4 votes

Modelling friction as a conservative force

Adding friction to a Lagrangian is not commonly taught. It is non-trivial, but not too difficult in the end. The key element is producing a "dissipation function" $D$ that we can use to ...
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3 votes
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Why is Hamilton's equations sometimes written with a gradient?

Your best guess is correct. Let's take the example of an $N$ particle system in $\text{3D}$. If I wanted to define my generalized coordinates as just the position of each particle, then one way I ...
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3 votes
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What do you think about this particularization of the Euler-Lagrange equation that resembles Newton's 2nd Law?

Yes, one may use d'Alembert principle to rewrite Newton's 2nd Law as Lagrange equations $$ \sum_{i=1}^N\underbrace{\left(\dot{\bf p}_i-{\bf F}_i\right)}_{\text{Newton's 2nd Law}}\cdot \delta {\bf r}_i ...
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2 votes
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Why the amplitude of monopole solution in Helmholtz equation is complex?

If there is a single monopole it does not matter whether $A$ is real or not but if you have two or more sources then their relative phases, and thus the phase of $A$, do matter. The same holds if the ...
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2 votes

When to apply $I_c \underline{\omega} = \underline{M_c}$?

The expression $ \underline{M}_c = \mathrm{I}_c \,\underline{\omega}$ is never correct. I think you forgot the time derivative of rotational velocity here. Also, the are Coriolis torques that relate ...
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1 vote
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Minimum energy required to behave like a turning point?

The basic problem you are running into is that different inertial observers do not agree on kinetic energy of objects or systems, but they do agree on changes in the system's kinetic energy. But on ...
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1 vote

Why is Hamilton's equations sometimes written with a gradient?

The simplest answer is, that it's a far more compact/applicable form of notation generalization, especially when dealing with various types of field theories (both classical or quantum). In general, ...
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  • 184
1 vote

Problem 6.3 from David Morin (classical mechanics)

Note that the Euler-Lagrange's equations for a set of $\{q_1,\dots, q_n\}$ generalized coordinates are valid if the $n$ coordinates are independent from each other. The $x$ coordinate of your problem ...
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1 vote

Modelling friction as a conservative force

For systems with a Hamiltonian formulation, the Poincaré recurrence theorem (PRT) would indicate that most trajectories will eventually evolve back to a state arbitrarily close to their initial ...
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