3
votes
1answer
121 views

Erroneous calculus of variations reference in V. I. Arnold's Mathematical Methods of Classical Mechanics?

The beginning of section 12, Calculus of variations (chapter 3, Variational principles) in V. I. Arnold's Mathematical Methods of Classical Mechanics (2nd edition, p. 55) reads: For what follows, ...
0
votes
0answers
35 views

Newton's method for the brachistochrone

Consider the potential $V(x,y)=-y$ and a particle at rest in the beginning of the coordinate system. We are going to examine the brachistochrone - the smooth curve of fastest descent. Assume we are ...
1
vote
0answers
49 views

Derivation of Euler lagrange for Yang Mills

I need someone to sketch the conventional steps(from variation to vanishing of arbitrary function chosen , etc, etc) of Classical Yang-Mills. If using exterior product product could you emphasize any ...
0
votes
1answer
78 views

Variational calculus applied to the strain energy functional in solid mechanics

The question is basically about when to apply the variational operator... Given the general functional representing the strain energy of a solid under a given stress state $\sigma$ and strain state ...
0
votes
1answer
33 views

Variation equations and integral invariants

I am reading Whittaker's Analytical Dynamics. This is chapter 10 Hamiltonian Systems. Paragraph 112 is Variation Equations. Let ${dx_r\over dt}=X_r(x_1,\dots,x_n,t),\quad (r=1,\dots,n)$ be a system ...
1
vote
0answers
58 views

The equvalence of the virtual work and the Hamiltonian equations

I am reading Whittaker's Analytical Dynamics. This is chapter 10 *Hamiltonian Systems&. Paragraph 109 is Hamiltonian Systems & Their integral invariants. Whittaker starts with the Lagrangian ...
0
votes
1answer
66 views

Deriving Hamiltons principle for conservative holonomial systems

I am reading Whittaker's Analytical Dynamics. This is chapter 9 Principle of least action and least curvature. The paragraph is 99 Hamilton’s principle for conservative holonomial systems. Let us ...
0
votes
1answer
68 views

How to include prescribed boundary conditions in the Ritz Method

Using the Ritz method to find the displacement field in structural analysis can be done as follows. $U$ and $V$ are recpectively the internal and external energy components of a given structural ...
2
votes
1answer
69 views

Interpretation of the variational principle for the Ritz approximation, solid Mechanics

Below $U$ and $V$ are recpectively the internal and external energy components of a given structural element: $$U+V=W$$ Expressing $U$ in terms of the strains $\varepsilon$ and the material ...
10
votes
1answer
304 views

Checking my understanding of $T^*M$ as a symplectic manifold and the links between the classical description of Lagrangians vs this invariant way.

I am working through a book titled "An introduction to mechanics and symmetry" by Marsden and Ratiu. I have written up a brief summary trying to solidify my understanding of the general principles. ...
1
vote
0answers
192 views

Closed Geodesics as minimisers of action functional

Suppose I have a Riemannian surface $(M,g)$. It's clear that closed geodesics are critical points of the length functional $l(\gamma)=\int\left|\gamma(t)^{\prime}\right|_{g(\gamma(t))}dt$ over the ...
3
votes
0answers
71 views

what is the domain of the Lagrangian of a surface embedding?

If we view our Lagrangian particle mechanics geometrically, then we describe a particle trajectory as a map from R to a manifold, and the Lagrangian $L(x,\dot{x})$ as a function on the tangent bundle ...