![]() ![]() To really keep this guy happy you probably need larger trackpad, but I’ve never felt the steam controller lacking in that sense personally. I don’t think it will be a big issue to accommodate, though would I think it forces the design towards you holding at least that side of the controller more palm up – as that seems the way to allow a larger ball and comfortable movements for the thumb (and you want a larger ball for the larger precision aim travel and so the curvature isn’t too extreme for the thumb to follow comfortably – either that or a pretty small ball were most inputs will be flick and brake with only a small amount of precision aim travel – which will probably work great too thinking about it (how I usually use my steam controller)).Īll that said I’m not sure it can really best the steam controller’s rather neat trackpads – best of all worlds, able to play joystick, trackball, trackpad, or button pads (the only big downside being the brain bending nature of having something that feels the same behaving in a completely different way, takes a while to get used to every time you change). The only challenge for a gamepad format is how big you can make the ball and still sink it into the controller enough to be comfortable to use – Though a larger ball gives a larger range of ‘precision aim’ before you have to take your thumb/finger off the surface it also will add mass that could make it a bit fatiguing to use, and you still need all the space for the ball inside. And shouldn’t loose out in precision aim over that method either, or at least not by enough to make the hassle of that thumb disk worth it… Trackballs have a few advantages – like you can flick them and if the friction is low enough/ the ball high enough inertia they just keep rolling – so going for those longer movements is easy. Seems backwards to have a thumb disk in that dish rather than a trackball arrangement to me. Very interesting, I’d love to try it, maybe it really will change my thoughts below. Posted in Peripherals Hacks Tagged game console, game controller, keyboard and mouse hack Post navigation If you are interested in working with him, be sure to get in touch! Modding controllers for fun and performance are great, but for gamers with prosthetic fingers it’s a requirement. ![]() is already working on V6, but feels close to the limit of his skills. ![]() For these situations, embedded a thumb stick on the back of the controller to allow for fast sideways movements using his middle fingers. While this “thumb mouse” has excellent precision, it can be a bit slow when you need to make large movements, like when performing 360° no scopes for the clips. It had slightly too much friction for ’s liking, so he embedded an array of small ball bearings in the surface using magnets. The concave surface was made from the diffuser of a large LED light bulb. On V4 he flipped the optical sensor around, embedding it in the controller, with a small circular “mouse pad” attached to his thumb. However, found that one’s thumb doesn’t work well for traversing a large flat surface, but works better with a concave surface. Mouse Pro Controller V1-3‘s main focus was to create the largest possible moving surface for an optical thumb mouse for precision aiming. His latest Mouse Pro Controller V5 features an inverted mouse riding on ball bearings. has been working to create a controller that’s the best of both worlds. Controller vs keyboard and mouse is one of the never-ending battles in the world of gaming, with diehard proponents on both sides of the fence. ![]()
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![]() If you have problems using a trainer in combination with Windows Vista, 7, 8 or 10 then make sure to run the trainer with Administrator rights and when needed in Windows XP or Windows 98 compatibility mode!.In ALL cases this is a FALSE ALARM as NONE of the Game Trainers GCW contain known malicious code! More info in the PC Games FAQ! Some Game Trainers are sometimes reported to be a Virus or Trojan, the most common is a keylogger called HotKeysHook or the file has been packed/protected with VMProtect or Themida and is recognized as Win32/Packed.VMProtect or Win32/Packed.Themida.In most cases using a No-CD or Fixed EXE will solve this problem! Some original games do not work when a certain application has been installed, like DAEMON Tools.When using Fixed Files make sure to use a Firewall which controls outgoing traffic, as some games call back to report the use of these modified files!.When this happens use the original EXE to play online, else you could find yourself banned from the game! Some No-CD/Fixed EXE files work fine in Single Player mode but are detected to be modified when trying to play online.Always make a backup of the files that are overwritten by the File Archive, as the original files are usually required to update the game to a newer version or to play Online!.Stronghold Legends SPECIAL EDITION BONUS MAPS.Stronghold Legends v1.2 (Collection) +7 TRAINER.Stronghold Legends: Steam Edition v1.3 +5 TRAINER.Stronghold Legends: Steam Edition v1.3 +13 TRAINER.Stronghold Legends: Steam Edition v20161219 +1 TRAINER.Stronghold Legends: Steam Edition v1.3 +8 TRAINER.Stronghold Legends v1.0 No-DVD/Fixed Image.Stronghold Legends v1.2 No-DVD/Fixed EXE.Stronghold Legends: Steam Edition v1.3 Fixed Files. ![]() Stronghold Legends: Steam Edition v20161029 Fixed Files.Game or Patch Questions? Visit FileForums ![]() ![]() Some features of this app even use memorization methods such as drawing, recording photos or using mnemonics. 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Unlike reminders, it doesn’t notify the user every time. It's helpful for keeping track of progress and planning ahead. You can even combine these methods for a greater effect! This app keeps a log of daily tasks and lets you set priorities. When performing everyday household chores or school projects, Boards and Lists are welcome tools. Both app users need to be online in order to connect to various projects. You can work on projects with Todoist Premium by collaborating with other people. Add notes to tasks to arrange and rearrange them quickly. You can customize each assignment by changing the name, priority level, due date, and more. These projects let you complete specific tasks related to being productive. Todoist Premium lets you create many projects today. You can easily change the name and color of each project by customizing them in the app. You can use this app to manage side hustles, school or work projects, and many more. There are so many wonderful apps and games in the market today- but most people use their smartphones all the time, which leads to less productivity. It allows people to keep track of a wide variety of things, including their personal, work and school schedule. Todoist Premium is a productivity app that's very popular among users today. This app is perfect for staying organized and on top of things. It's also easy to create new tasks and share them with other people. You can add notes to tasks and change the order of priority. This app is a phone-based version of its website counterpart. Each project can be customized to include different alarm options. You can create multiple projects with this app, including projects for hobbies, work, personal things and shopping. This app allows you to perform multiple projects at once and keeps you on top of things with your smartphone. With Todoist Premium, you have access to one of the most comprehensive productivity apps available today. These programs can be used for any purpose, such as school, work or personal matters. Today, you can download several helpful apps and websites that will help you create your to-do list. Get it right now and enjoy many features. Todoist Premium is the most productive app today. Use it for school, at the office or in your home. ![]() ![]() Customize this app to keep up with your work on a regular basis. This helps to get everything done more quickly and effectively. Additionally, you can work with others on shared projects. You can customize the name, due dates, notes and other aspects of a task when using this program. Creating these projects allows you to break up your work into smaller sections throughout the week or over a longer period of time. These include videos, work and school projects, chores or other miscellaneous notes. Projects in this folder allow you to create many different assignments inside a single folder. Todoist Premium is part of the complete package. The best to-do list app is Doist Premium, which was created by the company. Many people and businesses see the positive effect of productivity apps thanks to their availability. Some people still use default note-taking apps on their phones others use specialized apps. ![]() Today, many people use special productivity apps to manage their work, school and personal lives. ![]() ![]() Selected by LEGO® Ideas members (formerly known as CUUSOO). Ecto-1 vehicle measures over 3” (9cm) high, 7” (20cm) long and 2” (6cm) wide The perfect gift for fans of LEGO® bricks and Ghostbusters™ of all ages This item: Ghostbusters Plasma Series Ecto-1 Toy 15-cm-Scale Afterlife Collectible Vehicle, Children Aged 14 and Up E95575L0 75.00 75. Includes building instructions and exclusive Ghostbusters™ information booklet Build the model that LEGO® Ideas members requested LEGO: Ghostbusters 75828 ECTO 1 & 2 - BNIB - Adult Collector - 2016 Movie. Jump in Ecto-1 and track down some ghosts! Ecto-1 vehicle features Ghostbusters™ logo decoration, paranormal detection equipment, removable roof, tracking computer and seats for 3 minifigures Buy It Now 155 results for lego ecto 1 21108 Save this search Shipping to: 98837 Shop on eBay Brand New 20.00 or Best Offer Sponsored LEGO Ideas 21108 Ghostbusters Ecto-1, All Minifigures + Slimer & Marshmallow Man Pre-Owned 153 product ratings 71.00 41 bids 1d 18h left (Sat, 05:49 PM) +17. Includes 4 minifigures with proton packs: Peter Venkman, Ray Stantz, Egon Spengler and Winston Zeddemore So if there’s something strange in your neighborhood, strap on your proton pack and get ready to help Peter Venkman, Ray Stantz, Egon Spengler and Winston Zeddemore bust some ghosts! 4 minifigures with proton packs included. Filter 405 results for ecto 1 lego set LEGO Icons Ghostbusters ECTO-1 Car Set 10274 LEGO 209 239. This unique set also includes a fascinating booklet containing building instructions, selected images and behind-the-scenes details about the classic Ghostbusters™ movie. In-store pickup, ready within 2 hours Same Day Delivery Scheduled contactless delivery as soon as today Shipping Free with RedCard or 35 orders Exclusions Apply. It also features cool Ghostbusters™ logo decoration, removable roof, tracking computer and seats for 3 minifigures. ![]() Celebrate 30 years of ghost-busting action with the iconic Ghostbusters™ Ecto-1 car! Selected by LEGO® Ideas members (formerly known as CUUSOO), this fun and iconic vehicle from the blockbuster ‘80s movie is fully loaded with all the paranormal detection equipment needed to track down those ghastly ghosts. ![]() ![]() ![]() Take a look at this post for more ‘process’ photos e.g. So what do you reckon? Does the idea of fruit and cheese on a pastry tart float your boat? I hope you’re brave enough to give it a try. You can serve this puff pastry tart by itself, but I think a simple green salad and dressing on the side is the perfect accompaniment. What more could you want to get you right in the mood for spring? Or Autumn, if you happen to be where we were last week. Now that I’ve got my head around pears being the stars of the show in something savoury, that is!īut why not? The resulting taste is salty, sweet, crunchy, juicy, and tangy all at the same time. Doesn’t everyone love baby tomatoes, pesto and prosciutto?īut pear, feta and walnut seems like a bit more of a left-field choice for a tart topping, right? Well, maybe, but for me it really does work. The result is a posh pear dessert that will leave eaters in a quiet hush as they savor each ornate, little bite. The entire thing is then brushed with a bit of apricot jam or honey glaze. This baby tomato, pesto and prosciutto tart is one of the most popular recipes on this blog, and I’m not really surprised by that. Elise Bauer Bosc pears are smartly laid out over a layer of frangipane (a fancy name for almond filling) on a pâte brisée crust. They’re posh enough looking to serve to guests, but you won’t have to spend all morning in the kitchen making them = exactly what this lazy girl likes. I make them all the time, because they look impressive with hardly any effort. If you’ve never made a savoury puff pastry tart, you really are missing out. So, the next day for lunch I decided to skip the salad after all and throw together a quick puff pastry tart with toppings inspired by the flavours in the salad I’d enjoyed the most while away: classic pear, feta and walnut. We’d also gone on a bit of a salad binge while over in Perth, after months of ‘salad deprivation’ over here in wintery England, and that’s what we were both still craving.īut the weather didn’t seem to have warmed up one single degree while we’d been away. When we finally got home last week after our looooong flight from Australia, we both felt desperate to eat something other than sloppy gloop with rice and beyond-weird spongy things in packets. Serves 6.If you’ve got ready-rolled puff pastry in your freezer, and a few tasty odds and ends in the fridge, you’ll never be short of something to impress last-minute guests with! Serve the tarts warm with vanilla ice cream alongside. To serve, divide the tarts among individual plates and sprinkle with the pistachios. Transfer to a wire rack and let cool for 10 to 15 minutes.Ħ. ![]() Bake until the puff pastry is crisp and golden, about 25 minutes. Brush the edges of the pastry with the beaten egg and sprinkle with turbinado sugar. Arrange one pear half on each of the pastry rounds, gently pressing on the pear half to fan the slices but leaving the slices overlapping. Using a sharp knife, slice the pears vertically, stopping 1/2 inch (12 mm) from the stem end so the pear slices stay attached. Transfer the rounds to the prepared baking sheet.ĥ. ![]() Using a 5-inch (13-cm) round cutter, preferably fluted, cut out 6 rounds. To assemble the tarts, roll out the puff pastry to about 1/8 inch (3 mm) thick. Line a baking sheet with parchment paper.Ĥ. ![]() (For a deeper color and flavor, once the pears have cooled, cover and refrigerate in the poaching liquid for up to 2 days.)ģ. Remove from the heat and let cool to room temperature, leaving the pears in the poaching liquid. Add the pears, reduce the heat to a gentle simmer and cook, turning the pears occasionally, until very tender, about 1 hour. Bring to a boil, stirring occasionally to dissolve the sugar. In a small saucepan over medium heat, combine the wine, sugar, lemon zest and juice, star anise, cinnamon stick and ground cinnamon. Cut each in half lengthwise and, using a melon baller, remove the core from each pear half.Ģ. Peel the pears, leaving the stems intact. Divide the filling evenly between the two tart pans. Vanilla ice cream for serving (optional)ġ. In a medium bowl mix together the cream cheese, sugar, egg white, and vanilla until smooth and creamy.1/2 cup (2 oz./60 g) coarsely chopped toasted pistachios.1 sheet frozen puff pastry, thawed according to package directions.For a pretty presentation, use a leaf-shaped piecrust cutter or a paring knife to cut out a leaf shape, and attach it to the edge of the puff pastry round using a little bit of the egg wash. For the best results, choose an all-butter frozen puff pastry, which is available at most gourmet grocers. Using purchased puff pastry makes assembling these pretty autumn tarts a breeze. ![]() ![]() ![]() To learn the basic parts we will be using all semester: ![]() To a beginner the compound seems more complicated than it has to beīecause he does not recognize the parts. But to a beginner there appear to beĤ parts. Every chemist knows that it has two parts, a cation and an anion.Įvery chemist recognizes the two parts. Great simplification and you will be hopelessly confused the rest of the term.Ĭonsider the compound NH 4NO 3. Will study, the carbonate ion is never broken up, so it makes sense to treat itĪs a single entity rather than the sum of its parts.īut if you don't recognize carbonate ion, you won't be able to take advantage of this But it considerably simplifies things if we view theĬarbonate ion as a whole rather than as parts. What about the subscript inĬO 3 2-? Well, it means there are 3 oxygen atoms for eachĬarbon atom in the ion. Subscript in the formula: Na 2CO 3, 2 Na +įor each CO 3 2. To balance the charge of a carbonate ion. Has a charge of +1 and carbonate ion has a charge of -2, it takes 2 sodium ions The total charge must be zero for any compound. The superscript in each ion represents the charge, or oxidation state Sodium chloride NaCl Na + Cl - sodium carbonate Na 2CO 3 Na + CO 3 2- calcium chloride CaCl 2 Ca 2+ Cl - calcium carbonate CaCO 3 Ca 2+ CO 3 2. The cation comes first, both in the name and in the formula for Ionic compounds consist of two parts: a positive ion (cation) and a negative Please visit the new website at Metathesis Reactions Introduction ![]() From Caveman to Chemist: Metathesis Reactions This is the old Caveman Chemistry website. ![]() ![]() ![]() ![]() This includes a more gradual and smoother drag lever adjustment, improved cranking torque, increased durability and shape specifically designed for serious jigging anglers. Maxel is leading the way with ultra deep water reels for slow pitch jigging with the development of larger Rage models. The design and level of workmanship during the manufacturing process has taken the new models to a higher level. This may be the Mac Daddy of all slow pitch jigging reels for deep water and BIG fish! Rage 20H/HL are designed with Shallow Spool Non Clicking Alarm Single Drag System. The speed is also outstanding and it has the power to enable jigging in deep water. It has been tuned to deliver a more compact. Precision A new generation of specialized reel for slow jigging. Réglage du frein par levier hyper sensible, avec. SKU: Rage25HL Categories: Conventional reels, Reels Brand: Maxel. The modified EVA oval knobs are lighter weight and more user-friendly. The RAGE25H is constructed of aluminum and features nine sealed stainless steel ball bearings, a dual drag that puts out 17 lbs of max and a double ball bearing clutch anti-reverse system that. The New Transformer Modifications: - The side plate is now machined in an arc shape to provide more comfortable grip. Maxel Rage 25 Slow jigging Reel 2020 Machine one-piece aluminum frame Cold forged aluminum spool Compact and lightweight Precision machined stainless steel. Engrenages en acier inoxydable usinés avec précision. Fast, smooth and silent, the Maxel Transformer High-Speed Jigging Reel is designed to satisfy your need for a fast and efficient jigging reel. Caractéristiques : - Cadre en aluminium monobloc usiné. The reel foot is now fixed with four screws instead of two for a more secure attachment and the drag lever has been redesigned to produce a clicking sound. Le Rage 20H/HL est conçu avec une bobine peu profonde, une alarme anti-cliquement et un système de traînée unique. The additional ‘M’ position to the drag lever, between the ‘free’ and ‘strike’ settings, that brings significant advantages when using ultra-thin braided line, providing more precise control and minimizing line breakages. Slt, je voudrais savoir quel sont vos préférences entre un Shimano ocea jigger 2001 NR-PG et un Maxel Rage 25 HL pour faire du slow jigging dans Maximum 250 m monté sur une Maxel element 511. A name that has become synonymous with premium quality, Maxel continuously delivers when it comes to meeting expectations. Slt, je voudrais savoir quel sont vos préférences entre un Shimano ocea jigger 2001 NR-PG et un Maxel Rage 25 HL pour faire du slow jigging dans. Features : Machine one-piece aluminum frame Cold forged aluminum. The design and level of workmanship during the manufacturing process has taken the new models to a higher level. Maxel has been manufacturing high-quality big game trolling and deep water jigging reels since 2003. Rage 20H/HL are designed with Shallow Spool Non Clicking Alarm Single Drag System. Īrmed with a hyper-sensitive drag system which allows it to be filled with very thin PE braid -as fine as PE0.6. We are leading the way with ultra deep water reels for slow pitch jigging with the development of larger models Rage 60H, 90N ,and 90. It has been tuned to deliver a more compact appearance, lighter weight and smoother gearing and drag, also a lighter feel when cranking the handle. Precision! A new generation of specialized reel for slow jigging. Maxel Rage 25H Machine one-piece aluminum frame Cold forged aluminum spool Compact and lightweight Precision machined stainless steel gears Hyper-Sensitive. Precision A new generation of specialized reel for slow jigging. ![]() ![]() ![]() ![]() In other verses of the song, even more is promised, including candy. What a blast! Yes, those are the things promised for tomorrow if only you’ll just go to sleep tonight. But any singer can replace that name with any other.Ĭhocolate, milk, and cake. Oh, and the subject of the song below is Colas, which is a name, likely a nickname of Nicholas. The rest of the lyrics have more to do with baking than anything else. If you type “fais do-do” into Google Translate, it will spit out: “go to sleep.” So, it seems plausible that “go to sleep” and not “go do si do” is the aim of the tune. (Translation of dodo from the GLOBAL French-English Dictionary 2018 K. ![]() Almost like in English as “go,” as in go to sleep. But, then, one wonders what Fais has to do with “do si do”? The cooing “do-do” is likely a sweet little abbreviation of the French word for “to sleep,” which is “dormir.” Though other historians have called it the shortening of “do si do.” If this explanation is correct, then, that would further link the Cajun dance with the tune. The hope, some have said, was to get the kids to sleep soon because the worry was the husband would be dancing with someone else. Or, if not the entire song, they would repeat the phrase, Fais do-do, attempting to calm the children to sleep. Adults can say 'dodo' when they are trying to be cute. Elle y avait lair de mignonne -) Yes, 'dodo' is for children but I am 32 years old and a 25 year old french girl said to me 'je vais faire dodo' on the day we met. The Cajun dance party version is named after the song, which was used traditionally by the mothers attending the gathering, who would take their infant children to the “cry room” when, say, the babies were hungry or upset, and they would tend to the children and sing them the song. Les adultes peuvent dire 'dodo' quand elles essaient dêtre mignonnes. ![]() ![]() ![]() Vince Gill and Keith Urban also enlisted an array of acts - from Jason Isbell, the archetype of Americana singer-songwriters, to country-pop heavyweights Florida Georgia Line and Maren Morris - for an arena show benefiting the Country Music Hall of Fame and Museum. Later in the year pop culture blogs would zero in on Simpson's impassioned Facebook essay accusing the country music industry of neglecting its legacy after the announcement of an award named for Haggard, drifting from its moorings and diluting its traditions, criticism that wouldn't carry such an emotional charge in virtually any other genre. In the same month, self-styled outsider Sturgill Simpson released an acclaimed album embroidered with sprawling soul arrangements and psychedelic rave-ups. Jon Pardi - foremost in a group of twenty- and thirty-something neotraditionalists gaining momentum at radio for the first time this decade - highlighted the vintage leanings of his amiably shuffling single with a music video that literally stepped backwards in time. Chris Stapleton, held up by many as a throwback to rugged country masculinity, swept four major categories at the Academy of Country Music Awards. Dolly Parton, who decades ago perfected the art of packaging a downhome persona for glittery, discofied popular consumption, announced a back-to-roots tour and an album that placed an Adult Contemporary sheen on Appalachian folk. Charlie Daniels, newly elected to the Country Music Hall of Fame, did a round of interviews about what that honor meant. Merle Haggard's passing inspired countless remembrances. Consider the amount of influence wielded by exploratory pop affinities and fealty to established symbolism during the month of April alone - and how difficult it often was to tell the two orientations apart. ![]() This particular year, though, country's boundaries seemed stretched to new extremes. Built into the very foundation of the mainstream country music industry and its institutions is a tension between continuity and changeability that never really goes away. That's because country music is simultaneously a genre defined by its spelled-out relationship to tradition and a hit-driven radio format akin to the Top 40. But so is the imperative to let the music evolve with changing times and expand its reach in the process. The impulse to look back - to figures enshrined as icons, to sounds that conjure shared references, to historical dependability in general - is strong. In country music, there's no escaping the past. Even atonal music, which tends to be less popular among classical enthusiasts, has a kind of crossover niche, since it is widely used in filmmaking and television production scores "to depict an approaching menace", as noted by Charles Rosen.Dolly Parton, pictured with Kacey Musgraves, Reba McEntire and Jennifer Nettles (obscured), accepts the Willie Nelson Lifetime Achievement Award onstage at the CMA Awards in Nashville in November. For instance, Sacred Harp music experienced a spurt of crossover popularity as a result of its appearance in the 2003 film Cold Mountain, and bluegrass music experienced a revival due to the reception of 2000's O Brother, Where Art Thou?. ![]() In practice crossover frequently results from the appearance of the music in question in a film soundtrack. These covers were popular with a much broader audience. For example, in the early years of rock and roll, many songs originally recorded by African-American musicians were re-recorded by white artists such as Pat Boone in a more toned-down style, often with changed lyrics, that lacked the hard edge of the original versions. ![]() In some contexts the term "crossover" can have negative connotations, implying the dilution of a music's distinctive qualities to accommodate to mass tastes. If the second chart combines genres, such as a "Hot 100" list, the work is not a crossover. Freebase Rate this definition: 0.0 / 0 votesĬrossover is a term applied to musical works or performers appearing on two or more of the record charts which track differing musical tastes, or genres. ![]() ![]() Multiplying throughout by y ′ = d y / d x, There is a simple first integral of this equation. An Important First Integral of the Euler-Lagrange Equation This general result is called the Euler-Lagrange equation. ∂ f y, y ′ ∂ y − d d x ∂ f y, y ′ ∂ y ′ = 0. One point in the interval, and deduce that Since this is true for any infinitesimal variation, we canĬhoose a variation which is only nonzero near Δ J y = ∫ x 1 x 2 ∂ f y, y ′ ∂ y − d d x ∂ f y, y ′ ∂ y ′ δ y x d x = 0. Then integrate the second term by parts, remembering δ y = 0 at the endpoints, to get To make further progress, we write δ y ′ = δ d y / d x = d / d x δ y, Δ J y = ∫ x 1 x 2 ∂ f y, y ′ ∂ y δ y x + ∂ f y, y ′ ∂ y ′ δ y ′ x d x = 0. Then under any infinitesimal variation δ y x (equal to zero at the fixed endpoints) J y = ∫ x 1 x 2 f y, y ′ d x y ′ = d y / d x. To emphasize the generality of the method, we’ll just write General Method for the Minimization Problem Of course, this also means y ′ x → y ′ x + δ y ′ x where δ y ′ = δ d y / d x = d / d x δ y. ![]() That is, we want δ J = 0 to first order, if we make a change y x → y x + δ y x. To minimize the total area, so this is a different problem!) The soap film is not constrained in that way, it can stretch or contract (You might be thinking at this point: isn’t this identical to the catenary equation?Ĭhain has an additional requirement: it has a fixed length. Subject to given values of y at the two ends. J y x = 2 π ∫ x 1 x 2 y d s = 2 π ∫ x 1 x 2 y 1 + y ′ 2 d x The total area is given by integrating, adding Sequence of rings or collars, of radius y , and therefore area 2 π y d s. We need to find the function y x that minimizes the total area ( d s is measured along the curve of the Taking the axis of rotational symmetry to be the x -axis, and the radius y x, Propagates, its path traces out as “world sheet” and the string dynamics isĭetermined by that sheet having minimal area.) This problem is also closely related to string Neglecting gravity, which is a small effect). The minimum possible total area compatible with the fixed boundaries (and This problem is very similar to the catenary: surface tension will pull the soap film to A Soap Film Between Two Horizontal Rings: the Euler-Lagrange Equation Infinitesimal change in the curve of its position, subject to fixed endpoints,Īs a warm up, we’ll consider a simpler-butĬlosely related-problem. So, we’re looking for the configuration where the potentialĮnergy doesn’t change to first order for any Links, that is, or equivalently a continuous rope). The position of every point on the chain (in the limit of infinitely small Variables-it’s a function of a function, it depends on Hanging change isn’t just a function of a variable, or even of a number of The difference here is that the potential energy of the ![]() Infrequent in the problems we’re likely to encounter.) Is nonzero- what about x 3 near the origin? But such situations are ( Nitpicking footnote: Actually this assumes the second order term This method of solving the problem is called the calculus of variations: in ordinaryĬalculus, we make an infinitesimal change in a variable, and compute theĬorresponding change in a function, and if it’s zero to leading order in the Or internal friction, it will settle down again in the catenary configuration.įormally speaking, there will be no change in that potentialĮnergy to leading order if we make an infinitesimal change in the curve, y x → y x + δ y x (subject of course to keeping the length the Other words, if we nudge the chain somewhere, and its motion is damped by air Number of variables, and our problem is to minimize it with respect to arbitrary Here the potential energy is a function of a function, equivalent to an infinite Minimize a function with respect to a single variable, or several variables. J y x = 2 π ∫ x 1 x 2 y d s = 2 π ∫ x 1 x 2 y 1 + y ′ 2 d x, y ′ = d y / d x where we have taken the rope density and gravity both equal to unity for mathematical Minimizes the gravitational potential energy The catenary is the curved configuration y = y x of a uniform inextensible rope with two fixedĮndpoints at rest in a constant gravitational field. ![]() Technique developed can be successfully applied of a vast array of Here we’ll find howĪnalyzing that leads to a differential equation for the curve, and how the Second-order changes in the gravitational potential energy. Now going to look at a completely different approach: the equilibriumĬonfiguration is an energy minimum, so small deviations from it can only make Length of chain, the tension at the two ends and its weight, balanced. Shape of chain hanging between two places, by finding how the forces on a We’ve seen how Whewell solved the problem of the equilibrium ![]() |
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