What is Continuum: Definition and 142 Discussions

Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century.

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  1. S

    I Holographic principle in continuous spacetime?

    Can the holographic principle be applied to spacetimes and metrics that are (fundamentally) continuous/smooth? Or only to discrete ones?
  2. walkeraj

    I Is the Idea of a Continuum Always an Approximation to the Physical?

    Question: When thinking of continuums the most notable seems to be space-time but they also mark a simplification to reality like in continuum mechanics, often taught when learning the tensor calculus needed for general relativity. The question is that for general relativity when a geodesic...
  3. Ihsan

    Continuum Physics: Balance of Linear Momentum or Cauchy first equation of motion

    Hi, unfortunately, I am not getting anywhere with the following task and I try solve it
  4. G

    I “Interpreting astronomical spectra” -- Continuum origins

    I’ve been reading “Interpreting astronomical spectra” and am still confused about the micro processes contributing to the continuum. In particular,the sun’s photosphere. I’m not interested in the black body thermodynamics approach because some processes should cause this. Bound-bound...
  5. S

    A Moire potential depth from continuum model

    Dear All, I am trying to calculate the moire potential depth of transition metal dichalcogenide system. I have attached supporting material obtained from one of the thesis. Here they have describe the continuum model hamiltonian for TMDs homobilayer. My question is how to obtain the moire...
  6. yucheng

    A References: continuum approximation of discrete sums?

    Is there more references for how accurate is the continuum approximation to discrete sums? Perhaps more mathematical. What I've found: https://lonitch.github.io/Sum-to-Int/ https://arxiv.org/pdf/2102.10941.pdf Some examples are: Sum to integral $$\sum_{\mathbf{k}} \to 2 \left ( \frac{L}{2...
  7. A

    How much does the equivalent width of a line change by the introduction of 5% scattered light?

    How much does the equivalent width of a line change by the introduction of 5% scattered light? We know the equivalent width is defined as We know the equivalent width is defined as $$W = \int_{-\infty}^{\infty} \bigg(\frac{1-F_{\nu}}{F_c}\bigg) \, d\nu$$ where ##F_{\nu}## represents the flux in...
  8. JD_PM

    Euler Lagrange equations in continuum

    OK I've been stuck for a while in how to derive ##(1)##, so I better solve a simplified problem first: We work with Where $$\mathscr{L} = \mathscr{L}(\phi_a (\vec x, t), \partial_{\mu} \phi_a (\vec x, t)) \tag{3}$$ And ##(3)## implies that ##\mathscr{L}(\vec x, t)## We know that...
  9. M

    Types of thermodynamics processes that happen in solids or continuum medium

    What are the types of thermodynamics processes that happen in solids or continuum medium
  10. thaiqi

    Classical Which books are good for Lagrangian/Hamiltonian formulations for continuum?

    Can anybody suggest which books are good for Lagrangian/Hamiltonian formulations for continuum beyond The Classical Mechanics by Goldstein ( it seems a bit too complicated for my understanding.)?
  11. FireAP

    B Plotting the Space-Time Continuum: Is it Possible?

    How would one plot the space-time continuum graphically(if it were possible,obviously)?
  12. Q

    I The Continuum Hypothesis and Number e

    Summary: The Continuum Hypothesis and Number e Now, I must ask a very stupid question: When taking: $$\lim_{_{n \to \infty} } (1+\frac{1}{n})^n=e\\$$ the ##n## we use take its values from the set: ## \left\{ 1,2,3 ... \right\} ## which has cardinality ## \aleph_0 ##, which is equivalent...
  13. hilbert2

    A Exploring the 2-D Ising Model & Its Continuum Limit

    I've recently been reading about the 2-dimensional Ising model and its continuum limit from several sources, including https://webhome.weizmann.ac.il/home/fnfal/papers/Ising/lecture1.pdf https://webhome.weizmann.ac.il/home/fnfal/papers/Ising/lecture2.pdf As far as I understood it, the state...
  14. A

    I Discrete to continuum Gaussian function

    I have a question regarding a paragraph in "Radiation detection and measurement" by Knoll. In the chapter about the discrete Gaussian it states that "Because the mean value of the distribution ##\bar{x}## is large , values of ##P(x)## for adjacent values of x are not greatly different from each...
  15. A

    What is the Role of Continuum Mechanics in CFD Simulations?

    I am a Phd student working in a technological center. My work is related to CFD simulations by using OpenFoam coupled to discrete element method. I am very interested in mathematical and physical background of continuum mechanics both solid and fluids. I am loking forward to solve my interests...
  16. C

    Continuum mechanical analogous of Maxwell stress tensor

    Maxwell stress tensor ##\bar{\bar{\mathbf{T}}}## in the static case can be used to determine the total force ##\mathbf{f}## acting on a system of charges contanined in the volume bounded by ##S## $$ \int_{S} \bar{\bar{\mathbf{T}}} \cdot \mathbf n \,\,d S=\mathbf{f}= \frac{d}{dt} \mathbf...
  17. V

    Continuum mechanics in physics education

    I came across this article about the near absence of continuum mechanics in university-level physics education: http://www.troian.caltech.edu/papers/Gollub_PhysToday_Dec03.pdf I have wondered this issue myself: why is continuum mechanics mainly studied by engineers rather than physicists, even...
  18. S

    Definition(s) of entropy in continuum mechanics

    Those treatments of Entropy in continuum mechanics that I've viewed on the web introduce Entropy abruptly, as if it is a fundamental property of matter. For example the current Wikepedia article on continuum mechanics ( https://en.wikipedia.org/wiki/Continuum_mechanics ) says: Are other...
  19. T

    A Are all wave functions with a continuum basis non-normalizable?

    For example, I am following the below proof: Although the above derivation involves a projection on the position basis, it appears one can generalize this result by using any complete basis. So despite it not being explicitly mentioned here, are all wave functions with any continuum basis...
  20. S

    Engineering Continuum Mechanics or Design/Manufacturing?

    Hi, people of PF I'm trying to decide between concentrating on continuum mechanics or design/manufacturing for my master's degree. My goal is to ultimately work in the industry, so design/manufacturing seems to make a lot of sense. However at the same time, continuum mechanics (and physics in...
  21. T

    I Do you believe that continuum is Aleph-2, not Aleph-1?

    This negation of CH is based on Woodin's work: https://en.wikipedia.org/wiki/%CE%A9-logic Of course, you can only believe his result because you need to believe his axioms first. But for me it is really convincing for multiple reasons: 1. While it is, of course, a negation of CH, it does not...
  22. john baez

    Insights Struggles with the Continuum - Conclusion - Comments

    john baez submitted a new PF Insights post Struggles with the Continuum - Conclusion Continue reading the Original PF Insights Post.
  23. john baez

    Insights Struggles with the Continuum - Part 7 - Comments

    john baez submitted a new PF Insights post Struggles with the Continuum - Part 7 Continue reading the Original PF Insights Post.
  24. itssilva

    Continuum hypothesis and fractals

    Wikipedia: "The hypothesis states that the set of real numbers has minimal possible cardinality which is greater than the cardinality of the set of integers" ; i.e., let cardinality of integers = ℵ0 and cardinality of reals = ℵℝ; then there is no ℵ such that ℵ0 < ℵ < ℵℝ . But what about...
  25. john baez

    Insights Struggles with the Continuum - Part 6 - Comments

    john baez submitted a new PF Insights post Struggles with the Continuum - Part 6 Continue reading the Original PF Insights Post.
  26. john baez

    Insights Struggles with the Continuum - Part 5 - Comments

    john baez submitted a new PF Insights post Struggles with the Continuum - Part 5 Continue reading the Original PF Insights Post.
  27. john baez

    Insights Struggles with the Continuum - Part 4 - Comments

    john baez submitted a new PF Insights post Struggles with the Continuum - Part 4 Continue reading the Original PF Insights Post.
  28. E

    Covariant Characterization of Causality in Continuum: T^{ik}v_k

    Hi! Let ##T^{ik}## be the stress-energy-tensor, and ##v_k## some future-pointing, time-like four vector. How can I see that the object ##T^{ik}v_k## is future-pointing and not space-like? Thank you for your help!
  29. L

    Continuum Mechanics: Rigid Heat Conductor

    Homework Statement Determine the thermodynamic restrictions for a rigid heat conductor defined by the constitutive equations: \DeclareMathOperator{\grad}{grad}\psi = \hat{\psi}\left(\theta,\grad \theta, \grad \grad \theta\right) \\ \eta = \hat{\eta}\left(\theta,\grad \theta, \grad \grad...
  30. Pizza

    Multiple big bangs in one continuum - Question

    I have fond almost the same question here with some answers. Anyhow, I am not satisfied with the answers, or couldn't understand them well. Beside that, I would formulate the question a little bit different. Unfortunately the thread is already closed (from 2012). Provided there is only one...
  31. john baez

    Insights Struggles With the Continuum - Part 3 - Comments

    john baez submitted a new PF Insights post Struggles With the Continuum - Part 3 Continue reading the Original PF Insights Post.
  32. G

    Exercises in continuum mechanics

    Hello Please forgive me if i am not posting in the correct forum. Also you may find my English a bit rusty since i am basically French Ok so i want to solve some exercises in continuum mechanics . The first exercise states : we have a stress tensor in a Cartesian coordinate system with the...
  33. john baez

    Insights Struggles With The Continuum - Part 2 - Comments

    john baez submitted a new PF Insights post Struggles With The Continuum - Part 2 Continue reading the Original PF Insights Post.
  34. R

    What causes the liquid to rise in this picture?

    This is a diagram of a pitot-static tube. My question is however not related to its applications but rather, what causes the liquid to rise up the static tube? The static tube is at right angles to the fluid flow. I understand that this is a very basic question but I can't seem to get my head...
  35. john baez

    Insights Struggles With the Continuum - Part 1 - Comments

    john baez submitted a new PF Insights post Struggles With the Continuum - Part 1 Continue reading the Original PF Insights Post.
  36. N

    General Relativity and continuum mechanics

    Hello! I started learning relativity recently and I'm an engineering student. I can't stop drawing similarities between the nature of gravity and behaviour of continuous media in the field of continuum mechanics. Is there some direct connection and if so, has something like this been explored...
  37. T

    Observables vs. continuum and metric?

    Space in quantum mechanics seems to be modeled as a triplet of real numbers, i.e. a continuum. Same happens in special relativity. General relativity I do not know (nor field theories). And then we apply the Pythagorean theorem and triangle inequality and so forth... I have a few general...
  38. A

    Understanding Real-World Emission Spectra: Are Solids Similar to Blackbodies?

    Hi all, I've got two questions about the emissions spectrum from solids. Question #1: I feel like I have a reasonable understanding of line absorption and emission spectrum of low density gases based on transitions of electrons between discrete allowed energy levels in a gas. I'm trying to...
  39. shahid.arshad

    Continuum of Calculate pull force of magnet?

    Hi Guys, I am going to restart the discussion in the following thread https://www.physicsforums.com/threads/calculate-pull-force-of-magnet.162157/ I working on similar system where i am interesting to calculate the pull force of permanent magnet on nanoparticles in water suspension. I am not...
  40. S

    Is this evidence for a space-time continuum?

    I have always wondered if spacetime is actually smooth, or if it is granular such as a quantized spacetime. I remember learning that if you draw the smallest possible square around a circle, you will, of course, get a perimeter, 8 times the radius of the circle. Then if you draw in the...
  41. Sivasakthi

    Continuum mechanics in solids

    Hi all.. I am currently doing some works on the continuum mechanics. And trying to study the macroscopic behavior of solids ( for simplicity, taken homogeneous materials) upon the action of external force ( which is the stress; pressure). How is it possible to account for the changes that can be...
  42. Matt atkinson

    Streamlines - Continuum mechanics

    Homework Statement In Cartesian coordinates ##x##, ##y##, where ##x## is the horizontal and ##y## the vertical coordinate, the velocity in a small-amplitude standing surface wave on water of depth ##h## is given by; $$v_x = v_0 sin(\omega t) cos(kx) cosh[k(y + h)]$$ $$v_y = v_0 sin(\omega t)...
  43. Stoney Pete

    Is the Real Number Line Actually Infinite?

    Hi guys, Here is a little puzzle I have been wondering about. I can't solve it, perhaps you can help. We know there are infinitely many real numbers on the number line. Indeed, already between 0 and 1 there are infinitely many real numbers. So if a real number is a point on the number line...
  44. M

    Looking for Continuum Mechanics book using Einstein Notation

    I'm taking a course in continuum mechanics this semester and the instructor is using a set of notes to teach out of it, problem is, I don't really like them. Can anyone recommend an engineering/applied physics oriented introductory continuum mechanics textbook that uses the Einstein summation...
  45. Telemachus

    Continuum mechanics, physical interpretation of terms in balance eq.

    Hi there. I'm reading Gurtin's 'the mechanics and thermodynamics of continua', and working some exercises of his book. In the section 21: 'The first law: balance of energy', after the derivation of the balance equation, he uses an identity to rewrite the balance of energy. The balance of energy...
  46. B

    Integrating Discrete Spaces and Time: Implications for Continuum and Precision

    We know that in order to be integrated a function must be continuous. Does this imply that space and time must be a continuum? If they were considered discrete, say at the level of Planck's unit, would this affect the integrability of functions? It it would not, would it affect the precision...
  47. B

    Understanding the Incompatibility of Discrete Space and Time with Relativity

    Relativity and continuum I have read contrasting things about the issue. Can you explain briefly why the assumption of discrete space and time is absolutely incompatible with relativity? What is the main problem? Lorenz invariance or what? Thanks
  48. B

    Action Principles in Continuum Mechanics?

    Is there any book that does what Landau does in Fluid Mechanics and Theory of Elasticity, only using a Lagrangian/Action-principles the whole way through? I can really only find brief tiny descriptions like this one in books on other topics, is there nothing that does for fluids/elasticity...
  49. R

    Time average in a continuum probability distribution

    Homework Statement I have a question that looks so stupid that I have never dared to ask. If I want to measure the time average from t=0s to t=1s of a given f(t), the solution is compute the following integral: TA = 1/T*∫F(t)dt However, I have some doubts about this calculus.Homework...
  50. nomadreid

    Questions about real-valued measurable cardinals and the continuum

    Putting the following three statements together: (a) Assuming that the continuum hypothesis is false, the power of the continuum 2\aleph0 is real-valued measurable. (b) The existence of a real-valued measurable and the existence of a measurable (= real-valued measurable & inaccessible)...