razavy, 2010), and am reading the chapter on classical mechanics. I'm interested in part of their derivative of a generalized ...
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2
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$\int (f(x+\delta x) - f(x)) dx = \int \left ( \frac{df(x)}{dx} \delta x \right) dx$
From Landau and Lifshitz's Mechanics Vol: 1
$$
\delta S= \int \limits_{t_1}^{t_2} L(q + \delta q, \dot q + \delta \dot q, t)dt - \int \limits_{t_1}^{t_2} L(q, \dot q, t)dt \tag{2.3b}$$
$$\Rightarrow ...
user208739
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Derivation of generalized velocities in Lagrangian mechanics
So I know that: $$r_i = r_i(q_1, q_2,q_3,...., q_n, t)$$
Where $r_i$ represent the position of the $i$th part of a dynamical system and the $q_n$ represent the dynamical variables of the system ($n$ =...
user63248
2
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1
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222
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Real Lagrangian with complex variable
I have a general question concerning real valued Lagrangians that take complex arguments. I have seen in many works of physicists and lecture books where extremal problems are discussed in Lagrangians ...
- 151
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0
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260
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Partial derivative of $v$ w.r.t. $x$ in Lagrangian dynamics [duplicate]
In Lagrangian dynamics, when using the Lagrangian thus:
$$
\frac{d}{dt}(\frac{\partial \mathcal{L} }{\partial \dot{q_j}})-
\frac{\partial \mathcal{L} }{\partial q_j} = 0
$$
often we get terms such ...
- 11
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2
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Expansion in $\epsilon$ and $v^2$ dependence of the Lagrangian - Landau & Lifshitz's Mechanics [duplicate]
On page 4 of Landau & Lifshitz's Mechanics they say
$$L\left({v^\prime}^2\right) = L\left(v^2 + 2\bf{v \cdot} \bf{\epsilon} + \epsilon^2\right).$$ Expanding this expression in powers of $\...
- 11
4
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1
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Landau's derivation of a free particle's kinetic energy- expansion of a function?
I was reading a bit of Landau and Lifshitz's Mechanics the other day and ran into the following part, where the authors are about to derive the kinetic energy of a free particle. They use the fact ...
- 1,603
38
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5
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Equivalence between Hamiltonian and Lagrangian Mechanics
I'm reading a proof about Lagrangian => Hamiltonian and one part of it just doesn't make sense to me.
The Lagrangian is written $L(q, \dot q, t)$, and is convex in $\dot q$, and then the ...
- 559
2
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1
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Expansion of a function [duplicate]
In Landau-Lifschitz, following expansion is given,
We have,
$$L(v'^2)~=~L(v^2+2\textbf{v}\cdot\epsilon+\epsilon ^2)$$
expanding this in powers of $\epsilon$ and neglecting powers of higher order,
$$L(...
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