Maybe it is easy to write down the kinetic energy and potential energy. How could be use computer to solve the Euler-Lagrange equation automatically? I guess it is still hard to solve for the equation, and we use numerical simulation?
oadrian96
I'm in the process of writing a Smoothed Particle Hydrodynamics simulator for 418 so this was a very useful explanation.
0x484884
Thanks for listing this reference I think this stuff is really cool but I've never felt like I've understood Lagrangian mechanics or the principle of least/stationary action but this looks like it could help a lot.
It's interesting how a lot of these concepts that were discovered hundreds of years ago in physics are being so widely used in modern computer science. I've also noticed a lot of stuff in deep learning is inspired by or analogous to these ideas of minimizing some sort of 'energy' like quantity.
0x484884
Actually, this reminds me of some cool denoising algorithms and I was wondering if these are ever used with ray tracing to reduce the noise in the images without having to do too much sampling.
raymondx
Really interesting how this topic shows up in both robot kinematics/dynamics as well as graphics
keenan
@CMUScottie The final Euler-Lagrange equations you get via this approach will of course be identical to the Euler-Lagrange equations you obtain via any other derivation. The "ease" is just in the ease of deriving them in the first place. In fact, in many cases it's trivial to get your computer to derive the Euler-Lagrange equations from the Lagrangian---packages like Mathematica and Maple will trivially perform the necessary symbolic derivatives.
That being said, there are some very interesting computational possibilities available to you if you have a Lagrangian rather than just the Euler-Lagrange equations. Basically it becomes straightforward (i.e., just "plug and chug") in many cases to derive a so-called symplectic variational integrator. The short story is: rather than derive the Euler-Lagrange equations, then replace derivatives with numerical derivatives to get a time stepping scheme, you first discretize the so-called action integral (more or less: the integral of the Lagrangian over time) and then solve for critical points of this discrete action. The result is (often) numerical integrators that have nice conservation properties, e.g., conservation of energy, momentum, etc., over very long integration times. This is especially useful for animation, where you want to preserve the "liveliness" of motion, and may not care so much about accuracy.
Maybe it is easy to write down the kinetic energy and potential energy. How could be use computer to solve the Euler-Lagrange equation automatically? I guess it is still hard to solve for the equation, and we use numerical simulation?
I'm in the process of writing a Smoothed Particle Hydrodynamics simulator for 418 so this was a very useful explanation.
Thanks for listing this reference I think this stuff is really cool but I've never felt like I've understood Lagrangian mechanics or the principle of least/stationary action but this looks like it could help a lot.
It's interesting how a lot of these concepts that were discovered hundreds of years ago in physics are being so widely used in modern computer science. I've also noticed a lot of stuff in deep learning is inspired by or analogous to these ideas of minimizing some sort of 'energy' like quantity.
Actually, this reminds me of some cool denoising algorithms and I was wondering if these are ever used with ray tracing to reduce the noise in the images without having to do too much sampling.
Really interesting how this topic shows up in both robot kinematics/dynamics as well as graphics
@CMUScottie The final Euler-Lagrange equations you get via this approach will of course be identical to the Euler-Lagrange equations you obtain via any other derivation. The "ease" is just in the ease of deriving them in the first place. In fact, in many cases it's trivial to get your computer to derive the Euler-Lagrange equations from the Lagrangian---packages like Mathematica and Maple will trivially perform the necessary symbolic derivatives.
That being said, there are some very interesting computational possibilities available to you if you have a Lagrangian rather than just the Euler-Lagrange equations. Basically it becomes straightforward (i.e., just "plug and chug") in many cases to derive a so-called symplectic variational integrator. The short story is: rather than derive the Euler-Lagrange equations, then replace derivatives with numerical derivatives to get a time stepping scheme, you first discretize the so-called action integral (more or less: the integral of the Lagrangian over time) and then solve for critical points of this discrete action. The result is (often) numerical integrators that have nice conservation properties, e.g., conservation of energy, momentum, etc., over very long integration times. This is especially useful for animation, where you want to preserve the "liveliness" of motion, and may not care so much about accuracy.
The longer story can be found in the book Geometric Numerical Integration by Hairer, Wanner, and Lubich.
will we be using these physics formulas in our animation assignment?