Archive of posts in category “physics”
An example of non-uniform elements: heavy elastic rope
To illustrate the concept about non-uniform elements, we study a simple problem: suppose a uniform heavy elastic rope has mass , original length , and stiffness , and find the mass distribution and length of it when hung vertically. We can use the element method to solve this problem, but the elements are non-uniform in terms of length. The elements add up to get the total length .
- Categories: physics
- Tags: from zhihu
Kinetic energy, momentum, and angular momentum of rigid bodies
In this article, we will find that the inertia matrix naturally appears when we calculate the kinetic energy or the angular momentum of a rigid body. Then, we introduce the concept of principal inertia . We also study how the inertia matrix changes under translations and rotations and how those transformations may lead to conclusions that can help us simplify the calculation of inertia matrices.
- Categories: physics
- Tags: rigid body, linear algebra, classical mechanics, from zhihu
The image of a circular object through a thin lens
The image of a circle with radius and centered at through a thin lens at with focal length and centered at is a conic section with the focus being , the directrix being line , and the eccentricity being .
- Categories: physics
- Tags: geometrical optics, from zhihu
Mapping from Kepler problem to free particle on 3-sphere
There is a canonical transform of the Kepler problem which is the same as the problem of motion of a free particle on 3-sphere. The explicit formula of the transform as well as some links about this topic is written in the article. The explicit formula for is where is the position of the particle on 3-sphere (a 4-dimensional vector), is the momentum of the original particle in Kepler problem, is a vector perpendicular to the 3-dimensional hyperplane where lies, and .
- Categories: physics
- Tags: classical mechanics, canonical transformation, letter, kepler problem
Thoughts on a middle school thermal physics problem
To heat an object with hot water, if we divide the water into more parts and use each part to heat the object one after another, the final temperature will be higher. If the number of parts tends to infinity, then the final temperature will tend to the limit , where is the initial temperature of the object, is the temperature of the hot water, is the heat capacity of the object, and is the heat capacity of the hot water.
- Categories: physics
- Tags: calculus, number sequence, from zhihu
View of the world (physically rather than philosophically)
The view of the world… Physically! In this article, I tried to use mathematical language to describe models of the physical world. A view of the world should include: a space (actually spacetime) with some mathematical structure on it (whose points are events in the world), a symmetry principle describing the symmetry of the world, and a motion law to describe the physics and dynamics of the world. This article proposed models for Galilean, Einsteinian, and even Aristotelian worlds. Can you come up with even other worlds?
- Categories: physics
- Tags: imagination, long paper
Simulating a mechanical system using rpg_core.js
Continuing my last work of simulating a mechanical system using RGSS3, I made a new version using rpg_core.js, the game scripting system shipped with RPG Maker MV. This version is live on web!
- Categories: physics
- Tags: javascript, rgss, hamiltonian, calculus, ode, web
Simulating a mechanical system using RGSS3
Hamiltonian mechanics gives us a good way to simulate mechanical systems as long as we can get its Hamiltonian and its initial conditions. I implemented this simulation in RGSS3, the game scripting system shipped with RPG Maker VX Ace.
Use complex numbers as canonical variables
In this article, I try exploring an idea: using complex numbers to combine pairs of canonical variables into complex variables: . It turns out that we can write canonical equations , Poisson brackets , and canonical transformations in these complex numbers. Finally, I show two examples of using them in real problems: a free particle, and a harmonic oscillator.
- Categories: physics
- Tags: classical mechanics, canonical transformation, hamiltonian, complex, long paper