Hello everybody. Welcome to day one of our 30 day challenge here in the East Coast. I know that our Australia friends already started yesterday because of the time difference and I've already gotten so many responses, I mean we are basically at 400 people right now across the world doing this, which is incredible. Before we get into the subject at hand, I do want to tell you guys if any of your friends or anybody watching is just now learning about this challenge and you want to start now, it's never too late. It's never too late to start working on yourself and doing this kind of work. In fact, you are always the right time, the right place in the right time. So please don't hesitate to email me at shadowworkchallengeatgmail.com and I will send you the 30-day template. If you want to shift the dates, that's totally fine. This is just a template and it's something I put together that you can use going forward whenever you want to use it. And so if you want that template, just email me. No worries. Again, it's never too late for you to start working on yourself. Never. You know, as long as you're alive and breathing and you have a body, then now is the perfect time to start, right? It doesn't matter. You didn't miss the boat. I promise you you didn't miss the boat. As far as the gifts at the end of the month, it's going to be a raffle anyway. So it's going to be picked at random. So there's no like nothing you have to do to qualify.  And that's it because the names will be picked at random. There's a couple of things I wanted to talk to you guys about. First of all, I kind of wanted to tell you guys how I discovered Marnie Alton, the bar teacher that you worked with today. I don't know if I've shared this story before. So many people, especially in the support group on Signal, which again, that link will be down in the description box below if you want to join the support group, had never done bar before or had never done bar by Marnie Alton before. I think a lot of people were very shocked by how sore and how sweaty they got during this exercise because, of course, it looks easy. Usually if it looks easy, that means it's probably pretty hard. So how I discovered Marnie Alton. So back in 2016, I broke my sacrum. I actually broke my sacrum in a Mysore room, not here in Atlanta. I was actually in another state completely with a totally different teacher. And I got a bad adjustment and that I had my legs behind my head and my legs got pulled a little too far behind my head and it popped my sacrum. And so I broke it and that's a really hard place to have a break. You can't do a whole whole lot like there's no casting or anything like that. I maintained a practice throughout that whole time. I was a very modified practice, but I still got on my mat. I still moved and after a while, I was just really an emotional wreck over this break. I was having a really hard time regaining some stuff and it was bringing up a lot of frustration and a lot of anger for me towards the person who who broke my sacrum because this particular teacher is not a nice person. And so I think that kind of really like triggered me a lot, which was good, which is I needed that trigger. So I'm not saying that's a bad thing. But throughout this course, I was getting very emotional about not being able to heal my body. My arthritis started to get bad again. And one of my best friends who lives up in Canada, he's been on the show before Mike and Chris, he is a movement expert. He told me that I should try to incorporate some bar classes during the week. He had been he studies all sorts of movement. He's really fast and fascinated with the human body. He's in school to be an osteopath right now. He's also a raking master. I met him in India.  And he had just taken a bar course and he was like, I think you need to incorporate just a couple of classes during the week of bar just to really focus on restabilizing your core and your glutes to help your sacrum. And so I did that, I went to a few classes here in Atlanta and I wasn't really feeling the classes. It frankly was not that challenging. The classes here were not that challenging for me. So I kind of felt like that wasn't what I needed. And so I just kind of left it. And then right before lockdown happened, I was considering that again. And I had tried some Pilates and stuff too.  I'm just going to try this on YouTube and see if it's any different. And you know, can't hurt, right? It's 45 minutes, whatever. It can't hurt. And so I tried her bar and it was different than the other bar classes that I had done live in Atlanta. And I really liked it. I really liked her. I really liked her energy. I liked what she had to say. And so for during my healing process, I would do my six days a week of my Ashtanga practice, but a couple, two to three days a week. In the afternoon, I would do this 45 minute bar just to stabilize my core and my glutes. Well, around that time, I had reached out to her on Instagram regarding my sacrum because I just had some questions to see what her perspective was. And she was so kind. She got back to me immediately. We had a full on conversation. She was very generous with her time, very kind, very loving. And at that point, she was teaching out of her studio in Los Angeles. Well, when lockdown happened and I lost my Shala, she also eventually lost her studio as well. Of course, in Los Angeles, it was way more intense. And I remember when she lost her studio, she kind of made a statement about it was very sad. But she moved all of her classes onto an online platform called M, as in Marni, M backslash body, which I'll put a link to that in the description box below. And so what I did was I actually joined her platform. I do it month by month. And I started doing some of her full length classes and for harder classes with the weights and the movement ball. And it really targeted. I realized that it was making me so freaking strong. And my practice, my yoga practice, all of a sudden just jumped forward by leaps and bounds. Like something about the way Marni was teaching transfixed everything that I was working through in my practice. And so ever since then, I've been a huge supporter of hers. And I can tell that when she teaches her bar classes, especially off her own platform, where she has more, where she can talk more, she has Q and A's a lot. She has really, really studied energetic body. I can tell by the things that she says, even though she's coming at this for more of a workout versus a practice, she is very much incorporating the spirituality into the physical, into it. And I can tell by the way that she choreographs her own classes, what type of emotion she's trying to trigger in you. The ball work, it's really into the inner thighs, which you're not going to do the ball work on the YouTube. It's no equipment. But if you do do the workout website, you'll see you can get the ball and squeeze the ball. And it's really going to, that's going to trigger Moabunda. And so there was a lot of healing that took place for me with Marni Alton at that time too, because I lost my shawla. I started speaking out more about my political beliefs on YouTube. And I had other yoga teachers threaten me that they were going to try to get my authorization taken away because of my views. And so I was feeling very down about the community that I had been a part of for so many years. And I felt very uplifted by Marni Alton's community, even though I'm not vocal, like she has a platform, she has a group, all those things where they all know each other. I'm never really vocal, but just seeing how kind she is to her clients or students in the beginning of her life classes, she's always interacting with the people. She's so kind to people's children. In fact, one of a little girl made her a friendship bracelet and mailed it to her and she wears it now. She's just such a kind person and that kindness really, really was important to me at that time. And so not only was my body changing and healing and my practice was getting stronger, but I all of a sudden felt supported again by a community that I didn't even know. And so that's what I say about her. Now she's not the person that knows the intel that we know. She doesn't know about Mr. Teal, that kind of stuff. But in my opinion, she is already very ascended because she's healed a lot of herself. You can tell she has worked hard on herself in order to be able to now help other people. And so I'm a huge, huge, huge support of hers. Her method, what she's created has literally changed my body. It's literally changed me and I will forever, forever be so grateful to her for putting herself out there. And one of the prizes at the end of this month when the raffles is for a gift card to have access to her website for a month. So you too can, for taking all of her classes and all that kind of stuff. I have reached out to her on Instagram again. I wanna get her on the channel and talk with her about the spiritual body, but I don't know if she actually checks. When I talked to her before, I don't think she was as popular as she is now. And so I think she was checking her messages, but I don't think she actually checks her message. I don't check my Instagram messages at this point. And she has far more followers than I do. So I don't blame her if she's not checking me. That's overwhelming when it's just you. And so what I might do, I might give it like a month or so and I might just email her and see if I can get ahold of her through her business to see if she'll come on and talk more about her life. I did post one of her interviews on the community tab. I'll repost it down the description box below where she talks about kind of her come to Jesus moment and the work she's had to do in her life and the certain situations that have come up in her life that have brought her to this place. And so I thought that's really important to share because I think sometimes we do look at people. I mean, Marni Alton is beautiful. She's a beautiful woman. She's very fit. She's in her mid 40s and she looks like she's 25. She's from an interview I listened to of hers a while ago. She's been happily married for like 15 years. She just seems to have everything together. And so I think sometimes when we see people like that we just assume that everything's been easy for them and that life has been easy for them but that's never the case. No one gets out of this world alive. We all have our crosses to bear. She talks about growing up poor and having to like establish herself and figure out how to run a business. She talks about all the struggles that she went through. And so I'm gonna place an interview down in the description box below because those stories I believe are what connect us as humans. You know, as far as the physicality I always say don't compare somebody else's chapter 10 to your chapter one. Yes, Marni Alton is very, very, very fit. But she's been doing it for like 14, 15, 16 years. Same as me. I've been doing this for 16 years. Of course, of course, logically my level of fitness is gonna be higher than someone who is just now starting. Of course it is. That's just science. It doesn't make me better. It doesn't make me a winner. It just means that I'm in more of a place to help the people that have come behind me. And so those who are just starting now 10 years from now you're gonna be able to be in my position where you can put your hand out and help somebody else out. It's to whom much is given, much is expected, right? And so that's why I wanted to share her story is so that you can see someone like her that seems to just have a perfect life has gone through a lot to get where she is. And she's taken those adversities. She's taken those obstacles and she's healed herself through that friction and you can tell that through her interview and that's the opportunity we all have. Whatever obstacle that life gives us, it's not an obstacle, it's a puzzle. Actually, that's something Marnie Alton says. It's not an obstacle, it's a puzzle. It's just a puzzle, right? And you're the Puzzle Master. She said something once in one of her classes that I really, really liked because in our world, in this Truth or Community sometimes I think we think about that being awake means we know facts. And that's not true. Again, that's ito, that's outside knowledge. There is no difference. What I know and what a black cat knows is the exact same thing, we have the same knowledge. It's what we do with it that counts. And she said something in one of her classes, she was like, we don't fully wake up until we fully come into our body. And that is something that we've been talking about. That's something that Mary Magdalene speaks about in her Gospel. We will not ascend until we descend first. All right, which we go into our body and as Marnie Alton says a lot in her classes, we feel things here. We put our place, our bodies in a place to feel things. To feel things. One class she said, I'm paraphrasing. She said, there comes a point where this cage we think we've been in where we realize that cage isn't even locked. We can just push the door right at wide open. But we can't get to that point until we actually feel things. Until we actually come into our body. We descend into the knowing of our body. A lot of times I think that us when we embark on spirituality, we think it's so like ethereal and outside of us, but your body is the most spiritual instrument that you own. Being spiritual doesn't need talking to spirits. That's not what being spiritual means. Being spiritual means coming into your own spirit, which runs through your fascia, which runs down to your toes, which runs through your feet, which runs down up and down your spine. Talking to other spirits is just like talking to another human. There's nothing spiritual about it.  And we find that through the instrument of the spirit. What is the instrument of the spirit? It's the body. The body is the Shakti of this creation of the soul. It's like that's why I won't go to tarot card readers or any healer that is not working on themselves.  How can you channel me if you don't even channel yourself through your own pain, through your own body feeling? Okay, and so that is... Anyway, so what Marnie Alton said was very, very accurate. She also talks a lot about being the watcher, which is a huge yoga theory about just observing yourself, observing yourself. And that's what I really, really, really like about her. And I'm so glad that so many of you guys had such a good experience with her YouTube or 45 minute YouTube. That is the main bar YouTube you're gonna do over the course of this 30 days. There's another 30 minute one. I've put up a first as well from YouTube. Now, tomorrow, November 2nd, you're gonna be doing the exact same workout. And I did this intentionally. Sometimes there's gonna be a repeat of exercises. I do this and I picked everything I did in this, in this challenge I did intentionally. It was not random, but a lot of thought into how I wanted the workout scheduled to look like with my knowledge of energetic body. I did it in a certain pattern, okay?  That's a good thing. It worked. That means your body responded. It was supposed to respond, right? And so you're gonna have a different experience tomorrow with the exact same workout that you experienced today. And that experience is gonna bring you a new information. For some of you, you're probably gonna feel better doing it again because you know what to expect. And maybe you'll feel more grounded in your body. Maybe the soreness will bring you to a more grounded place into your body, more of a place of honesty and vulnerability in the workout. For some of you, it's gonna trigger more emotions.  And so that's why we're just experimenting to see how that affects you. If you shake in the workout, good. That means it's working. If you're sweating profusely in the workout, good, that means it's working, right? These are good things. I'm curious to see again for those who did it early in the morning, how did it affect your day? Do you have a better day today? Do you feel, does your mind feel clearer? Does your body, does your nervous system actually feel more relaxed during the day because you worked out this morning? Is, are you able to handle work better?  Are you able to handle traffic better? Because your nervous system is more relaxed. Another thing that came up, and I'm so happy, one of our subscribers is a personal trainer over in the UK, I won't say her name, but she had a question about us and she has been working out a lot replacing some of the workouts I've scheduled with some of the workouts that she normally does. And this is always gonna be your choice, but I noticed the workouts that she was doing were more cardio based. And I explained this in the comment section and we're gonna go deeper into this later in the challenge when most of you guys have gotten more adapted to exercising. Back in like the 80s and the 90s, cardio was like the only way people really thought of when they thought about losing weight because it's a big calorie burner, right? So I think a lot of us, I know this was something that I had ingrained in my head because my mother was a runner, I was addicted to running for a long time, the Jazzercise, the Zumba's, all those kinds of things. And as fun as those are, what we now know scientifically about cardio is that it's not ideal for weight loss, it's also not ideal for emotional body information and this is why. When we go into aerobic activity consistently, so if you're doing like hardcore aerobics for an hour, six days a week, what's gonna happen to your body? This is how amazing our bodies are. So your body isn't necessarily gonna register that you're on the elliptical machine or that you're on the treadmill or that you're in an aerobics class. What it's gonna start to register is that you're in fight or flight and so it's going to start to behave as if you were in a war. Now we are in a war, but we're in an information war but I'm talking about like a physical war and this is because our bodies are designed with the nerves to also protect us, right? We'll talk about that with back bending with the nerve endings in the front of the chest and what tends to happen with back bending a little bit later on in the challenge. So what tends to happen when people get into the habit of doing intense extreme aerobics a lot is for a lot of people, they're gonna have a mad calorie burn but they're also gonna be extremely hungry during the day and start taking in more calories and then their body isn't going to flush the appropriate calories. Their body is actually gonna start to slow down the metabolism because the body thinks you're in a war and so what happens after a war historically is a famine and so your body is gonna start to hold on to the fat and hold on to what it needs to so that when you hit famine, it has something to feed off of, okay? Usually people who do hardcore cardio especially if their disposition is Vata end up with higher anxiety because of that and so it's more of like burning out your muscles versus activating them. Now, when we look at the opposite of that like if we look at the bar classes and the yoga what's really happening is called anaerobic. So all this hardcore cardio is aerobic, the bar, the yoga is anaerobic. Now what happens with Marnie Alton's bar and with yoga, Ashtanga is you start with like five minutes of cardio because that if you do a little bit, the good thing about the cardio is it's gonna get the heart rate up which is gonna cause your blood to flush your system more and wake your body up. That's why Marnie Alton only does it for like five minutes. That's why in Ashtanga we only really get the heart rate up in the beginning of the sun salutations. It's to wake the blood flow up. It's not to put you into an aerobic state but just to wake up your prana, okay? But then when we get into the anaerobic we're building muscle and so when you're doing the yoga poses that you're holding or you're doing the squats and bar or squeezing the ball, you're an anaerobic. And this is why the anaerobic is so much better emotionally and physically for you than the aerobic because the anaerobic is going to pull off of your stored energy, okay? So the aerobic is just gonna grab for any energy it can't where the anaerobic is gonna be, it's not gonna be in a stressed out state. The aerobic is gonna put you in a stressed out state anaerobic is not. And so the body is gonna pull from stored energy. Well, what is stored energy? Physically that looks like fat, right? It looks like cellulite. That's what's gonna pull from to give it energy, okay? But energetically what the stored energy is is stored emotion, it's the stuck emotion. So over here in the aerobic we're not even touching the stuck emotion because we're too busy trying to survive. But in the anaerobic the body can survive and so it's gonna start to pull apart all the stuck stuff. That's why a lot of times dancers and yoga people have very long lean muscles is because of the anaerobic work because the body is pulling from stored energy and not frantically looking for any energy it can find. That's also why you find with bar and yoga the appetite typically doesn't increase. In fact, it might decrease because the body is not in fight or flight. The nervous system is getting a chance to relax itself into the work at hand. That's why in bar and yoga you're gonna be presented with opportunities to actually feel the emotions versus running from the emotions. So sometimes when I have a student that's a big time runner and it's having a hard time getting running away are not doing it as much. I'll say, well, where are you actually running from? What are you running? Because you're literally running. What are you running from? Right? With that being said, the cardio, the kick box cardio is gonna be offered. It is cardio, but I've heard many personal trainers who understand this say, if you're gonna do cardio just do it once a week. Once a week, that's all you need for your heart to get that exercise and everything else should be anaerobic, okay? That's why people lose a lot of weight doing yoga and doing bar too because they're going into the anaerobic. The dance with Richard Simmons is my favorite. I love it. But the aerobics that he offers you isn't as intense as some of the other stuff is. And so the body's gonna be able to relax into and actually have fun, okay? And that's why I picked Richard Simmons for the dance stuff because it's not gonna be putting your body in a stressed out state, yeah? So I just wanted to go ahead and go over that. Sometimes now if you're new to exercise this is not gonna be something that you're gonna be struggling with right now but if you're not new to exercise, with everything positive, there's also a negative. With everything negative, there's also a positive. This is the polarity, right? So with exercise, there's also the propensity people have to use exercises as escapism. And I know what that looks like. So that's like gardening. Like if you say, oh, I garden, no, you're escaping because you're not coming into your body. You're actually looking at something outside of your body, right? Using exercise as an intellectual spiritual modality means that you're allowing yourself to actually feel things in your body. You're not distracting yourself with something outside of your body, yeah? So like gardening is fun and it's good for you to get in the sun and be in nature but it's not, it should not replace your work. It's just something extra, you know? Like playing with your kids in the swimming pool is great exercise but it's not spiritual work, right? It's great exercise. It's good for your health but it's not spiritual work, yeah? And for a lot of people I think especially in the Western world, we've viewed exercise in one category where it's like for our physical health which is a great reason to exercise, don't get me wrong. Being physically healthy and using exercise to keep yourself healthy is amazing. It's a great reason to do it but it shouldn't be the only reason. Are we do it to abuse ourselves? We, there's a form of bulimia where it's over exercise where you binge and then you purge on exercise, right? A way to burn calories to punish ourselves to get into very tight genes and that's another reason why I told you guys at the beginning of this challenge, I don't want you weighing. I really don't want you weighing yourself. I don't, I think weighing ourselves takes us away from our body and it gives us a tool to judge our body from something that's outside of our body.   The only time I've weighed myself as of late is when my weight got really, really low and I got below 100 pounds because there was some health concerns, yeah? So, but beyond that, I let my body go to the weight that it wants to be at. And for me, I'm about five, four, my mom says I'm five, five, whatever. My weight is usually around 110, 115 pounds. That's where my body is comfortable and that's where it's been comfortable for most of my adult life. That's where my body functions the best is in that weight range. Now I am also Vata.  It doesn't matter. That's just your body disposition. There is not a one size fits all when it comes to this. And so if you, I can tell in my body now, so if I feel like bloated or if I feeling underweight, I can feel it in my body. My body, I've gotten to the point where my body can now tell me when something's not right, okay? And so I want you guys to get into that versus what the scales say, okay? I hope that makes sense. And I want you to, I don't want you, if you're using exercise solely as a way to lose weight then stop exercising, just watch what you eat because that should not, because if you're doing it that, if you're doing it for that reason and that reason alone, then you're gonna end up abusing yourself in the exercise. I want you, regardless of what your weight looks like, I want you to use the exercise as a way to feel things. When you're in the bar and the muscles, and the muscles are burning, hold on my second guys, that was my update with Tamara video downloading. Well, when you're in the bar, when you're in the bar and your muscles are burning, I don't want you to think about calories. I want you to bring your mind's eye into the quadricept, into that area of the body and feel it, be there, lean into that sensation.  New body is making, that's what my teacher says. New body is making, but old body has to break first. We have to get rid of the old. The opposite of war isn't peace. The opposite of war is creation. So we have to have controlled demolition of the old patterning. The body is the minefield. So the thoughts you've thought in your life has what caused your body to be in the stated sense. And so we have to work with both the mind and the body to reset patterns. And that physical sweat and burn is what's controlled demolition of the old so you can build the new. It's beautiful, it's amazing, it's wonderful. And so I hope that makes sense.  I'm really excited. Somebody in the group said, this is the thing I'm most afraid of. And I responded, this will probably be the thing you end up loving the most. And if I can do cold therapy, anybody can, cold therapy has literally changed my life. I struggle with inflammation because I have arthritis because I'm Vata. And I can tell and see it in my body when my joints are inflamed. And the cold therapy has helped me so, so, so much. It also, if you watch when Haas interviews on cold therapy, I placed one of his interviews in the challenge. He talks about how it treats depression. It treats migraine headaches. There's so much that cold therapy treats. And all I'm asking for is just five minutes. So after you do your workout, you get in the shower. You can make the shower warm, shampoo, clean yourself.   Let it feel and just breathe into it and just lean into it and allow it just to feel how it's hitting the cold waters, hitting your skin and going into your skin and allowing the inflammation to kind of calm down. Because at night, one of the bonus challenges is taking a hot bath, a hot therapy at night with the, with the salts, the magnesium salts. And so it's just five minutes. Anybody can do anything for five minutes. And over the course of the month, month, I'm going to be very curious to see how much the cold therapy has actually affected you. Changed your mood, changed your outlook and everything like that. So anyway, guys, leave me your thoughts down in the description box below. Tell me how you're doing, how you like to Marni. And once again, if you have any friends or family or if you're just now watching this, it's never too late to start. Just send me an email at shadowworkchallenge at gmail.com.  You are the storm. By doing this, as Tamara says, which will be released later. By doing this work, you're going to get more in touch with you, with your gut instinct. And that's the last thing that the controllers wants you to do. The controllers want you to be stuck, like, like controllable, you know. And the more you take your power back by really feeling your body, your body's your information highway, it's the best instrument. It's the one thing you're never going to leave in this life is your body.  So get cozy with it. Ask your beautiful body, what it wants you to know. What do you want me to know, right? And if you don't get an answer right away, just wait. The answer will come. It will come. This is not a one-time thing. This is a lifetime. It's like, but Tabi Joy said, with yoga practice, one year, no good, 10 years, no good, whole life, whole life.  Welcome to your new life. All right, guys. And of course, be proud of yourself when you see the muscles start to develop and the stomach gets smaller. I mean, that's the physical sign that the work is actually working. So anyway, I love you guys and I hope you're all doing good and I'm really, really, really proud of you. Please share these videos. I am heavily shadow banned. I don't know if you guys noticed that. Like it is laughable how shadow banned I am. Obviously we're doing something very right on this channel if they shadow banned me this much. So make sure you're sharing these videos out to people so that other people can join the challenge and take their power back too. All right, you guys, love you all and I'll talk to you soon. Bye.


So wat I'm going to do now is show you some something Beautiful, some beautiful mathematics. We can't do much in physics or we can't start this discussion properly as far as the gradients and Hamiltonians are concerned if we don't understand the concept of the minimum. Something is at a minimum and that is how nature decides what to do. Nature decides that the shortest distance between two points in a clean space is going to be a straight line, how does it decide that. is must be something underlying this decision in nature is the concept of the minimum and you've got to understand it through the concept of the calculus of variations so how do we do that it's going to be many boards and i'll have to have to wipe the board in between let's just have a look at this i have the x y plane and i have two points which is a starting point and a final point and there's various paths i can take through this so you could in space there's my x and there's my y i'm going to go from one point to the other point on some path you can imagine i can go to andromeda galaxy and back and there will still be one path if andromeda is is there at least if i can just go in a two-dimensional plane there is a straight there is many lines i can take between this but i want the shortest possible distance between those two the shortest possible distance now one way i can measure this any of the paths that i've that i can possibly choose is just to divide it into tiny little segments of say a specific length delta s so let's blow that up a little bit if i were to blow that up it's almost going to be a straight line you can most most lecturers would skip this button go straight just to the to the to the integral but this is good so this is going to be this arc length it has a length but it will have from that point to that point a delta y and a delta x so the length of delta s is very nearly going to be what well the square root of that percent this is by theory so that will be delta x squared plus delta y squared very beautiful and we know if i can let delta x approach what's limited the limit take the limit as this approaches zero this is going to become smaller this line segment will become shorter and right in the end i'm going to have this differential tiny little tiny little distance ds which is going to be the square root of dx squared plus dy squared we can all see that and i can rewrite this just want to show you how nice you can rewrite it of course you're going to say ds is going to equal the square root of i can take a dx squared out which will leave me one plus dy squared over dx squared it's beautiful i've just taken this out as a common factor if i would want to multiply those back in there of course i'm going to land back up there in other words ds i can write as the square root of one plus let's just call it y prime y with respect to x squared that's very ugly i should do that that you can see is the whole it's dy squared and dx squared okay so that and i'm just calling it y prime dx that's what i'm doing so that will be the length of an infinitesimally small little line segment there i've zoomed in so much but it actually is a straight line so i go from approximate to absolutely equal to and that is in the limit as delta x goes to zero and what i can i do how do we do integration if i have to just sum all of these together i take the integral of that so if i were to take the integral of both sides what am i left with s the total distance of this arc so this is this uh a line integral it's going to be this definite integral i'm going from x one to x sub x sub one to x sub two of the square root of one plus y prime square dx so there's my line integral okay and this is setting up distance but this needn't be distance i can set up many things in physics i might want to know what the shortest time is so there's going to be some path chosen that represents the shortest time i can have many things in here i can have not just distance but shape what would be the minimum surface area of something and i can do that by just setting up an equation for surface area okay so in what we do in in calculation of various calculations of variations is we're just going to call this i that is this group for not only length but surface area time anything is going to be something that i can now plot on a so there's needn't be distance in distance we can have something else but it's going to go from some initial point which i'll call x sub one and some final point which i'll call x sub two okay just to refer to the fact that there's something on one of the axes of some function and if i look at this function very carefully it's actually a function of x y and y prime with respect to that axis so it's from one point to other point on that axis so it's the f of x y and y prime now there's no x in there but that one might as well have been x to the power zero so there is and there's y there's no y just a normal y in there but that might also have been y to the power zero in effect this is a distance one so there was no x and y in there but there could be so there's a function of x y and y prime so anytime we can set this up it's called the it's called the functional this is called the functional that i'm going to do something and it's going to give me some scale of value and i'll clean the board and we're going to use this concept to prove to you that in your pleidian space you know it's not some funny space this flat space that the shortest distance between that point and that point is going to be a straight line we're going to develop this to prove that that is indeed so and through doing that hopefully i'll show you that that is what nature does and that's what we can do we're going to use the calculus of variation to show that there is something in here that is a minimum distance and or a minimum time or a minimum surface area and that is what nature chooses i'm going to clean the board and we'll carry on the board is clean let's have a look so there we go now imagine i'm going to construct something so imagine i knew the answer the equation for that curve between the two points which is the minimum imagine i knew that and in that solution was the y of x small y so of x now let's construct another function we'll call it capital y of x and that's going to include this small y of x plus some constant multiple of a second function called eta of x as most textbooks will do it this way and i'm going to pause the video here and just insert another video i made which is a screencast of how this is actually the absolute truth before i do that i just want to make two things abundantly clear and we're going to set this up this is like a magician setting up something specifically so that it works out for him or her and we're going to set this situation up so that it does work for us so there's for instance my one point in my other point and and we all know intuitively this is going to end up being a straight line but what if it isn't so i have this curve there and write at this x sub one and write at this x sub two i'm going to choose another curve and that's my curve eta of x such that at these points x one and x two it has a y value of zero and i can choose many of these just by changing this constant multiple eta if eta is one this would be one curve if eta is 1.5 it will be another curve if eta is negative three this will be another curve and this was my two answer which is a secret up till now but imagine i knew what it was and if i just add this at every point i just add these two then i was going to have something like this but you can imagine this capital y of x this capital y of x is is just these two added to each other but you imagine they are multiple for every eta of x i can choose any constant which will give me an infinite number of these little curves which means added to the real answer i can get many others and remember single variable just normal variable calculus if i just had a curve if i had a curve and i wanted to know the minimum the maximum well all these extreme points extreme would be at spots where the derivative the first derivative was going to be zero that is a minimum or maximum and i knew it was a minimum if i took the second derivative and if that answer was at that point was larger than zero i knew that to be a minimum and that's the same thing we're going to do to this we want a minimum we're going to take the derivative of this and set it equal to zero remember this is how i set up my distance i had s equals the definite integral going from x1 to x up to of the square root of one plus y prime square dx that was my distance i want to take the derivative of that and i want to set it equal to zero and if i do that i'm going to get to the minimum point and that's what we want i want to construct something that this line falls exactly on top of that line because if i do that there'll be some minimum and that will be the true result that i get there i'm ahead of myself so this is what we set up i'm going to pause the video here and insert that a little bit just to show you that this is an actual fact you can set this up many ways you can set this up and this is not this is not some trickery here that is mathematically unsound it's very sound you can set this up and i'll show you in the video so here we go let me show you i want to construct this capital y of x in terms of small y of x plus some other function eta of x and that eta of x i want to be able to multiply by some constant so i've set things some things up here let's have a look at them let's just take some of these away and let's just bring them back one by one so imagine just that this is my y of x i've made it a straight line it needn't be a straight line here's my eta of x and i'm just drawing in these two straight lines just to note that i've constructed eta x in this instance it is something here there we go there's my epsilon which i've called in here and it's this minus x minus four square plus one it's this little curve and i'll show you now why i put the n in there so what i'm saying is one thing i've got to construct this so that at x one and x two which is this spot here and that spot there i want that to correspond here to the same values that's why i've got the two straight lines and the fact that the y value for eta of x down here has got to be zero and zero at both points that's the way i want to set this up so what i have now is the capital y of x and the capital y of x between x one and x two has got to be the addition of these two and this is what we have right up here this is this green note though that because eta of x is zero at this point i'm not adding anything to small y of x and at that point i'm not adding anything to small y of x but there is the addition of these two curves at every point on the x axis between x one and x two if i added this value to that value i was going to get that top value that's what i'm saying but look at this if i put eta in front of it which here i've called in it doesn't matter what size i make eta it's just some constant multiple what epsilon i mean my small little my n here is epsilon sorry for that it doesn't matter what value i make epsilon it's a constant multiple it is a constant multiple of eta of x and these points still correspond to my little y of x and my little y of x there so it doesn't matter i i now have this almost well i can say indefinite number of capital y of x every time that i change epsilon i'm going to have a different capital y of x look at the green curve there are so many of them look how many of them there are that means to get from one point point one to point two there are many many paths that i can take and i've got to find the shortest one basically that's why i'm setting up this problem i want some minimum value en de problemen die we deal with obviously minimum arc length between these two points but just to show you that i can construct this capital y of x in the manner that i've done on the board and it makes absolute sense and you can just see it beautifully in this illustration good i hope you enjoyed that now this is beautiful mathematics stay with me you'll see where this is going so what we suggesting you remember that was my equation and i'm going to take the derivative of that and set it equal to zero that will give me the minimum value that's all we're going to do but i don't have this anymore i've changed it that refers to what i had in me the the real answer was going to come from that was going to come from that the distance the real distance was going to have some equation to some value to it but now i have this so what i en actual fact that now is a new function in x capital y in capital y prime that is what i have now so by constructing this addition i've changed it from x small y small y prime to x large y y prime so in some suggestion of this would be remember that was the square root of one plus y square y prime square the square root of one plus y prime square but if i do this to it i'm going to have this as my f of x i have this and i'm going to just do some preliminary work and you'll see you'll see shortly why i'm doing this preliminary work if i if i if i had this function imagine let me just write it outside the y of x equals the y of x plus epsilon eta of x what if i were to take the derivative of this with respect to this constant can you take a derivative with respect of constant of course you can because what is four other than four x to the power zero okay if i were to take the derivative with respect to x i can still take the derivative with respect to x that's probably not the best way a best way to to explain it let's let me rather do this can i say dy capital y of x d epsilon can i say that now i don't necessarily want to do that it's going to be a bit typical let's do the partial derivative of partial derivative with respect to that constant well it doesn't appear there so that's going to equal zero and i as i said i could have written this with an x to the power zero and what you can see this as basically i suppose is this the product of two functions so i can take the derivative of this times that and this times the derivative of that so i can actually have the the epsilon dy epsilon dy epsilon eta x plus eta times d and down whatever you want to pronounce it of eta x so what are we going to have here we're going to have the fact that's zero that is one so we will definitely have that so that'll be the eta x and there's no eta in there so the partial derivative of that with respect to epsilon so is zero zero times epsilon is zero so the derivative of capital y of x with respect to epsilon is just eta of x let's take y prime of x well it's just going to be that prime and you can put you can put the examples in here and just show it to yourself and what is going to happen here well it's with respect to x that i'm taking but i'm taking this derivative that is a function of x so this is very easy this is going to be epsilon eta prime of x in what way to happen if i took the partial derivative of y prime with respect to to epsilon okay there's definitely a partial derivative there's nothing in there that's going to be a zero and exactly the same thing has happened there it's going to happen here so we just going to have eta prime of x so that's all we going to have so these are the two equations it's some preliminary work i probably shouldn't have bought it in here because you want to why is he doing this why is he doing this let's clean the board and we'll carry on let's carry on now remember this thing back i had my equation which was this one plus y prime square square root of that what were to happen if i took the derivative of this of this function well i would have had to use the chain rule so when i had the f of x y and y prime i was going to have to use the the chain rule nothing changes if i have this if i have that function i'm also going to have the chain rule but because i have changed this into that i have a new i have a new function i which i now want to make a function of epsilon that is now going to be x of 1 to x of 2 of the f of x capital y capital y prime dx okay i have now changed when i went from my previous solution to this solution i have now made this a function of eta because i could get many y of x's by simply just just changing this constant so this now becomes a constant of this function and if i wanted to take the derivative of this if i wanted to take the derivative of this with respect to the epsilon why did i say it all the time you know what i mean when i write this when i write this with respect to take this derivative with respect to epsilon what is going to happen now that is why i chose things very carefully i also want to do it at a point where epsilon equals zero and that's why i set things up so remember what i showed you on the screencast that's why i chose things one of the reasons and there's another reason it's coming up i wanted to because of that equals zero then my large y of x this one will be exactly equal to this one they'll fall on top of each other and i'll have my solution okay and another thing is remember when i took the derivative when if that is zero i have the minimum so this is the secret to this whole thing i'm going to take the derivative so i've changed it to a function of eta i'm going to take the derivative with respect to eta exactly where eta is zero and if i set it equal to zero i know i have a minimum that is the calculation the calculus of of variations now just as we had here i would have had to use the chain rule let's use the chain rule of this because if i take the derivative of this it's actually this if i take the derivative of of i with respect to eta it's the same as is doing my integral and taking the derivative of this inside remember your first year calculus so what we're saying here that is going to equal the different integral going from this to that and i'll have to apply the chain rule so i'm actually going to have die f die small let's do that yeah die f die capital y times del y del epsilon please we must not make a mistake here and i'm going to have f with respect to y prime times y prime over epsilon all of that dx so don't get confused here this is just the chain rule of a function of any function remember if if you were to take the chain rule of something if you would take the first derivative of something as simple as this it would be very easy to do this was going to be to the power half you bring the half forward and that becomes negative a half and then you do the derivative of the inside function as well which is going to be to y prime so it's easy to do so for that for any function f of x y and y prime i was going to use the chain rule so yeah i'm also going to use the chain rule nothing different and that's exactly that's exactly what i have there and remember just a board before i did this and i did this so i know what this is this was remember what this was or that was this eight of x remember that rewind in your c and what was this it was eight of prime of x okay so that works for me and this has got to equal zero because if i set it equal to zero then i have a minimum okay we know we at once it's going to be a minimum it's going to be an extreme but let's call it a minimum for now because that is exactly what it is okay so we are there what i want you to realise is one thing i can do this as integration by parts remember the integration by parts i just have to decide which one is u and which one is v prime and that's very easy to do let's choose my v prime equal to this eight of prime of x that makes my v equal eight of prime eight of x and if this was that then that must mean this one is u di f di capital y prime and that makes my u prime the d dx of di f di y prime now i can just simply do integration by parts remember that will be this times that minus minus the integral of this one times that one remember integration by parts but i want to show you something very specific here as well and that's why i set things up i was a magician and i set things up that they work out specifically for me i said i want to do this around a point where eight equals zero and what's going to happen when eight equals zero when eight equals zero capital y of x and small y of x are going to be exactly the same thing they're going to be exactly the same thing so what we're going to have is the following so remember when i have the integral of the sum of two functions i can just do them separately so yeah i'm going to have x sub one to x sub two of del f del y i'm making it a small y now because i'm setting things up way epsilon equals zero so then i will have capital and lower case y y of x is being exactly the same times the eta of x dx plus now for this side i'm going to do this integration by parts so it's u times v so it's d these two so it's di f di y prime times eta of x and from x sub one to x sub two minus the integral of then from x sub one to x sub two of these two so it's d dx of del f del y prime eta oh yeah what was that eta of x dx and that's going to equal zero i hope you can still see that now remember also what i did here i let these two points be exactly the same so if i were to plug those in today i was going to get zero so this whole turn disappears this whole turn disappears and what am i left with this integral minus this integral is going to equal zero i can put them all together in the same integral because this is the difference between two functions and i can take these two out is a common factor and what am i going to be left with going from x sub one to x sub two of what well we're going to have del f del y minus the d dx of del f del y prime all of that times the eta of x all of that dx is going to equal zero in that low and behold is the way we were trying to get this is a thing of beauty now i want to set this up for any eta of x any eta of x because for a different integral to equal zero that means this inner part this integrand has got to be zero but i want it to be for any function of zero so any arbitrary function of eta and so that can't be zero it's being multiplied by this this has got to equal zero and that is the Euler Lagrange function and it states the following by f by y of del f del y minus the d dx of del f del y prime equals zero there we are where is the thing of beauty and remember when my f was this this was my f of x y and y prime if i were to take the derivative of this with respect to y and i took the derivative of this with respect to y prime and then the derivative of that and set it equal to zero i was going to get the answer for the shortest distance between those two points but i generalized it was i so it might be the shortest time it might be the shortest surface area i can use the Euler Lagrange equation to set up many physical problems if i can just through my physical problem develop this and we look at examples in the next video i might just show you proved you then that the shortest distance between two points is a straight line by solving this Euler Lagrange but this Euler Lagrange equation right here is absolutely a thing of beauty en we set it up right from here right from the start we used the simple fact that the first derivative we set it to zero you are going to get an extreme point and that extreme point is going to be a minimum value for you it's always going to work out in this instance to be a minimum let's call it a minimum instead of an extreme point and you're going to land up with this this is a real true a thing of mathematical beauty now let me prove to you that the shortest shortest distance between two points on the euclidean surface is just a straight line nature will always choose a straight line so you'll remember my equation for using integration a line integral for the length of any arc is that is my Euler Lagrange equation so let's just do that this is my if this is my function if function of x y and y prime let's get if time y what is that going to equal of just this bit well there's no y in there so that's just going to be zero that's a simple one what is time if by y prime well there is a y prime in there so i better do this properly and i've got to do it we said it before i've got to do it by the following method so what is that going to be i'm going to bring the half forward so that was going to be a half times again we we we were still going to have this square to the power negative for half and then i still have to multiply that by the inner which is going to be two times y prime that's really first year calculus except that this is a partial derivative but there is y prime in there so these two are going to go so what am i left with i'm left with y prime over the square root of one plus y prime squared that's what i'm left with and so they are left with zero so i have minus ddx of that which is now just this y prime over the square root of one plus y prime squared is going to equal zero which i can multiply both sides by negative one so this will disappear simply when is the derivative of any function equal to zero when that function is a constant isn't it so all of this all of this is some constant in other words y prime equals some constant times the square root of one plus y prime squared i can square both sides if i square both sides i get good of this and i square my constant which leaves me with another constant i can multiply this constant in there so i'm going to see plus some constant plus some constant of that i can bring this over to one side which means y prime i can get on its own which is then going to leave me with one one minus this squared remember i've squared this now to get rid of that and that is going to still equal some constant i'm going to divide this into that side which just gives me another constant and in the end i'm going to take the square root of that so y prime is going to be some constant in the end just run through that i've done in a very ugly fashion there but just do it for yourself you'll notice very quickly that y prime does equal some constant in when is the derivative how can a function of x y prime of x how can that be a constant well the only way that that can be is if the y of x equals ax plus b because if i were to take the derivative of that that would come forward so that's just my constant a is that x and a so that falls away and what is that an equation of a straight line proof that the shortest distance between any two points in euclidean two-dimension euclidean plane is a straight line how did i solve that i set up my differential here my equation here which is my my differential a and i used the calculation of the calculus of variations to set up the the grand jouallier equation i used to the grand jouallier equation and i proved that the shortest distance is there so if i set something else of the any physical system i'll have to set up this function f and if i can set that function of f with its time with its surface area i can use the grand jouallier equation and that will always always give me the minimum that is why i can set up a physical system where i can look at time what would be the shortest time i can set up the surface area and the most famous one of those is if you have two rings one smaller one larger and you put them in some of this bubbly soapy water and you bring them out and you'll separate them and that will form a very specific pattern between those two soapy bubble circles okay and nature chooses that and there's a reason nature chooses that because that will give you the minimum surface area that whole surface area of the of the soapy fluid material between these two you've got to make that a minimum and minimum and you can do it for volume as well why is this one a volume a sphere and nature does that because it is a minimum and some minimum is chosen en lastly from that who will come this fact if you have a ball that has to roll down a plane that has to roll down what is going to be the shortest time for that ball to roll down is it going to be a path that goes straight down in time or is it going to be one of these one of these parts that is called the brachistocroon problem brachie which means short chrome referring to time shortest time there's a reason why this ball will roll down will get from there to there faster than the ball rolling down that path there's a reason for that because nature can choose the minimum time that's what nature does it must choose the minimum time to get from there to there and that is why it will choose a specific path that will be the shortest path to get down you just have to work out en if of x for this problem