# Questions tagged [variational-principle]

any of several principles that find the physical trajectory of a system by minimizing or maximizing some value computed over the proposed path (for instance geometric optics can be reproduced by insisting on a minimum time principle).

585 questions
78 views

37 views

### Intuition behind the use of the Principle of Stationary Action in Classical Field Theory [duplicate]

Whilst studying Field Theory and after checking numerous sources it appears that people always just state the action without providing some sort of motivation/intuition as to why we should/can use the ...
43 views

22 views

### Deriving Canonical Transformation from Generating Function using Principle of Stationary Action

In Hamill's "A Student's Guide to Lagrangians and Hamiltonians", section 5.2, the equations for a canonical transformation $(q,p) \to (Q,P)$, induced by the generating function $F(q,Q,t)$ are derived ...
26 views

### Attaining extrema when a stationarity condition has no solution

I was wondering if someone could shed some light on the following for me: If a stationarity (maximizing or minimizing) condition has no solution inside a particular domain, then how do we reason that ...
55 views

### Paths of least action and loops in time

In the book Quantum Field Theory for the Gifted Amateur link: https://books.google.ca/books?hl=en&lr=&id=nIk6AwAAQBAJ&oi=fnd&pg=PP1&ots=JZjwG_qDt5&sig=...
103 views

### Need help with Gauss Bonnet gravity

I am trying to find the equations of motion for a coupling Gauss-Bonnet gravity. The action is : \begin{equation} S=\int{d^4x \sqrt{-g}\left[\frac{R}{2k}+f(\phi)(\alpha R^2+\beta R_{\mu\nu} R^{\...
222 views

### Why relativistic Lagrangian doesn't simply equal kinetic minus potential energy $L=T-V$?

As the question above, I wonder why the relativistic Lagrangian is written as: $$L=-mc² \sqrt{1-\frac{v²}{c²}} - V ~=~-\frac{mc^2}{\gamma} -V~?$$ I know that the kinetic energy of a relativistic ...
38 views

### Transformation of ADM parameters under diffeomorphisms

I am trying to prove the invariance of the ADM formalism under (infinitesimal) diffeomorphisms. I have checked Wald and other textbooks on the subject but have been unable to find expressions for how ...
36 views

### Lagrangian for first order equation of motion? [duplicate]

Let us have the following equation of motion (it might not necessarily correspond to a physical system): $$\dot{x} + a \cdot x + b \cdot x^2 + c=0.$$ I would like to deduce the corresponding ...
18 views

### How can we predict how a system evolves using the stationary action principle even though we need to specify the final state? [duplicate]

The stationary action principle states that a system evolves between a fixed initial and fixed final configuration in such a way that the action is stationary. But isn't the final configuration what ...
49 views

### Rigid Body Equations in terms of Body Coordinates by Hamilton's Principle

I sought-for the equations of motion of an unrestrained rigid body. The equations of motion are readily available in the literature, but my concern is to derive them by Hamilton's principle. ...
75 views

### Euler-Lagrange Equation, Shortest Path On Sphere

The equations for spherical polar coordinates are $$x = r \sin(\theta) \cos(\phi) \\ y = r \sin(\theta) \sin(\phi) \\ z = r \cos(\theta)$$ Now, consider a path expressed as $\phi = \phi(\theta)$...
75 views

### Approximating ground-state energy without using variational principle

Given the Hamiltonian for one dimension harmonic oscillator: $$H=-\frac{\hbar^2}{2\mu}\frac{d^2}{dx^2}+\frac{\mu\omega}{2}x^2 ,$$ I need to calculate the approximate ground state energy using the ...
51 views

### Fermat's principle: when does light actually travel along the local maximum of accumulated phase? [duplicate]

In class we learned that Fermat's principle dictates that light travels either along a local minimum or a local maximum of the accumulated optical phase, but the professor only gave examples of local ...
35 views

### Is Entropy a monotonically increasing function of Gibbs Free Energy/ Helmholtz free energy/ Enthalpy?

Entropy can be axiomatically taken as a monotonically increasing function of internal energy $(E) ,$ from where "energy minimum principle" can be deduced, and this can be stated using variational ...
71 views

### D'Alembert's principle and equation of motion

Is obtaining proper equation of motion from D'Alembert's principle a mere coincidence or there is some logic behind this? This is asked because while we are finding the equations in a regime where ...
207 views

### What is the physical content of the principle of least action?

Say the world is governed by the Principle of Least Action (or Hamiltonian mechanics) and let's not worry about quantum mechanics too much. Independently of any Lagrangian or Hamiltonian, does that ...
49 views

44 views

### How can the action can describe a movement? What is the argument behind? [duplicate]

We define the action of a system as $$S(q)=\int_{t_1}^{t_2}L(t,q(t),q'(t))dt,$$ where $q(t)$ is the evolution of the system and $L$ is the Lagrangien. How can a stationary point of $S$ can describe ...
20 views

### Geodesic curve definition [duplicate]

Do we have a choice in defining the covariant derivative by the use of a set of coefficient functions(Christoffel gammas)? If so, could we then say that these coefficient functions need not to ...
47 views

### How does Fermat's principle of least time come from this statement? [duplicate]

In Wikipedia Fermat's Principle is stated as: A ray of light prefers the path such that there are other paths, arbitrarily nearby on either side, along which the ray would take almost exactly the ...
65 views

122 views

### Curve for fastest time down a ramp [duplicate]

I came across a physics experiment video showing three balls released from a point A, going down three different kinds of ramps leading to a point B (https://www.youtube.com/watch?v=61S0KW7e-rc) ...
85 views

### Example in which light takes the path of maximum optical length [duplicate]

According to the modern version of Fermat's principle,"A light ray in going from point A to point B must traverse an optical path length that is stationary with respect to variations of that path.".Is ...
144 views

### How to deal with explicit time dependence of the Lagrangian?

Clearly, if the Lagrangian in explicitly time dependent, the Euler-Lagrange equations being satisfied does not extremise the action. I am unclear as to how to deal with systems with an explicitly time-...
103 views

### Deriving the geodesic equation using a Lagrange multiplier to fix affine parametrisation

The geodesic equation can be derived using the action $$S_0 ~=~ \int d\tau \sqrt{-\dot{x}_\mu\cdot \dot{x}^\mu}\tag{1}$$ (I am using the (-+++) convention and $\dot{x} = \frac{dx}{d\tau}$). To ...