# 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).

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### Action functional in the formalism of symplectic manifolds with Hamiltonian

Call Hamiltonian a symplectic manifold $(M, \omega)$ equipped with a distinguished Hamiltonian $h \in \mathcal C^\infty(\mathbb R \times M)$. Wikipedia 'Tautological form' page has a section about ...
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### What will happen if we take up Cartesian Coordinates in Lagrangian Formulation instead of Generalized Coordinates?

Why do we actually need generalized coordinates? Is it a mathematical manipulation only or does it serve physical purpose? And will principle of stationary action stay valid if we use cartesian ...
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### Connection between different kinds of “Lagrangian”

Being a physic student I first heard the term: "Lagrangian" during a course about Lagrangian mechanics; at that time this term was defined to me in the following way: For a classic, non ...
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### Least Action in General Relativity

For an affinely parameterised geodesic we can form the Lagrangian: $$\mathcal L = g_{ab}\dot x^a\dot x^b = \text{constant}$$ The Lagrangian is constant by the fact that the geodesic parallel ...
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### Can you write and solve the EM field Lagrangian density without reference to the EM potential? [duplicate]

Is it possible to write the Lagrangian density for the EM field and charges $$L=-\frac{1}{4\mu_0}F^{\mu \nu}F_{\mu \nu}+j^{\mu}A_{\mu}$$ only in terms of the Electromagnetic Tensor and current vector? ...
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### Lagrangian stationarity to find amplitude of perturbed system eigenfunction

Say we have a system that satisfies the Helmholtz wave equation $\nabla^{2}h+k_{0}^{2}h=0$ and some boundary conditions over the boundary of the system volume $V$ in 3D space. Let the field solution ...
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### Is there a minimization principle for Hamiltonian? [duplicate]

Consider a point particle in $n$ dimensions. For a Lagrangian $\mathcal L(\mathbf{q, \dot q}, t)$, we have that $\mathbf q(t)$ is a feasible trajectory for times $t_0<t<t_f$ iff it extremizes ...
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### How is the relationship between the old and the new canonical variables justified?

In Classical Hamiltonian Mechanics, a canonical transformation of the phase-space coordinates $(p,q,t) \to (P,Q,t)$ is such that the general form of Hamilton's equations is followed and Hamilton's ...
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### Hamilton's principle for fields

According to Goldstein, Hamilton's principle can be summerized as follows: The motion of the system from time $t_{1}$ to time $t_{2}$ is such that the line integral (called the action or the action ...