# Tagged Questions

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### Why does Principle for least action hold for classical fields [duplicate]

Let $\mathscr L (\phi(\mathbf x), \partial \phi(\mathbf x))$ denote the Lagrangian density of field $\phi(\mathbf x)$. Then then actual value of the field $\phi(\mathbf x)$ can be computed from the ...
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### Internal potential energy and relative distance of the particle

Today, I read a line in Goldstein Classical mechanics and got confused about one line. To satisfy the strong law of action and reaction, $V_{ij}$ can be a function only of the distance between ...
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### Derivation of Lagrangian?

I know that the Lagrangian $L$ is defined to be $T-V$, i.e. the difference between kinetic energy and potential energy. Also the Action $S$ is defined to be $\int Ldx$ and from this we can derive ...
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### Translation symmetry and the non-conserved momentum in Viscous fluids

Even though a viscous fluid has a translation symmetry (invariance) for its Lagrangian , it still 'waste' Linear momentum. How come ?, isn't the rule that every symmetry yields a conservation law ?
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### Lagrangian to Hamiltonian

I'm having some problems with an assignment where I have to state the Hamiltonian from the kinetic energy $T$ and potential energy $U$. These are as follows: ...
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### Are the Hamiltonian and Lagrangian always convex functions?

The Hamiltonian and Lagrangian are related by a Legendre transform: $$H(\mathbf{q}, \mathbf{p}, t) = \sum_i \dot q_i p_i - \mathcal{L}(\mathbf{q}, \mathbf{\dot q}, t).$$ For this to be a Legendre ...
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### How can I tell that circular motion is a solution for a particle confined to the surface of a cone?

I'm working on a problem where a particle of mass $m$ is confined to the surface of an inverted half cone (and is circling downwards due to gravity), with the cone's half angle $\alpha$. I chose to ...
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### Why friction force is force of constraint?

My understanding about constraint force is that it is a force which limits the geometry of particle's motion. For example, situations such as the particle trapped in a track or limited in domain can ...
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### How does isotropy of free space imply $L(v^2)$ for a free particle? [duplicate]

From Mechanics; Landau and Lifshitz, it's stated on page 5: Since space is isotropic, the Lagrangian must also be indpendent of the direction of $\mathbf{v}$, and is therfore a function only of ...
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### $\cos^{2}(\phi)$ in the kinetic energy term of the Lagrangian is one?

I'm doing some homework in Classical Mechanics, and is about to write out the Lagrangian of a system. But, when I check the answer from my teacher, something is missing. The kinetic energy I'm using ...
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### From Lagrangian to equations of motion [closed]

I have a given Lagrangian: $$L= e^{st}\cdot\frac12\cdot(mv_y^2-ky^2)$$ And are asked to identify the equations of motions, the constants of motions and physical system. Without the exp-time-term, ...
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### Pendulum with a rotating point of support from Landau-Lifschitz

I found this problem in Landau-Lifschitz vol.1 (Mechanics) A simple pendulum of mass $m$, length $l$ whose point of support moves uniformly on a vertical circle with constant frequency $\gamma$. ...
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### Lagrangian formalism and Contact Bundles

In his Applied Differential Geometry book, William Burke says the following after telling that the action should be the integral of a function $L$: A line integral makes geometric sense only if ...
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### Do we need inertial frames in Lagrangian mechanics?

Do Euler-Lagrange equations hold only for inertial systems? If yes, where is the point in the variational derivation from Hamilton's principle where we made that restriction? My question arose ...
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### What is the neatest way to describe a “non-autonomous” (lagrangian) system?

The configuration space of a system of particles $(m_i,x_i)$, $i=1,\dots,n$, subject to constraints $$\Phi (x)=0,\qquad \Phi\colon \mathbb R^{3n}\to \mathbb R ^{3n-k},\qquad x=(x_1,...,x_n),$$ if the ...
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### Is there a better choice of coordinates for a bead on a rotating helical wire?

A bead of mass $m$ is threaded around a smooth spiral wire and slides downwards without friction due to gravity. The $z$-axis points upwards vertically. Suppose the spiral wire is rotated about the ...
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We can derive Lagrange equations supposing that the virtual work of a system is zero. $$\delta W=\sum_i (\mathbf{F}_i-\dot {\mathbf{p}_i})\delta \mathbf{r}_i=\sum_i ... 1answer 137 views ### Is there a Lagrangian whose Euler-Lagrange equation is the gradient? I am trying to recast a problem I am working on in terms of Lagrangian mechanics. I am in the following situation. Suppose I have a function f:X \rightarrow \mathbb{R} (a field). In the its ... 1answer 118 views ### Clarifying constraint forces in Lagrangian dynamics In the Lagrangian formulation, the addition of constraint forces that are unknown can be done with Lagrange multipliers, which allows for the forces to be found. Taking k constraints of the form ... 5answers 310 views ### Noether Theorem and Energy conservation in classical mechanics I have a problem deriving the conversation of energy from time translation invariance. The invariance of the Lagrangian under infinitesimal time displacements t \rightarrow t' = t + \epsilon can be ... 2answers 110 views ### Landau's argument for dependence of Lagrangian on magnitude of velocity In chapter 1, of Landau-Lifshitz Mechanics' book, Landau through isotropy and homogeneity of space and homogeneity of time proves that the Lagrangian must depend of magnitude of velocity of the ... 1answer 74 views ### How to check \renewcommand{\vec}[1]{\mathbf{#1}} \vec{v'}\cdot\vec{V} and \vec{v}'^2 are time derivatives of some other functions? From Landau, Lifshitz Mechanics p.127 \renewcommand{\vec}[1]{\mathbf{#1}}L'=\frac{1}{2}m(\vec{v}'^2+\vec{v'}\cdot\vec{V}+\vec{V}^2)-U  He states that "\vec{V}^2(t) can be written as the total ... 0answers 26 views ### Expansion of L(v^2 + 2\vec{v}\cdot\vec{\epsilon}+\epsilon^2) [duplicate] How can I find the expansion of the Lagragian (it it only dependent on v^2) L(v^2 + 2\vec{v}\cdot\vec{\epsilon}+\epsilon^2) in powers of \vec{\epsilon} ? (From L.Landau, E. Lifshitz, Mechanics , ... 2answers 333 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 ... 1answer 51 views ### Classical Mechanics & Coordinates [closed] What is the meaning generalised coordinates in Classical Mechanics? How is Lagrangian formalism different from Hamiltonian formalism? How are they related to Hamilton's Principle? How are they ... 1answer 60 views ### Stationary action with maximized action [duplicate] I would like to ask for an example (a lagrangian) both in classical and quantum level for which the action is maximaized (rather than minimized). What is special in these cases? 1answer 69 views ### Transforming a lagrangian to hamiltonian and vice versa I am not refering to Legendre transform, but to something more simple. In analytical mechanics, the Lagrangian can be described as L=T-V, and the Hamiltonian is if the Lagrangian doesn't explicitly ... 1answer 186 views ### Constraints of massive relativistic point particle in hamiltonian mechanics I try to understand constructing of Hamiltonian mechanics with constraints. I decided to start with the simple case: free relativistic particle. I've constructed hamiltonian with constraint: ... 1answer 114 views ### Why can we assume independent variables when using Lagrange multipliers in nonholonomic systems? I'm studying from Goldstein's Classical Mechanics. In section 2.4, he discusses nonholonomic systems. We assume that the constraints can be put in the form f_\alpha(q, \dot{q}, t) =0, \alpha = 1 ... 8answers 2k views ### What's the point of Hamiltonian mechanics? I've just finished a Classical Mechanics course, and looking back on it some things are not quite clear. In the first half we covered the Lagrangian formalism, which I thought was pretty cool. I ... 1answer 191 views ### Using Lagrange's Equations with Generalized forces I am a bit confused on how this works. For instance if I wanted to look at an object moving in 2 dimensions only subject to gravity (and assuming that the potential is just mgy), I get that my ... 0answers 183 views ### What's the physical intuition for symplectic structures? I always thought about symplectic forms as elements of areas in little subspaces because of the Darboux theorem, however I cannot get the physical intuition for it and for the hamiltonian vector ... 1answer 94 views ### Hamilton-Jacobi formalism and on-shell actions My question is essentially how to extract the canonical momentum out of an on-shell action. The Hamilton-Jacobi formalism tells us that Hamilton's principal function is the on-shell action, which ... 1answer 68 views ### Why does a particle fall in a straight line? In Lagrangian Mechanics we choose the path of least action. Given a uniform gravitational field, and a particle of finite mass; and fixing two points the start & end-point we consider all paths ... 0answers 28 views ### How can a transversality condition be invoked to reduce the Euler-Lagrange equation? I asked this question regarding the Euler-Lagrange equation at MSE and have gotten no response. I will ask it here too. I think I might have more luck here since the E-L equation is at the core of ... 0answers 73 views ### How to analyze this constraint question Let \gamma be a smooth curve in the plane, and introduce curvilinear coordinates q_1,q_2 on a neighborhood of \gamma; q_1 is the direction of \gamma and q_2 is distance from the curve. ... 1answer 176 views ### Classical Mechanics - Equation of motion, Lagrangian, Newtons 2nd Law [closed] I really don't even know where to start with this question any help would go very very far. http://imgur.com/g4KxNY5 1answer 112 views ### Can the Lagrange Multipliers depend on the coordinates? When dealing with Lagrange multipliers to solve systems with constraints we usually have two ways if the constraints are holonomic: Differentiate the constraint and add the appropiate term to the ... 2answers 314 views ### How can you solve this “paradox”? Central potential A mass of point performs an effectively 1-dimensional motion in the radial coordinate. If we use the conservation of angular momentum, the centrifugal potential should be added to the original one. ... 0answers 83 views ### A discrete approach to the catenary I'm trying to work out a model for the system above, that is, N particles of unitary mass subject to the constraints:$$1=\varphi _i(\mathbf r _1,\mathbf {r}_2,...,\mathbf r _n)=|\mathbf ...
I have two planes, one characterized by equation $$\phi_1=f(x)-z=0$$ and another $$\phi_2=\alpha y-z=0$$ where $\alpha$ is arbitrary. In their line of intersection(we assume it exist and is continous) ...